Please contact Johanna Franklin with any questions.

## Fall 2023

### Wednesday, September 13, at 1 p.m.

**Speaker:** Eric Rowland, Hofstra University**Title: **Powers of a sequence satisfying a recurrence**Abstract: **This summer I mentored two students in research projects in the NYC Discrete Math REU at Baruch College. One of the projects was to understand the complexity of powers of sequences defined by recurrences. For example, the Fibonacci sequence satisfies the recurrence F(n + 2) = F(n + 1) + F(n) of size 2. If we square each term of the Fibonacci sequence, this new sequence also satisfies a recurrence, but the size goes up to 3: F(n + 3)2 = 2 F(n + 2)2 + 2 F(n + 1)2 - F(n)2. How does the size change as we keep increasing the power? If we start with a sequence satisfying a recurrence of size r, what are the possible sizes of the recurrence for the mth power?

### Friday, September 29, at 4 p.m.

**Speaker:** Nara Yoon, Adelphi University**Title:** Mathematical modeling of cancer treatment**Abstract:** Despite the development of many treatment methods, it is still hard to cure cancer due to its complexity in terms of cellular heterogeneity, genetic/epigenetic mutations, micro-environment, etc. In this talk, I will introduce several mathematical modeling researches that explore the effect of various types of treatments. In the researches, models are developed in different ways (i.e., deterministic or stochastic model, differential or difference equation), depending on the specific research topics/objectives. Based on the best therapeutic prognosis simulated by each model, we will discuss the direction of therapeutic plans.

### Wednesday, October 11, at 1 p.m.

**Speaker:** Adam Mata, Warsaw University of Technology**Title:** On finite representations of Convex Geometries**Abstract:** Convex Geometries, as structures, were introduced to capture the notion of convexity in a discrete manner. Typically, a finite Convex Geometry is introduced as a pair *(X, α)*, where:

*X*is an arbitrary, non-empty set,*α*is a closure operator on*X*with a property:

*q **∊** α (K **∪** {p}) → p **∉** α (K **∪** {q})* (Anti-Exchange),

where *p, q **∊** X*, *p ≠ q*, *K **∊** P(X).*

There are many equivalent representations of Convex Geometries, e.g. by means of:

- families of lower sets of finite chains,
- meet-irreducible elements,
- implicational bases.

We introduce (finite) Convex Geometries and their crucial properties, with focus on their representations. Then we survey translations between the representations and their computational complexity.

During the presentation I show some results obtained together with Kira Adaricheva and Sylvia Silberger from Department of Mathematics at the Hofstra University, as well as with Anna Zamojska-Dzienio from the Faculty of Mathematics and Information Science at the Warsaw University of Technology.

### Wednesday, October 25, at 1 p.m.

**Speaker:** Ximena Catepillán, Millersville University**Title:** Ethnomathematics and Kinship Systems**Abstract:** In this talk, the following examples will illustrate how kin relationships and mathematics can be connected. The first two originate in the Warlpiri and the Aranda tribes. These tribes are the most traditional aboriginal groups in Northern Australia. The last example is from the tribes in the Malekula and Ambrym Islands of the republic of Vanuatu located in the southwestern Pacific Ocean.

### Friday, November 10, at 4 p.m.

**Speaker:** Andrew Lee, St. Thomas Aquinas College**Title:** Equivariant dimensionality reduction on Stiefel manifolds with applications**Abstract: **Nonlinear dimensionality reduction often begins by thinking of a high-dimensional data set *X* as living on, or near, a manifold *M*. One can then leverage extra structure on *M* to obtain a lower-dimensional embedding of the data that preserves relevant structure. We consider the case where *M* is a Stiefel manifold of orthogonal *k*-frames in **R**^{n}, which has a naturally occurring action of the orthogonal group *O(k)*. In this talk, we present a method for dimensionality reduction in Stiefel manifolds which respects this group action, and discuss applications to both synthetic and real-world data. This is joint work with Harlin Lee, Jose Perea, Nikolas Schonsheck, and Maddie Weinstein.

### Wednesday, December 6, at 1 p.m.

**Speaker:** Liam Lang, Stony Brook University**Title: **Applied mathematics for the analysis of neural systems**Abstract: **In the context of neuroscience, metastability refers to populations of neurons spontaneously and abruptly transitioning from firing at one set of rates---called a metastable state---to another in a coordinated manner. These metastable dynamics underlie a wide range of brain functions, and their study involves a variety of applied mathematical techniques. In this talk, I will go over the main findings of my thesis research concerning sequences of metastable neural states during decision-making, and then discuss the math behind the two major tools I utilized: Hidden Markov Models (statistical models of the brain) and spiking neural networks (biophysical models of the brain).

## Past Seminars

### Spring 2023

### Wednesday, February 22, at 1 p.m.

**Speakers:** Kate Poirier, New York City College of Technology, CUNY**Title:** Intersecting Loops on Surfaces and String Topology**Abstract: **String topology is the study of algebraic operations arising from intersecting loops in a space. The Goldman Lie bracket for surfaces is an example of such an operation that intersects two loops to get one loop. In this talk, I will introduce loops on surfaces, define the Goldman Lie bracket, and describe what this algebraic operation remembers about the underlying surface. With the help of lots of pictures, I will also discuss analogues of the Goldman Lie bracket in higher-dimensional spaces, more general string topology operations, and the algebraic structures these operations give rise to.

### Wednesday, March 8, at 1 p.m.

**Speaker:** Michael Cole, Hofstra University**Title:** The Uniqueness of Inverse Laplace Transforms**Abstract:** In courses on differential equations and engineering mathematics, Laplace transforms are often introduced for the purpose of providing a convenient way to solve certain differential equations.

For a function *f(t)* that satisfies suitable hypotheses, its Laplace transform *F(s)* is defined by the improper integral from 0 to infinity of *f(t) e-st dt*.

The idea is to translate questions about the original function *f(t)* into hopefully easier questions about the Laplace transform *F(s)*. When applied to a linear differential equation, the Laplace transform converts a differential equation in *f(t)* to a purely algebraic equation in *F(s)* that is easily solved. The laborious part of the procedure is to recover the function *f(t)* after its Laplace transform *F(s)* has been found.

Students are always told without a shred of proof or explanation that inverse Laplace transforms are unique. Thus it is claimed that if two functions *f(t)* and *g(t)* have the same Laplace transform, then* f(t) = g(t) *except possibly for isolated values of *t*. To my intuition, the claim is plausible, but not obvious. How does one prove it? There is a direct construction of an inverse Laplace transform (the __Mellin__ transform) which provides an immediate proof, but the __Mellin__ inversion formula requires nontrivial complex analysis to define and understand. The proof is only easy after a lot of theory of complex analysis and Fourier transforms has been established. Other proofs are also known, but they again require advanced methods.

The purpose of my talk is to present a more elementary proof of the uniqueness of inverse Laplace transforms. I will spend 20 minutes or so explaining some facts about Laplace transforms, some of which will be familiar to students of math, physics, or engineering who have taken courses that mention Laplace transforms. Then I will discuss the so-called Dirac delta function, which is an important concept used in physics and engineering as well as in pure mathematics. I will then present my proof and briefly discuss other proofs in the literature.

### Wednesday, April 5, at 1 p.m.

**Speaker:** Kerry Ojakian, Bronx Community College (CUNY)

**Title:** Markov processes and graph labelings

**Abstract:** We motivate various graph labeling questions by their connection to questions about Markov processes.

We consider a basic Markov process on an *n* vertex graph, i.e., an entity moves randomly through the graph, according to proscribed edge probabilities. For each vertex, we consider the expected proportion of time the object will be there, and can thus associate a length n probability vector with the Markov process; i.e., its stationary vector. Our first fundamental question is this: Given a graph, which stationary vectors can be achieved by adjusting the edge probabilities of the Markov process? There is a nice condition for determining which vectors are possible (the answer is essentially buried in the literature). For the main part of our investigation, we consider the "rank vector" derived naturally from the stationary vector; for example, if the stationary vector is [5, 9, 8, 7], then its corresponding rank vector is [1, 4, 3, 2]. Our second fundamental question is this: Given a graph, by adjusting the probabilities of the Markov process, what rank vectors are achievable? Using standard Markov theory, we reduce this question to a graph labeling question. Given an edge labeling, we can determine a length *n* vector by assigning each vertex the sum of the numbers on its incident edges, then from this vector we derive a rank vector. We ask: Which rank vectors are achievable by such an edge labeling? The graph labeling question turns out to be equivalent to our second fundamental question, so we focus on this graph labeling question. We answer the question in special cases, and give a nice condition which we conjecture to fully answer the question. Behind the scenes, these questions are motivated by an interest in associating centrality measures (i.e., betweeness centrality, closeness centrality, etc.) to natural graph processes (such as a Markov processes).

This is joint work with David Offner.

### Wednesday, April 19, at 1 p.m. - ** (Canceled) **

**Speaker:** Waseet Kazmi, University of Connecticut**Title:** What is Computability Theory?**Abstract: **There are certain process in mathematics that one feels intuitively are effective, while others are not. Computability theory gives us an important tool to study the effective content of various areas of mathematics, by identifying effective with computable. In this talk, I will give an introduction to computability theory discussing various topics such as computable functions and sets, computably enumerable sets, and how we can assign a noncomputable set a degree of noncomputability.

### Wednesday, May 3, at 1 p.m.

**Speaker:** Todd Bichoupan, Hofstra University**Title:** Closure systems and implication bases**Abstract:** A closure system is a family of sets that is closed under set intersection and includes a top element. The family of closed sets can often be represented compactly with an implication basis – a set of rules (called implications) that all members of the system must satisfy. I will discuss the connection between implication bases and the so-called essential sets of the underlying closure system, and I will show a form of independence between implications corresponding to distinct essential sets. My result resolves a 2014 conjecture by K. Adaricheva and J.B. Nation about the right sides of optimal bases – implication bases with the smallest possible size. In addition, I will discuss properties of implication bases associated with convex geometries – special types of closure systems satisfying the anti-exchange property. In particular, I will discuss the problem of optimizing implication bases for convex geometries and the problem of determining when the closure system associated with an implication basis is a convex geometry.

### Fall 2022

### Friday, September 16, at 4 p.m.

**Speakers:** Kira Adaricheva, Johanna Franklin, and Eric Rowland**Title:** I know what you could do next summer**Abstract: **This summer, Kira Adaricheva and Eric Rowland were faculty mentors in the New York Combinatorics REU program, which ran at Baruch College in Manhattan; Johanna Franklin was a faculty mentor in the entirely virtual Polymath Jr. undergraduate research program. These are the stories of these three intrepid professors and their students.

Kira Adaricheva mentored a two-student team who addressed the problem of convex geometries representation and found some partial solutions. Building on the work of the first Polymath program in 2020 with 17 undergraduate students, which catalogued all convex geometries on 4- and 5-element sets with respect to their representation by circles on the plane, the team addressed an open problem prompted by that work: whether such representation can be found for all convex geometries of convex dimension 3. Convex geometries are ubiquitous structures that appear in many areas of finite discrete mathematics, and their infinite versions develop in areas of algebra and logic.

Johanna Franklin mentored 24 students who studied different aspects of Kolmogorov complexity, a tool for formally saying how difficult it is to describe a finite binary string. Some of them used Kolmogorov complexity to define a partial order on infinite binary sequences and then studied the properties that this order has, and some of them extended the concept of Kolmogorov complexity to allow "mind changes."

Eric Rowland mentored two students who studied pattern avoidance in words. For example, the word 0110 contains the pattern 11 as a subword but avoids the pattern 00. There is a nice theorem that gives conditions for two patterns to be avoided by the same number of words. But are there natural bijections between those two sets of words?

### Friday, October 7, at 4 p.m.

**Speaker:** Matthew Jura (Manhattan College), joined by Joseph Canavatchel**Title:**The Computability of Edge-Magic **Z**-Labelings of Countable Graphs: A Summer Undergraduate Research Project**Abstract: **An edge-magic **Z**-labeling of a countable graph *G* having vertex set *V* and edge relation *E* (with magic constant *k*) is a bijection λ: V ∪ E →** Z** such that for every edge *vw* in *E*, we have λ(*v*)+λ(*w*)+λ(*vw*) = *k*. Over the summer of 2022, Joseph Canavatchel, an undergraduate student at Manhattan College, worked on a research project with faculty member Matthew Jura, investigating the computability of such labelings for computable graphs. Intuitively, a function from **N** to **N** is computable if there is an algorithm which computes it on each element in its domain. We think of a graph with vertex set **N** as being computable if there is an algorithm to decide whether any two vertices are adjacent. In this talk, we will summarize our undergraduate research experience and present the results that we have discovered.

### Friday, October 21, at 4 p.m.

**Speaker: **Richard Gottesman (Hofstra University)**Title:** Primes, Gaussian integers, and sums of two squares**Abstract: **In this talk, we will discuss how to do number theory with the Gaussian integers.

A Gaussian integer is a complex number whose real and imaginary components are both integers. The number 5 is no longer prime in the Gaussian integers since it can be factored as (2+*i*)(2-*i*). We will describe which prime numbers remain prime in the Gaussian integers. As an application, we will prove that every prime number which is one more than a multiple of 4 is a sum of two perfect squares. If time permits, we will discuss what happens if one tries to generalize this approach. The challenges one faces when doing so hint at some of the major problems in modern number theory.

### Friday, November 4, at 4 p.m.

**Speaker:** Mónica Morales Hernández (Adelphi University)**Title:** Some historical, artistic, and engineering aspects of Navier-Stokes**Abstract: **The Navier-Stokes equations (NSE), named after Claude Navier (French engineer) and George Stokes (Anglo-Irish physicist), are known for describing the motion of fluids such as water, fire, and smoke. NSEs have many popular applications such as modeling of airflow motion around an object, weather modeling to predict tornadoes, how to generate clean energy, and ocean current behavior. Other not-so-popular but important applications, such as modeling pollution in an area and modeling how blood flows through the human body, will be discussed. In this talk, we will go over some historical aspects that led to the development and formulation of the NSEs. We will review some interesting mathematical characteristics of the equations that could make you a millionaire. We will also cover some engineering applications, as well as explore some of the art where the NSEs are intrinsically present.

### Friday, November 18, at 4 p.m.

**Speaker:** Paul Baginski (Fairfield University)**Title:** Abundant numbers, semigroup ideals, and nonunique factorization**Abstract:** For any positive integer *n*, if *d1=1, d2, ..., dk=n* are the divisors of *n*, then we can sum those divisors to get a new number σ*(n)=d1+d2+...+dk*. Since *d1**=1* and *dk**=n*, for any *n>1* we have σ*(n)≥n+1*. A positive integer *n* is abundant if σ*(n) > 2n*; *n* is perfect if σ*(n) = 2n*; and otherwise *n* is deficient. Both the set *H* of abundant numbers and the set *H** of non-deficient numbers are closed under multiplication, making them subsemigroups (in fact, semigroup ideals) of *(***N***,×)*. As a result, we can consider how elements of *H** (or *H*) factor into irreducible elements of *H** (resp. *H*). As it turns out, non-deficient numbers (or abundant numbers) do not factor uniquely into products of irreducible non-deficient numbers (resp. irreducible abundant numbers). We describe the factorization theory of these two semigroups, showing that they possess rather extreme factorization behavior with respect to standard measures of nonunique factorization. These two semigroups are special cases of a more general phenomenon that occurs with semigroup ideals of factorial monoids. We will describe some of the algebraic and arithmetic theory of these semigroup ideals and mention some additional arithmetic functions that yield extreme factorization properties.

### Friday, December 2, at 4 p.m.

**Speaker: **Melissa Newell (Iona University)**Title:** Taxicab Geometry: An Exploration of an Alternative Distance Metric**Abstract:** The traditional method to measure the distance between two points is to measure the length of the straight line that connects them. However, there are many other ways that we can think about distance. In this talk, we will discuss an alternate distance measure called the Taxicab metric, which is based on the idea that in a city built on a grid system any movement will need to be either east-west or north-south along the streets of the city. The goal of this talk is to explore the ramifications of changing the distance formula on other features we are familiar with in geometry and to discuss a few relevant applications. Finally, we will discuss how the distance formula can be further modified to better fit the realities of a given situation and how these modifications would continue to change our understanding of the resulting geometry. This will be an interactive talk with many opportunities for exploration!

### 2021-2022

### Wednesday, September 8, at 1 p.m.

**Room:** Roosevelt 213**Speaker:** Johanna Franklin, Eric Rowland, and Zoran Sunic**Title: **What can you do in 67 summers?**Abstract: **In the 2021 Polymath Jr. summer research project, 16 faculty members supervised over 300 students in an entirely virtual undergraduate research program. These are the stories of three of these intrepid professors and their students.

Iteration of polynomial maps is full of wonderful surprises, as witnessed by the complex beauty of their Julia sets. Zoran Sunic's group considered a certain class of polynomial maps and studied sequences of graphs that approximate the dynamics of the maps on their Julia sets.

Eric Rowland’s group looked at infinite words that contain no block of letters that appears twice consecutively. Guessing the structures of some of these words required computing several hundred thousand letters.

Johanna Franklin's group studied two traditional abstract strategy games: Tapatan and Picaria. They conducted a combinatorial analysis of the game boards, determined optimal strategies for these games, and evaluated the successes of other strategies using Markov chains. Then they created variations on these games and analyzed these variations similarly.

### Wednesday, September 29, at 1 p.m.

Contact Johanna Franklin for the Zoom link.**Speaker:** Enrique Treviño, Lake Forest College**Title:** A trio of research projects with undergraduates**Abstract:** We will talk about some research projects with undergraduates regarding happy numbers and a project about parking functions. A number *n* is happy if after iterating the sum of squares of digits function, the number lands at 1. For example, 133 is happy because it goes to 1^{2} + 3^{2} + 3^{2} = 19 which goes to 1^{2} + 9^{2} = 82, and then 82 → 8^{2}+2^{2} = 68 → 6^{2} + 8^{2} = 100 → 1. Happy numbers have been generalized to different bases, different sums of powers, and even different number systems. Our project covers two generalizations, one to "fractional bases" and another to the factoradic representation of a number.

Changing gears, consider *n* cars *C*_{1}, *C*_{2}, …, *C _{n}* that want to park in a parking lot with parking spaces 1, 2, …,

*n*that appear in order. Each car

*C*has a parking preference α

_{i}_{i}∈ {1, 2, …,

*n*}. The cars appear in order, if their preferred parking spot is not taken, they take it, if the parking spot is taken, they move forward until they find an empty spot. If they don't find an empty spot, they don't park. An

*n*-tuple (α

_{1}, α

_{2}, …, α

_{n}) is said to be a parking function, if this list of preferences allows every car to park under this algorithm. For example (2, 1, 1, 2) is a parking function while (4, 3, 3, 2) is not. In our project we study a couple of generalizations where we introduce randomness.

### Wednesday, October 6, at 1 p.m.

**Speaker:** Meng-Che "Turbo” Ho, California State University Northridge**Title:** Word problem for groups**Abstract: **The word problem for groups was proposed by Max Dehn in 1911, along with the conjugacy problem and the isomorphism problem. Much like Hilbert's Tenth Problem for solving Diophantine equations, it (likely) was expected that such problems are solvable. It turns out that they are not.

In this talk, starting from the definition of the presentation of a group, we will talk about the history of the word problem and its relationship with logic. We will then discuss some more recent developments of the word problem, using the framework of formal language theory. If time permits, we will turn our attention to the isomorphism problem and discuss its relationship with the classification of groups.

This talk assumes only a minimal background in group theory and no background in logic.

### Wednesday, October 20, at 1 p.m.

**Speaker:** Jessica Oehrlein, Fitchburg State University**Title:** Math in the Sky: Studying the Stratospheric Polar Vortex**Abstract: **Earth's atmosphere is a complicated system to understand. We might care about everything from clouds in Long Island right now to average global temperature in a century. The state of the atmosphere depends on lots of other factors, like the ocean or our carbon dioxide emissions. And it's really hard to do experiments on the atmosphere. So how do we study this system, and where does math come in?

In this talk, I'll focus on one atmospheric phenomenon, the stratospheric polar vortex, a region of cold air surrounded by west-to-east winds over the poles during winter. We'll look at how people have tried to understand the polar vortex through three main tools: theory, observations, and models. We'll see how these all work together and explore the role that math plays in each of them.

### Wednesday, November 3, at 1 p.m.

**Speaker:** Berit Nilsen Givens**Title:** Spread of Infection in a Network**Abstract:** The spread of an infection across a network depends on the degree of interconnectedness of the network and on the level of contagiousness of the infection. We explore what happens as networks grow or shrink in size, along with how the infection rate *p* affects the spread. We first consider a complete graph as a model of a “bubble” of people who interact freely. Then we explore a variant of star graphs, which could model a teacher interacting with several disjoint classes of students. Using graph theory and combinatorial reasoning, we show that limiting the size of gatherings and reducing the value of *p* can dramatically slow the spread of a highly contagious disease like COVID-19.

### Wednesday, December 1, at 1 p.m.

**Speaker:** Catherine Cannizzo, Stony Brook University**Title:** A mirror into the higher dimensional world**Abstract: **We live in a three dimensional world. If we consider time as a fourth “coordinate”, we have four dimensions. These four dimensions are known as “space-time.” In physics, string theory conjectures that, at scales much smaller than an electron (of a similar order of magnitude of the universe to the atom), there are small strings vibrating. For the theory to work, the strings must vibrate in 6 compact extra dimensions, for a total of 10 dimensions! It turns out that two geometric models for the strings give the same physics. These pairs opened up mathematicians to the notion of “mirror symmetry” which gives us a lot of interesting information about geometric spaces. In this talk, I will define a “geometric space”, give some examples, and illustrate how to visualize higher dimensions as well as touch on the connection to my research in homological mirror symmetry.

### Wednesday, March 2, at 1 p.m.

**Speaker:** sarah-marie belcastro**Title: **Topological Graph Theory... and YOU!**Abstract: **This presentation will introduce the field of topological graph theory. Along the way, YOU will get to work on some elementary problems and present your insights, so that YOU will acquire a sense of how mathematicians approach topological graph theory questions.

### Wednesday, March 16, at 1 p.m.

**Speaker:** Pawel Pralat, Ryerson University**Title: **Semi-random process**Abstract: ** The semi-random graph process is a single-player game that begins with an empty graph on *n* vertices. In each round, a vertex *u* is presented to the player independently and uniformly at random. The player then adaptively selects a vertex *v* and adds the edge *uv* to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible.

During the talk, we will focus on the following problems: a) perfect matchings, b) Hamilton cycles, c) constructing a subgraph isomorphic to an arbitrary fixed graph *G*. We will also consider a natural generalization of the process to *s*-uniform hypergraphs.

### Wednesday, March 30, at 1 p.m.

**Speaker:** Wesley Calvert, Southern Illinois University**Title: **Telling One Thing From Another**Abstract: **When are two things the same? In mathematics, that judgment is generally difficult. The answer, of course, depends on what moves we are allowed to make in the game, and for what purpose we care about the sameness. Floating point arithmetic in computers, checking the isomorphism of graphs, and saying what a machine learning algorithm is supposed to be doing will provide some of our examples for thinking about the broader problem.

### Wednesday, April 13, at 1 p.m.

**Speaker:** Anna Zamojska-Dzienio, Warsaw University of Technology and Iowa State University**Title: **Barycentric algebras and their applications**Abstract: **Real convex sets can be presented algebraically with binary operations given by weighted means, the weights taken from the (open) unit interval in the real numbers. The class of convex sets is a quasivariety (a class defined by certain implications) and generates the variety (a class defined by identities) of barycentric algebras. In this talk, we focus on the latter class, presenting properties of its members, examples, and applications, mainly in biology for the modelling of complex systems that function on (not necessarily comparable) different levels, but also in computational geometry for investigations of (barycentric) coordinates.

### Wednesday, April 27, at 1 p.m.

**Speaker:** Joshua Crisafi, Hofstra University**Title:** The Lucas Congruence mod p^2 with the Multinomial Function and Legendre Polynomials**Abstract:** The multinomial function is an extension of the binomial function to multiple variables and with a slight change in its coefficients. It is known to support a Lucas Congruence mod p for all inputs, for any prime p, and for any number of variables, but it only supports a Lucas Congruence mod p^2 for specific inputs. This talk will explore exactly which inputs will and will not have the multinomial function support a Lucas Congruence mod p^2 for any given number of arguments. The results in this talk will explain exactly which points have the binomial function support a Lucas Congruence mod p^2 as well. It will also include several conjectures about Lucas Congruences mod p^2 for certain values of the Legendre Polynomials.

### 2020-2021

### Friday, September 4, at 3:30 p.m.

Speaker: Kira Adaricheva, Hofstra University

Title: Convex geometries representable by at most 5 circles on the plane

Abstract: In a unique project led by Adam Sheffer from Baruch College (NYC), 8 mathematicians from universities across the US worked with about 300 undergraduate students from around the world in 12 research projects in PolyMath style, during two months of summer 2020.

One of the projects guided by Kira Adaricheva (Hofstra University) and involving 20 students investigated the open question of whether there exists a convex geometry (a closure system with anti-exchange axiom) on a 5-element set that has convex dimension less than 6 and is not representable by circles on the plane, an open question from a publication in Discrete Mathematics in 2019.

The question was answered in the positive because several geometries that do not have representation were found with convex dimensions = 4 and 5. Moreover, all 672 geometries on a 5-element set were either represented by circles on the plane or classified into one of possible non-representable class.

It was also proved that one of these classes comprising 7 geometries was not representable due to the Triangle Property. The results of this project will appear on the arXiv in September.

### Friday, September 18, at 3:30 p.m.

Speaker: David Nacin, William Paterson University

Title: The Power of Padovan

Abstract: The Fibonacci (or Pingala) numbers are the single most famous number sequence in music, art, and architecture. They arise naturally in both a spiral of squares and a ratio involving proportions of rectangles. In this talk, we question how natural the choices in both these constructions are, and focus on a different sequence which arises as a result of both a different spiral and a different question about proportions of rectangles. We construct this new sequence in both ways and provide ample evidence for its importance alongside its more famous cousin.

We will show how our new sequence surprisingly appears in Pascal's triangle and then prove several identities without either words or numbers, by considering colorings corresponding to sums of entries. We will examine the uses of this sequence in architecture, particularly in Hans van der Laan's Monestary at Vaals. Finally we will present recent research on the growth properties of this sequence, presenting a counterexample to a conjecture and seeing how far we can push the limits of that counterexample.

### Wednesday, September 30, at 11:30 a.m.

Speaker: Erin Craig, Stanford University

Title: Predicting hospital readmissions from doctors' notes

Abstract: Data science in medicine is an exciting and impactful application of math and computer science, with many open areas of research. We will motivate this with a brief discussion of electronic health records and the challenges they present. Then, we will give a gentle introduction to common neural network architectures for working with natural language (Word2Vec and CNN) and show how we can use them to predict hospital readmissions from doctors' notes.

### Friday, October 16, at 3:30 p.m.

Speaker: Edinah Gnang, Johns Hopkins University

Title: Broadening the Linear Algebra Toolkit and Applications

Abstract: The quote “Mathematics is the art of reducing any problem to linear algebra.” by William Stein wonderfully articulates the importance of Linear Algebraic techniques in Pure Mathematics. We motivate and sketch a broadening of the scope of the algebra of matrices to an algebra of hypermatrices and constructs.

### Friday, October 30, at 3:30 p.m.

Speaker: Jennifer Chubb, University of San Francisco

Title: Distance functions on computable graphs

Abstract: An infinite graph—one made of nodes and edges—is computable if there is an algorithm that can decide whether or not a given pair of nodes is connected by an edge. So, for example, the internet is a computable graph which is, for all intents and purposes, infinite. Now, given two nodes on a connected, computable graph, a natural question to ask is, What is the length of the shortest path between them, i.e., the distance between the nodes? Of course, for a connected graph we can always find such a path and determine its length, but the question of finding a shortest path is harder, and is not in general something we can compute for infinite computable graphs.

In this talk, we will see what it means for something to be non-computable via a classic example called the halting problem. Next we will see that it's possible to encode non-computable information into even very simple computable mathematical objects. Finally, we will see how non-computable information can be encoded into the distance function, the function which outputs the shortest distance between two nodes, of a given graph. So, the answer to What is the shortest path between two nodes? Well, we may never know for sure.

This work is joint with Wesley Calvert and Russell Miller.

### Wednesday, November 11, at 11:30 a.m.

Speaker: Miranda Teboh-Ewungkem, Lehigh University

Title: A Mathematical Study of a Complex Disease, Malaria, with Focus on the Within Mosquito Life Cycle

Abstract: Malaria is a complex disease involving three interacting populations: the Plasmodium parasites, the agents that cause the disease; the female Anopheles mosquitoes, the agents responsible for spreading the parasite and hence malaria from human to human; and the humans, trying to stay healthy!!!! Part of the parasite's life cycle, the asexual part, is spent in humans while the sexual part is spent in mosquitoes. Successful transmission of the parasite to humans requires that a susceptible female mosquito feed on two distinct humans—one infected with the parasite and the other susceptible, at two distinct sequential time points. In addition, the parasite must be in its transmissible form in the mosquito at the latter feeding. The bottlenecks involved in the process illuminates how the parasite, driven by the need to survive, has captured the evolutionary and reproductive needs of the mosquito to ensure the parasite's survivability. In this talk, I will present the first ever mathematical model of the within-mosquito life-cycle component of P. falciparum parasites which accounts for the developmental stage transformations of the parasites from ingested gametocytes to sporozoite formation. The model will consider the action and effect of blood resident human-antibodies, ingested by mosquitoes during a blood meal, in inhibiting gamete fertilization. Model analysis and simulations will be used to explore the question of whether it is possible to control and limit oocysts development, the precursors of sporozoites, and hence sporozoite development within a mosquito by boosting the efficiency of antibodies that can be ingested during a blood meal, as a pathway to the development of transmission-blocking vaccines.

### Friday, February 12, at 4 p.m.

**Speaker: **Zoran Sunic, Hofstra University**Title: **If there is miso, I’ll have gazpacho, otherwise –- minestrone!**Abstract:** This is a conversation I recently had with the waiter at a nearby restaurant:

- I would like to start with a soup.

- Excellent, which soup would you like?

- I see that you have only miso, gazpacho, and minestrone on the menu.

- Yes, sir, the choice is limited, but each of our soups is delicious. However, I am sorry to point out that we might have run out of miso, I would need to check.

- Well, no need to check, I can decide right now –- if there is still miso, I’ll have gazpacho, and if there is no more miso, then minestrone, please.

In this talk, we will:

(1) explain why this conversation makes perfect sense,

(2) look at other examples that make (no) sense, and

(3) indicate why making choices is, provably, difficult.

### Friday, February 26, at 4 p.m.

**Speaker:** Erika Ward, Jacksonville University**Title: **Geometry of Gerrymandering: Why Redistricting is so Difficult and How Mathematics Can Help**Abstract: **How we group people to elect representatives has a tremendous effect on who gets elected. How does it work? How does Gerrymandering happen? Mathematicians are harnessing geometry to create tools to combat the problem... but even deciding what's fair is more difficult than it sounds.

### Friday, March 12, at 4 p.m.

**Speaker:** Elizabeth Gillaspy, University of Montana**Title:** Volterra's Function and Other Counterexamples: or, Why Hypotheses are Important**Abstract:** On the surface, Volterra’s function *V*(*x*) seems to violate the Fundamental Theorem of Calculus. *V*(*x*) is differentiable on the interval [0,1], but the integral of the derivative *V*'(*x*) on that interval does not equal *V*(1) − *V*(0)!

In this talk, I’ll explain what Volterra’s function is and why it does not, in fact, contradict the Fundamental Theorem of Calculus. (The answer has to do with hypotheses...) Along the way, we’ll see a number of other, seemingly contradictory, examples from Real Analysis: a function *g*(*x*) that is differentiable but *g*'(*x*) is not continuous; a subset of [0,1] that contains no intervals and yet has length 1/2; and perhaps the most contradictory example of all, the Cantor set. By the end of the talk, I hope to have convinced you that the hypotheses of a theorem are at least as important as its main result.

### Wednesday, March 24, at 1 p.m.

**Speaker:** Salvatore Giunta, Babson College**Title:** Recent Work on p-value Alternatives: A Literature Review**Abstract:** *p*-values have long been the standard metric used to determine whether or not data from an experiment is statistically significant. In recent years, *p*-values have faced increasing criticism, and many consider them to no longer be sufficient to evaluate data and a contributor to the ongoing Replication Crisis in science. This culminated in the American Statistical Association’s 2016 statement about proper use of *p*-values in statistical analysis. In this presentation, we examine the strengths and weaknesses of *p*-values and introduce some alternative methods of determining statistical significance. In particular, we discuss confidence intervals, Bayesian Statistics, likelihood ratios, *d*-values, and *w*-values. We also examine other non-metric based adjustments to current practice including flexible thresholds, pre-registered studies and random auditing. Finally, we discuss some implications for teaching statistics at the secondary and university level.

### Wednesday, April 21, at 1 p.m.

**Speaker:** Kameryn J. Williams, University of Hawaiʻi at Mānoa**Title:** Incompleteness and the universal algorithm**Abstract:** In 1936 Alan Turing introduced a mathematical model of computation. The Turing machine has since become the standard way to formalize the notion of computability. Many incompleteness results can be recast as statements about Turing machines. For example, Kurt Gödel’s second incompleteness theorem can be equivalently stated as saying that whether certain Turing Machines halt depends upon in which model of arithmetic they are ran. In this talk I will present a particularly striking instance of this phenomenon, Hugh Woodin’s universal algorithm. Woodin produced a single Turing machine which can be made to output anything so long as you run it in the right model of arithmetic.

This talk is self-contained and does not assume any background in computability theory nor mathematical logic.

### Wednesday, May 5, at 1 p.m.

**Speaker:** Hakim Walker, Harvard University**Title: **Cinco de Maya: A Crash Course in Maya Mathematics**Abstract: **May 5th is commonly known as Cinco de Mayo, a date in Mexican history that commemorates their victory over the French empire at the Battle of Puebla in 1862. To honor the occasion, we’ll be discussing the mathematical achievements of one of Mexico’s oldest civilizations, the Maya, who lived in southern Mexico and central America as early as 4,600 years ago.

In this crash course, we will introduce the Maya numeral system, which is simple enough to only require three symbols, yet powerful enough to represent large numbers easily and efficiently. We’ll observe how to perform basic arithmetic operations in their system (such as addition and multiplication), as well as how to extend their system to capture mathematical ideas that the Maya may have never used (such as rational and real numbers). Then, we’ll broaden our scope and take a brief tour of numeral systems throughout history and around the world, looking at their motivations, benefits, and disadvantages. By studying these numeral systems comparatively, we can improve our understanding of our own system of numbers, appreciate what makes the Maya system unique and useful to this day, and gain a deeper understanding of civilizations and cultures, past and present. Also, if time permits, we will see what makes the number 252 special.

This talk is intended for people of all mathematical backgrounds. No prerequisite knowledge is required.

### Friday, May 7, at 4 p.m.

**Speaker:** Joseph Ronzetti, Hofstra University**Title: **A statistical analysis of mana-positive mana rocks**Abstract: **I studied the effects that mana-positive mana rocks have on the number of turns it takes for my deck to win in Magic The Gathering. Mana is a resource within the game that allows the players to play their other cards. These mana rocks enable the user to produce more mana earlier in the game and see use in a variety of decks. In this project, I look at the cards' effects individually and in categories based on their unique drawbacks.

There are several different factors to consider when judging each of these mana rocks' effectiveness in the game. I measure this quality by the kill turn in the trial. The kill turn is the term used for the turn when I can win the game. First, I discuss general relationships between the kill turn and the number of mana rocks in the deck, my starting hand, and on the battlefield. I then analyze how unique aspects of the individual mana rocks may affect the kill turn. Finally, I examine the effect each mana rock has individually on the kill turn.

### Many Cheerful Facts — Summer 2020

**Date: Monday, June 8, at 3 p.m.**

Speaker: Eric Rowland

Title: Cheerful facts about Pascal's triangle

Abstract: Pascal's triangle comes up everywhere in mathematics. Pascal studied it in the 1600s in connection with probability, but it had been described in other parts of the world at least a thousand years earlier. In the 1870s, Édouard Lucas studied Pascal's triangle from a number theory perspective. He obtained a beautiful formula for the remainder left after dividing a given number in Pascal's triangle by a prime p. Variants and generalizations of this formula have been actively investigated ever since, including a new result this year that uses some hidden rotational symmetry in Pascal’s triangle.

**Date: Monday, June 29, at 3 p.m.**

Speaker: Johanna Franklin

Title: Mathematics Wrapped in a Mystery Inside an Enigma

Abstract: The Enigma machine was used by the German military to encipher their communications during World War II. I'll talk about the mathematics that made the Germans believe the Enigma was secure, the practical and mathematical reasons they should not have, and how the Allies were able to crack it by exploiting its weaknesses.

**Date: Monday, July 6, at 3 p.m.**

Speaker: Stefan Waner

Title: The boundaries of mathematics

Abstract: The mathematicians Alan Turing and Kurt Gödel each precipitated a kind of existential crisis in mathematics; specifically in the theory of computation and in the foundations of mathematics respectively. Each exposed a startling limitation in the power of these disciplines that reverberated way beyond mathematics and indeed throughout the philosophical and intellectual world: the unsolvability of the so-called Halting Problem in computing theory and Gödel’s First Incompleteness Theorem, on the existence of true statements in mathematics for which there is no possible proof. What I will try to convey is roughly based on Chaitin’s argument on the equivalence of versions of these two results. It is that same argument I will present here. It is neither precise nor rigorous—the proof of the pudding is in the details I will simply gloss over—but the conceptual relationship is startling and understandable at a level assuming no knowledge of either formal mathematical systems or recursive function theory.

**Date: Monday, July 20 at 3 p.m.**

Speaker: Steve Warner

Title: Strips, rectangles, and limits

Abstract: One of the first topics that math students often struggle with is the ε–δ definition of the limit of a real-valued function. In this talk, I will present an approach to such limits that is more visual in nature than the traditional approach, but still rigorous. This approach has the benefit of avoiding mysterious Greek letters and absolute value inequalities, while providing a single framework for both finite and infinite limits.

**Date: Monday, August 3, at 3 p.m.**

Speaker: Abe Mantell

Title: Linear regression via recursion (i.e., without calculus!)

Abstract: The least-squares-best-fit linear function is initially developed in the usual way, but rather than use multivariable calculus to minimize the sum of the squares of the deviations, precalculus mathematics is used to obtain recurrence relations for the slope and y-intercept. These recurrence relations can then easily be entered into a spreadsheet to show convergence to the true solution. Moreover, the recurrence relations can be solved to yield the familiar regression formulas for the slope and y-intercept.

### 2019–2020

**Date: Friday, September 27, at 3:30 p.m.**

Room: Roosevelt 201

Speaker: Michael Cole, Hofstra University

Title: Orthogonal Polynomials in One or More Variables

Abstract: PDF

**Date: Wednesday, October 16, at 11:30 a.m.**

Speaker: Yotam Smilansky, Rutgers University

Title: Patterns and Partitions

Abstract: A colored partition of a set in Rd is its representation as a disjoint union of subsets, referred to as tiles, where each tile is also assigned a color. In the talk, we will consider sequences of colored partitions defined using multiscale substitution rules on finite collections of colored prototiles. In the substitution process, which generalizes a construction first introduced by Kakutani, tiles of maximal volume in a given partition are replaced by colorful patterns consisting of rescaled copies of colored prototiles, thus defining the next partition in the sequence. Tiles that appear in the process are modeled by a flow on a directed weighted graph, and distributional and statistical questions on sequences of partitions are reformulated as questions on the distribution of paths on graphs. Under a natural incommensurability assumption, special properties of the poles of the Laplace transforms of graph counting functions imply various explicit statistical results. In addition, computer experiments reveal the beautiful patterns in which these poles appear in the complex plane, patterns which seem to be closely related to Diophantine properties of the generating substitution rule.

**Date: Wednesday, October 30, at 11:30 a.m.**

Speaker: Stefan Waner, Hofstra University

Title: Online resources for elementary math classes

Abstract: Steve Costenoble and I have developed online adaptive tutorials as games—as well as numerous other online resources—to address many of the well-documented challenges facing mathematics instructors in elementary math classes. Viewing the classroom experience as a game is more than simply a convenient metaphor, but allows the designer of educational products to deploy numerous features borrowed from student experience in computer games and, more importantly, to present the entire classroom experience to the student explicitly in terms with which they are familiar from their experience outside the classroom, and to which they know how to respond effectively.

**Date: Wednesday, November 6, at 11:30 a.m.**

Speaker: Manon Stipulanti, Hofstra University

Title: A way to extend Pascal's triangle to words

Abstract: Pascal's triangle and the corresponding Sierpiński's triangle are well-studied objects and have connections with different areas in science. The main ingredient of this presentation is the link between them. I will first recall it and then exploit it to present a way of extending both objects to the area of combinatorics on words.

Combinatorics on words is a relatively new domain of discrete mathematics, which focuses on the study of words and formal languages. In this context, a finite word is simply a finite sequence of letters, or symbols, that belong to a finite set called the alphabet. For instance, 01101 and 01 are two finite (binary) words over the (binary) alphabet {0,1}. A language is a set of words. For instance, we let {0,1}* denote the set of all finite words over {0,1}. The binomial coefficient of two finite words u and v over some alphabet is the number of occurrences of v as a subsequence of u. For example, the binomial coefficient of 01101 and 01 is 4. This concept, which generalizes binomial coefficients of integers, has been widely studied for the last thirty years or so. Knowing the definition of the Pascal's triangle with binomial coefficients of integers, its extension to binomial coefficients of words seems somewhat natural.

**Date: Wednesday, November 13, at 3 p.m.**

Room: Breslin 103

Speaker: Jim Thatcher, University of Washington-Tacoma

Title: Electoral Districting in more than Euclidean spaces: travel-time considerations for communities of interest

Abstract: With the coming 2020 census, new electoral and representational districts will be drawn across the United states. This talk discusses the historic and current role cartography, here the literal drawing of lines, plays in that process. Specifically, it discusses the different mappings that can be produced using various more-than- and non-euclidean metrics for distance and association. Travel-time across districts is presented as one means of measuring equity, access, and representation within voting and representational districts.

**Date: Friday, November 15, at 3:30 p.m.**

Room: Roosevelt 201

Speaker: Mike Chinbayar, Hofstra University

Title: Pascal's Triangle and Finding the Expected Value

Abstract: In a recent paper, Spiegelhofer and Wallner demonstrated that the nth row of Pascal's Triangle, when put under the 2-adic valuation, typically follows a normal distribution. In addition, they discussed the possibility of generalizing their result for some prime number p. We show a possible conjecture on what that general result may be.

**Date: Wednesday, November 20, at 11:30 a.m.**

Speaker: Kawkab Abid, Hofstra University

Title: Digit patterns of the Collatz function in base 3

Abstract: We study the patterns of the Collatz function in base 3. When the Collatz function is evaluated, we observe an array consisting of 0, 1 and 2. We prove several theorems describing the patterns that form.

**Date: Wednesday, January 29, at 11:30 a.m.**

Speaker: Célia Cisternino, University of Liège

Title: Properties of the alternate base representations

Abstract: Depending on the considered base, numbers can be represented differently. Émilie Charlier and I introduced the new theory of the alternate bases representations. In this generalization of the beta representations, we use a p-tuple β1, β2, ..., βp as a base to represent any real number in [0,1]. During this presentation I will first recall the theory, already widely studied, of the beta representations. Then, some properties of the alternate base representations will be stated as well as the generalization of Parry's Theorem.

**Date: Friday, February 7, at 3:30 p.m.**

Speaker: Peter Winkler, Dartmouth/MoMath

Title: When Can You Avoid Backward Steps?

Abstract: Suppose the whole computer system at Hofstra requires upgrades, but all changes must be done without reducing service below a certain level. If this can be done, can it be done without at some point *down*-grading some component? Inspired by a still-open problem in metric topology, we develop a model for answering *some* questions of this form. The consequence is a set of general conditions under which optimal scheduling can be done without backward steps. Among the applications are observations about searching for a lost child in a forest, and a fast algorithm for scheduling multiple processes without overusing a resource. This is joint work in part with Graham Brightwell (LSE) and in part with Lizz Moseman (NSA).

**Date: Wednesday, February 26, at 11:30 a.m.**

Speaker: Marina Jacobo, Hofstra University

Title: Extremal Graphs With Large Rank Numbers

Abstract: A k-ranking of a graph G is a function f : V(G) → {1, 2, ..., k} such that if f(u) = f(v) then every uv path contains a vertex w such that f(w) > f(u). The rank number of G, denoted χr(G), is the minimum k such that a k-ranking exists for G. The rank number is a variant of graph colorings. It is known that given a graph G and a positive integer t the question of whether χr(G) ≤ t is NP-complete. The characteristics of any n-vertex graph whose rank number is equal to n−1 or n−2 is known; in this talk we extend this question to n−3. Also, we examine the extremal graphs such that their rank number is equal to n, n−1, n−2 and n−3.

**Date: Wednesday, March 4, at 11:30 a.m.**

Speaker: Lara Pudwell, Valparaiso University

Title: What's in your wallet?!

Abstract: You may associate this title with credit card commercials, but it is also an invitation to some interesting probabilistic mathematics. A Markov chain is a model for analyzing sequences of events where each event only depends on the result of the previous event. In this talk, we will explore Markov chains in general, and use them to answer the particular question "what is the most likely distribution of coins to have in your wallet?".

**Date: Friday, March 13, at 3:30 p.m. [This talk has been canceled.]**

Speaker: Emily Gunawan, University of Connecticut

Title: Cluster algebras and binary subwords

Abstract: We establish a connection between binary subwords and perfect matchings of a snake graph, an important tool in the theory of cluster algebras. Every binary expansion w can be associated to a piecewise-linear poset P and a snake graph G. We will construct a tree structure called the antichain trie which is isomorphic to the trie of subwords introduced by Leroy, Rigo, and Stipulanti. We then present bijections from the subwords of w to the antichains of P and to the perfect matchings of G. This is joint work with Rachel Bailey (https://arxiv.org/abs/1910.07611).

**Date: Friday, April 3, at 3:30 p.m.**

Speaker: Benjamin Gaines, Iona College

Title: Playing to Win: Winning Strategies in the Game of Cycles

Abstract: A combinatorial game is a two player game that has a well-defined ruleset and no element of chance. This means that if both players play optimally, the winner can be determined before the game even begins. The Game of Cycles is a new combinatorial game played on any simple connected planar graph, introduced by Su (2020). In this talk I will introduce the basics of combinatorial game theory, the rules for the Game of Cycles in particular, and discuss results we have found about which player has a winning strategy on various classes of gameboard. This is joint work with Ryan Alvarado, Maia Averett, Christopher Jackson, Mary Leah Karker, Malgorzata Aneta Marciniak, Francis Edward Su, and Shanise Walker.

**Date: Wednesday, April 15, at 11:30 a.m.**

Speaker: Michael Cole, Hofstra University

Title: Cubic and Quartic Analogues of the Trigonometric Functions

Abstract: PDF

**Date: Wednesday, April 22, at 11:30 a.m.**

Speaker: Brandon Crofts, Hofstra University

Title: An Exploration and Generalization of the KRC Sequence

Abstract: For the sequence defined as KRC(n) = "the amount of integer triples (a,b,c) which satisfy a2 + 2bc = 0 where a,b,c are bounded by n," previous algorithms were simplistic and costly. These inefficiencies were due to their recursive nature. An algorithm not relying on recursive principles was theorized, developed, and partially optimized. Upper and lower bounds of this function were considered, as well as attempting to write this function algebraically.

From this point, the sequence defined as KRC3(n) = "the amount of integer triples (a,b,c) which satisfy a2 + 2bc = 0 where a,b,c are bounded by n," was brought to the forefront of the project. Our previous algorithm was adapted and updated to fit. Further optimizations were discovered, and a general algorithm was written for KRCP(n, p) = "the amount of integer triples (a,b,c) which satisfy a2 + pbc = 0 where a,b,c are bounded by n, and p is prime."

**Date: Wednesday, April 29, at 11:30 a.m.**

Speaker: Briana Schmidt, Hofstra University

Title: Defining Long Island's Sea Breeze Event Using Hofstra Data and D-Basis Algorithm

Abstract: Weather prediction is important. From daily weather forecasts to hurricane paths, we rely on it on a daily basis. Studying one localized weather event, like the Long Island sea breeze event, can improve small-scale weather predictions and pave the way to more accurate predictions of large-scale storms. Hear about the process of defining the sea breeze event using data collected at Hofstra. Future goals include a new application of the D-basis algorithm, developed at Hofstra Math Department, to predict the sea breeze event. My supervisor was Dr. Adaricheva and we collaborated with Dr. Bernhardt.

**Date: Wednesday, May 6, at 11:30 a.m.**

Speaker: Daniel Dimijian, Hofstra University

Title: Searching the Apéry numbers efficiently

Abstract: The Apéry numbers constitute a sequence with many interesting number-theoretic properties. However, terms in the sequence quickly become very expensive to compute, so it can be difficult to study experimentally. In part of an attempt to determine a nice formula for the p-adic valuation of arbitrary terms in the sequence with regard to certain primes p, we searched the sequence for values that satisfied a particular rare property. In order to make the search computationally feasible, we employed several optimization strategies and attempted to split the sequence into chunks to be searched independently and in parallel.

### 2018–2019

**Date: **Friday, September 21, at 3:30 p.m.**Speaker**: Russell Miller, Queens College, CUNY**Title**: Noncomputable Functions and Unsolvable Problems**Abstract**: The *Turing machine*, as defined by Alan Turing in 1936, is widely agreed to be an accurate if slightly idealized representation of what we mean when we speak today of "a computer." The definition is rigorous and has proven extraordinarily useful in mathematical logic. Unexpectedly, though, it also gave rise to the notion of a *noncomputable function*: a function (often just from **N** to **N**) which must exist, according to reasonable axiomatizations of mathematics, yet which cannot be computed by any Turing machine.

We will investigate these ideas from a slightly different angle, considering functions on the real numbers **R** instead of on the naturals. Here it is possible to impart a reasonably intuitive understanding of how an innocent-seeming function could fail to be computable. We will not spend any time on the details of Turing's definition, since it is safe to assume that today's students have a good intuition about what a computer is. Instead, we will discuss which functions on **R** can be computed, and how the representation of the real numbers (as decimals or other ways) may be relevant.

**Date**: Friday, September 28 at 3:30pm**Speaker**: Corrin Clarkson, Courant Institute, NYU**Title**: A pictorial introduction to the curve complex**Abstract**: Curves on surfaces play an important role in low dimensional topology. The curve complex is a way of geometrically encoding the relationships between such curves. I will use pictures to describe the construction of this complex and then relate some interesting facts about its structure.

**Date**: Friday, October 19 at 3:30pm**Speaker**: Chris Hanusa, Queens College, CUNY**Title**: The Making Of Mathematical Art**Abstract**: In this talk I'll be sharing some recent 2D and 3D art derived from mathematical concepts and created using a computer and 3D printing techniques. I'll discuss my inspiration, my methods, and the math that lies behind a number of my pieces. Come learn how it works and help me to generate new mathematical art right before your eyes!

**Date**: Wednesday, October 24 at 11:30am**Speaker**: Nick Bragman, Hofstra University**Title**: Probabilistic determinants of sign pattern matrices**Abstract**: Sign pattern matrices are matrices where the only possible entries are +, – and 0. We say that two sign pattern matrices are equivalent if they can be obtained from one another solely through signature equivalence, where sign pattern matrix *B* is equivalent to *A* if *B* = *AS*, for some signature matrix *S*. We begin by focusing on the probability of nonnegative determinant for each equivalence class, intertwining ideas connected to (–1,1)-matrices and (0,1)-matrices. We use this numerical analysis to quickly identify many matrices with probability 1/2 of nonnegative determinant, and show that this is not exhaustive via failure of the converse.

This research has been conducted throughout the past year at both Hofstra University and The College of William & Mary's REU.

**Date**: Friday, November 9 at 3:30pm**Speaker**: Robert Rand, University of Maryland**Title**: Provably Correct Quantum Programming**Abstract**: Quantum computing is hard, not only because of the challenges of building quantum devices, but also due to the challenges of programming them and having our programs run as intended. In this talk we introduce QWIRE, a tool that allows us to write quantum programs and mathematically prove that they have the desired behavior. These proofs are mechanically checked by the Coq proof assistant, guaranteeing that our programs meets their specifications.

**Date**: Wednesday, November 14 at 11:30am**Speaker**: Moshe Cohen, Vassar**Title**: An introduction to line arrangements and the search for Zariski pairs**Abstract**: A line arrangement is a finite collection of lines in the plane. We consider the projective plane where every pair of lines intersects exactly once, so that parallel lines intersect at a point "at infinity" (and then we need to include a half-circle of points "at infinity").

We can study a projective line arrangement using algebra and geometry by looking at equations of lines as in high school algebra. We can study this using combinatorics by looking at the points that are intersections of lines. We can study this using topology by looking at the complement -- the leftover space. We can ask if the combinatorial information forecasts the topological information of the complement.

When this does not occur, that is, when the combinatorics does not predict the topology, we obtain two different geometric arrangements; we call this a Zariski pair. There is no such pair of up to nine lines. Examples have been found with thirteen lines by Rybnikov in 1998 and with twelve lines by Guerville-Balle in 2014. Together with Amram, Sun, Teicher, Ye, and Zarkh, we investigate arrangements of ten lines. Together with four undergraduate students, we investigate arrangements of eleven lines.

This talk is accessible to those without backgrounds in combinatorics, topology, or algebraic geometry.

**Date**: Friday, February 8 at 3:30pm**Speaker**: Brian Katz, Augustana College**Title**: How do mathematicians believe?**Abstract**: Love it or hate it, many people believe that mathematics gives humans access to a kind of truth that is more absolute and universal than other disciplines. If this claim is true, we must ask: what makes the origins and processes of mathematics special and how can our messy, biological brains connect to the absolute? If the claim is false, then what becomes of truth in mathematics? In this seminar, we will discuss beliefs about truth and how they play out in the mathematics classroom, trying to understand this thing we call the Liberal Arts.

**Date**: Wednesday, February 20 at 11:30am**Speaker**: Amita Malik, Rutgers University**Title**: Sporadic Apéry-like sequences**Abstract**: In 1982, Gessel showed that the Apéry numbers associated to the irrationality of ζ(3) satisfy Lucas congruences. In this talk, we discuss the corresponding congruences for all sporadic Apéry-like sequences. In several cases, we are able to employ approaches due to McIntosh, Samol-van Straten and Rowland-Yassawi to establish these congruences. However, for the sequences often labeled *s*18 and η, we require a finer analysis. As an application, we investigate modulo which numbers these sequences are periodic. We also investigate primes which do not divide any term of a given Apéry-like sequence. This is joint work with Armin Straub.

**Date**: Friday, February 22 at 3:30pm**Speaker**: Lionel Levine, Cornell University**Title**: Will this avalanche go on forever?**Abstract**: In the abelian sandpile model on the *d*-dimensional lattice **Z***d*, each site that has at least 2*d* grains of sand gives one grain of sand to each of its 2*d* nearest neighbors. An "avalanche" is what happens when you iterate this move. In arXiv:1508.00161 Hannah Cairns proved that for *d*=3 the question in the title is algorithmically undecidable: it is as hard as the halting problem! This infinite unclimbable peak is surrounded by appealing finite peaks: What about *d*=2? What if the initial configuration of sand is random? I'll tell you about the "mod 1 harmonic functions" Bob Hough and Daniel Jerison and I used to prove in arXiv:1703.00827 that certain avalanches go on forever.

**Date**: Wednesday, March 27 at 11:30am**Speaker**: Nick Bragman, Hofstra University**Title**: Limiting densities of the Fibonacci sequence modulo *pn***Abstract**: The Fibonacci sequence mod *pn*, where *p* is prime, is periodic. Therefore, it is natural to ask what proportion of Fibonacci residues is attained modulo *pn*. As *n* goes to infinity, this proportion converges. It is already known that the limiting density of the Fibonacci sequence modulo powers of 11 is 145/264. We look to determine the limiting density of the Fibonacci sequence with respect to general primes *p*. We see that this question is split into two cases, dependent on whether *p* is congruent to 1 or 4 mod 5 or congruent to 2 or 3 mod 5. For primes congruent to 1 or 4 mod 5, we give a method for computing the density. We also discuss the difficulties of the case where *p* is congruent to 2 or 3 mod 5, which arise from the fact that the extension **Z***p*[*x*] /〈*x*2 – 5〉 is nontrivial.

**Date**: Friday, March 29 at 3:30pm**Speaker**: Heidi Goodson, Brooklyn College**Title**: Vertically Aligned Entries in Pascal's Triangle and Applications to Number Theory**Abstract**: The classic way to write down Pascal's triangle leads to entries in alternating rows being vertically aligned. In this talk, I'll explain and prove a linear dependence on vertically aligned entries in Pascal's triangle. Furthermore, I'll give an application of this dependence to number theory. Specifically, I'll explain how a search for morphisms between hyperelliptic curves led to the discovery of this identity.

**Date**: Wednesday, April 3 at 11:30am**Speaker**: Doron Zeilberger, Rutgers University**Title**: Quicksort**Abstract**: A novel approach, using experimental mathematics, to "analysis of algorithms" will be introduced, using Quicksort as a case study.

**Date**: Wednesday, April 10 at 11:30am**Speaker**: Angel Pineda, Manhattan College**Title**: The Mathematics of Medical Imaging: What Is Essential Is Invisible to the Eyes**Abstract**: Medical imaging began in 1895 when Wilhelm Roentgen took the first x-ray image of his wife's hand. Since Roentgen's discovery that electromagnetic waves could be used to see inside the human body, there have been many exciting discoveries in medical imaging, including how to image using many x-ray projections (CT scans), using sound (ultrasound), using magnetic spins (MRI), and more recently using near-infrared light (optical tomography). Mathematics has been a partner in the development of these imaging techniques. Calculus, linear algebra, Fourier transforms, partial differential equations, scientific computing, and statistical inference are only some of the mathematical and statistical tools which play an important role. In this talk, we will give an overview of the past, present and future of medical imaging and its partnership with mathematics. Even though the talk will include some advanced mathematics, statistics and machine learning, it will be accessible to undergraduate students.

**Date**: Friday, April 19 at 3:30pm**Speaker**: Thomas Dickson, Lehigh University**Title**: An Intro to Stochastic Calculus**Abstract**: Brownian motion describes the seemingly random movement of a particle immersed in a liquid. The particle's trajectory is unpredictable, hence, its trajectory is modeled by a random process. We will discuss how to perform calculus on a Brownian motion (integrate with respect to something stochastic, or random) and explore its applications in probability and differential equations. In particular, we will analyze the long term behavior of solutions to select stochastic differential equations.

**Date**: Wednesday, April 24 at 11:30am**Speaker**: Lisa Schmelkin, Hofstra University**Title**: Association Rules in Analysis of Medical Data**Abstract**: The bases of implications and their optimization is an active line of research. This topic relates to association rules that describe dependencies between variables and databases. In our talk, we will touch upon these theoretical components and their applications in data analysis. The Lattice Upstream Targeting Algorithm (LUST) is a new software designed by Prof. J.B. Nation (University of Hawaii), which targets partial order and association rules in gene expression data. In particular, it has been successful in identifying metagenes connected to different types of cancer. During the semester, our work involved automating the comparison of candidate metagenes produced by the LUST algorithm, a time-consuming process which, up until this point, has relied entirely on the user.

**Date**: Wednesday, May 1 at 11:30am**Speaker**: Justin Cabot-Miller, Hofstra University**Title**: The D-Basis Algorithm and Applications in Medicine and Beyond**Abstract**: How does one describe a relational database? What do gene expressions say about the likelihood of survival outcomes? Over the past couple of years, there's been the development of the D-basis algorithm to find the rules which answer the questions above. It uses closure operators and concept lattices to retrieve a certain type of rules from large sets of data. These are of the form "if these attributes are present, then this outcome occurs." While these are typically known as Association Rules, holding true in at least one part of the input, we only retrieve the set of Implications, rules which are universally satisfied. Through systematic data analysis and heuristic development, we are finding reliable ways to retrieve important rules which hold almost everywhere. We're developing applications of this method for use in both cancer research and weather data analysis.

### 2017–2018

**Date**: Wednesday, September 13 at 11:30 a.m.**Room**: Roosevelt 213**Speaker**: Catherine Pfaff, UC, Santa Barbara**Title**: Symmetries, Outer Space, & the Outer Automorphism Group of the Free Group**Abstract**: The symmetries of a polygon form a group. This group acts on the polygon by rotating it and flipping it. This basic idea of studying a group as symmetries of an object extends far beyond polygons. My favorite group is the outer automorphism group of the free group. Through a myriad of colorful pictures I will introduce this group and the object, Culler-Vogtmann Outer Space, that it acts on.

**Date**: Wednesday, September 27 at 11:30 a.m.**Room**: Roosevelt 213**Speaker**: Michael Cole, Hofstra University**Title**: The Mathematics of Gravitation and Eclipses**Abstract**: This talk will contain a mix of mathematics, physics, and astronomy. We begin with a derivation of Kepler's laws using vector calculus. Then tides will be discussed. There is a mathematical derivation of the basic facts about lunar tides that is quite simple and should be better known. Next astronomy: e.g. the layout of the solar system and some facts about the moon's rather complex orbital motion about the earth. We will study how periodicities of the moon's orbit about the earth and the orbit of the earth-moon system about the sun gives rise to the so-called "saros cycle" that describes the timing of lunar and solar eclipses.

**Date**: Wednesday, October 4 at 11:30 a.m.**Room**: Roosevelt 213**Speaker**: J. B. Nation, University of Hawai'i**Title**: How Aliens Do Math**Abstract**: We use a fanciful tour of the solar system to provide a gentle introduction to Universal Algebra. All major planets, plus a few Kuiper Belt objects, are included for the same low fare.

**Date**: Friday, October 6 at 3:30pm**Room**: Roosevelt 213**Speaker**: J. B. Nation, University of Hawai'i**Title**: A Primer of Quasivariety Lattices**Abstract**: This talk develops the theory of lattices of quasivarieties in a very general context. The lattice of subquasivarieties of a quasivariety can be represented as the lattice of closed algebraic subsets of an algebraic lattice with operators. This representation is used to develop new restrictions on the equational closure operator. This is joint work with Kira Adaricheva, Jennifer Hyndman and Joy Nishida.

**Date**: Wednesday, October 18 at 11:30 a.m.**Room**: Roosevelt 213**Speaker**: Dan Turetsky, University of Notre Dame**Title**: How hard is it to tell if two things are the same?**Abstract**: If I have two groups, how hard is it to tell if they're isomorphic? If I know they're isomorphic, how hard is it to find an isomorphism between them? Is it easier if I look at fields instead of groups? How about linear orders? These are the sorts of questions computable model theorists think about.

This talk will provide a gentle introduction to the field of computable model theory. We will cover the necessary concepts to make sense of the above questions, and we'll discuss some of the answers.

**Date**: Wednesday, October 25 at 11:30 a.m.**Room**: Roosevelt 213**Speaker**: Neil J. A. Sloane, Rutgers University and The OEIS Foundation**Title**: What Comes Next After 2, 4, 6, 3, 9, 12, 8, 10? - Confessions of a Sequence Addict**Abstract**: The On-Line Encyclopedia of Integer Sequences (or OEIS, oeis.org) is a free web site that contains information about 300,000 sequences, and is often called one of the most useful mathematical sites on the Web. I will discuss some classic sequences (van Eck, Gijswijt, Queens in Exile, etc.) and some very recent sequences from geometry, number theory, and the theory of computing. There will be music, movies, and a number of unsolved problems.

**Date**: Wednesday, November 1 at 11:30 a.m.**Room**: Roosevelt 213**Speaker**: Genevieve Maalouf, Hofstra University**Title**: Conjugacy Class Graphs of Dihedral and Permutation Groups**Abstract**: In this talk, we combine the study of group theory and graph theory by generating a graph with a group. If we take a group, G, we construct the graph Γ(G) by computing the conjugacy classes of G–Z(G). A node is produced by every conjugacy class and labeled with the cardinality of the class, ci. Lastly, an edge connects two vertices if gcd(ci,cj)>1. We say Γ(G) is the conjugacy class graph generated by G. The main focus of this talk is to classify all graphs of Γ(D2n×D2m) and to study the completeness of Γ(Sn×Sm). This work was done at the 2017 Missouri State REU and is joint with Taylor Walker (Tuskegee University) under the advisement of Les Reid (Missouri State University).

**Date**: Wednesday, November 15 at 3:00 p.m.**Room**: Roosevelt 110**Speaker**: John Goodrick, Universidad de los Andes**Title**: Counting integer points in polytopes with an extension of Presburger arithmetic**Abstract**: Fix some polytope P in Rd whose vertices have integer coordinates. Then for any positive integer t, one can ask to compute the number fP(t) of points in the lattice Zd that lie within the t-th dilate of P. By a theorem of Ehrhart, the function fP(t) is always a polynomial. If the vertices of P are rational (i.e. in Qd instead of Zd), then the function fP(t) is no longer necessarily polynomial but it is a quasi-polynomial: there is a number m and polynomials g1, ..., gm such that fP(t) = gi(t) whenever t is congruent to i modulo m.

In this talk, we will review the classic theory of Ehrhart polynomials and present a generalization (based on recent joint work with Tristram Bogart and Kevin Woods): if f(t) is the function which counts the number of integer points within a bounded region of Rd which is defined by a formula using addition, multiplication by the parameter t, inequalities, and quantifiers over variables from Z (but not over the domain of the variable t), then f(t) is quasi-polynomial for all sufficiently large values of t. We call such families "parametric Presburger families" in analogy with the logical theory of Presburger arithmetic. We will also present some new applications of this result.

**Date**: Wednesday, December 6 at 11:30 a.m.**Room**: Roosevelt 213**Speaker**: Zoran Sunic, Hofstra University**Title**: Context-free orders on free groups**Abstract**: We provide countably many orders on the free group such that, for each order, the set of positive words forms a context-free language. On the other hand, we show that there is no order on the free group with set of positive words that forms a regular language. Thus, as Einstein would say, things should be made as context-free as possible, but not regularer than that.

**Date**: Wednesday, February 14 at 11:30am**Room**: Roosevelt 213**Speaker**: Eric Rowland, Hofstra University**Title**: Formulas for Primes**Abstract**: Is there a formula that always produces primes? Fermat thought he found one; he conjectured that 22n + 1 is prime for all n ≥ 0, but he was wrong (this time). The answer depends on what we mean by a "formula". It turns out there is an expression for the nth prime using ordinary arithmetic functions! There are also simple functions/recurrences that generate primes. There is even a polynomial whose set of positive values is precisely the set of prime numbers. However, on closer inspection these formulas say less about prime numbers than they do about translating mathematical statements into others, and it's the clever translation that makes them interesting.

**Date**: Friday, February 23 at 3:30pm**Room**: Roosevelt 213**Speaker**: Fanny Shum, Courant Institute / NYU**Title**: Brownian Motion: Its History and Application**Abstract**: Brownian motion is used in many disciplines, such as mathematical physics, probability, and mathematical finance. We will look into the brief history of the development of brownian motion, also referred to as the Wiener process, and its significance in the mathematical field. In addition, we will discuss some of its applications.

**Date**: Wednesday, March 7 at 11:30am**Room**: Roosevelt 213**Speaker**: Jonathan Farley, Morgan State University**Title**: The Many Lives of Lattice Theory: An Expository Talk about Geometry, Topology, and Stanley**Abstract**: Modern lattice theory, the abstract study of order and hierarchy, was reborn at Harvard in the 1930's, a creation of Professor Garrett Birkhoff. His colleague Gian-Carlo Rota wrote, citing a prediction of I. M. Gelfand, that "lattice theory will play a leading role in the mathematics of the twenty-first century". Using the g-Theorem on polytopes, Anders Bjorner proved a result about how the number of totally ordered subsets of a finite distributive lattice grows as the subsets increase in size. He then asked in 1997 if that result could be proven combinatorially. At "the other end of the galaxy," one finds Priestley duality for distributive lattices, a way of understanding distributive lattice-ordered algebraic structures by means of topology. One day, on an airplane crossing the Atlantic, I saw these two notions collide.

**Date**: Wednesday, March 14 at 11:30am**Room**: Roosevelt 213**Speaker**: Josh Hiller, Adelphi University**Title**: Some simple mathematical models for cancer incidence and relative risk**Abstract**: Multistage models of carcinogenesis form the backbone of mathematical oncology. In this talk I will give a historical review of some of the most well known variations of this large class of models. I will attempt to place each model within its own bio-epidemiological context and theory. I will also go over some new relative risk results derived from two new (simplified) models based on generalized Erland processes.

**Date**: Wednesday, April 4 at 11:30am**Room**: Roosevelt 213**Speaker**: Taylor Ninesling, Hofstra University**Title**: Direct and Binary Direct Bases for One-set Updates of a Closure System**Abstract**: The different representations of closure systems lead to an alternative method for discovering association rules in a database. I will describe some of the different representations and how they relate to databases. We will discuss the notion of an implicational basis and the different minimality conditions we can subject them to, with the final goal of discussing methods for updating an existing basis with the addition or removal of a record.

**Date**: Friday, April 13 at 3:30pm**Room**: Roosevelt 213**Speaker**: Stephen Melczer, University of Pennsylvania**Title**: Lattice Path Enumeration and Effective Computation in Enumerative Combinatorics**Abstract**: The problem of enumerating lattice paths in cones with a fixed set of allowable steps has a long history dating back at least to the 19th century. This talk focuses on the interaction between the kernel method, a powerful collection of techniques used extensively in the enumeration of lattice walks in restricted regions, and the relatively new field of analytic combinatorics in several variables (ACSV). In particular, the kernel method often allows one to write the generating function for the number of lattice walks restricted to certain regions as the diagonal of an explicit multivariate rational function, which can then be analyzed using the methods of ACSV. This pairing is powerful and flexible, allowing for results which can be generalized to high (or even arbitrary) dimensions, weighted step sets, and the enumeration of walks returning to certain boundary regions of the domains under consideration. In the process, we will survey some decidability results in asymptotic and enumerative combinatorics. There are no high-level prerequisites for the talk, which should be accessible to upper year undergraduates.

**Date**: Wednesday, April 18 at 3:00pm**Room**: Barnard 101**Speaker**: David Rosenthal, St. John's University**Title**: Large scale notions of dimension**Abstract**: Large scale geometry, also known as coarse geometry, has grown into a vibrant subject in recent years due to the important role it plays in several fields, including high-dimensional manifold topology, geometric group theory, non-commutative geometry and related areas of analysis. One of the most well-known results linking large scale geometry to geometric topology is Guoliang Yu's result that a group with finite asymptotic dimension and a finite model for its classifying space satisfies Novikov's conjecture on the homotopy invariance of higher signatures. Since then, much interest in asymptotic dimension and its connections to high-dimensional manifold topology, geometric group theory, and non-commutative geometry ensued. In this talk I will discuss certain large-scale notions of dimension, namely, asymptotic dimension (introduced by Gromov) and decomposition complexity (introduced by Guentner, Tessera and Yu). The basic properties of these notions will be presented, along with several examples and open problems along the way.

**Date**: Wednesday, April 25 at 11:30am**Room**: Roosevelt 213**Speaker**: Susan Hermiller, University of Nebraska**Title**: Algorithms for groups of piecewise-linear functions**Abstract**: The group G of piecewise-linear homeomorphisms of [0,1] that fix the endpoints includes many important subgroups, including a particularly important group known as Thompson's group F. For finitely generated "computable" subgroups H of G (including F), we use properties of the generating set to build algorithms that solve a variety of problems. This is joint work with Collin Bleak and Tara Brough.

**Date**: Friday, April 27 at 3:30pm**Room**: Roosevelt 213**Speaker**: Gent Gjonbalaj, Hofstra University**Title**: The Description of Convex Geometries of Dimension 2**Abstract**: A convex geometry is a closure space with the anti-exchange property. In the paper by Edelman and Jamison in 1985 they introduced the parameter of convex dimension of a geometry G as the minimal number of linear subgeometries defining G. Czedli in 2014 and Richter and Rodgers in 2017 have shown that any convex geometry of dimension 2 can be represented by segments on a line. In our work, given a closure operator of a convex geometry, we try to determine whether it has convex dimension 2. This is joint work with Dr. Adaricheva.

**Date**: Wednesday, May 2 at 11:30am**Room**: Roosevelt 213**Speaker**: Élise Vandomme, LaCIM / UQAM**Title**: Critical exponent of balanced words**Abstract**: Over a binary alphabet it is well-known that the aperiodic balanced words are exactly the Sturmian words. The repetitions in Sturmian words are well-understood. In particular, there is a formula for the critical exponent (supremum of exponents e such that xe is a factor for some word x) of a Sturmian word. It is known that the Fibonacci word has the least critical exponent over all Sturmian words and this value is (5+√5)/2. However, little is known about the critical exponents of balanced words over larger alphabets. We show that the least critical exponent among ternary balanced words is 2+√2/2 and we construct a balanced word over a four-letter alphabet with critical exponent (5+√5)/4. This is joint work with N. Rampersad and J. Shallit.

**Date**: Friday, May 4 at 3:30pm**Room**: Roosevelt 213**Speaker**: Brian Zilli, Hofstra University**Title**: Some properties of reversible cellular automaton rules**Abstract**: A cellular automaton is a simple model of physics that applies a local rule at every time step. A cellular automaton rule f is invertible if there exists another cellular automaton rule g such that f(g(x)) = x for all bi-infinite words x. In a 1991 paper, D. Hillman formalized a characterization of reversible rules. We implemented this characterization in Mathematica to study the structure of the group of reversible cellular automaton rules under composition.

**Date**: Wednesday, May 9 at 11:30am**Room**: Roosevelt 213**Speaker**: Genevieve Maalouf, Hofstra University**Title**: Infinitely Many Stable Marriages**Abstract**: In 1962 the Gale-Shapely Algorithm was produced in order to solve the Stable Marriage problem in the finite case. It is already well known that the algorithm will terminate in a finite number of steps and always produce stable marriages. Will this algorithm work in the infinite case? Is it ever impossible to develop such an algorithm? If there are infinitely many men and women, it is not too hard to see that it is not necessary that everyone be matched. With this in mind, we would like to discuss the possibilities of a semi-stable pairing. First, we find the conditions needed to always produce a semi-stable pairing. Then, we discover that it is possible that no semi-stable pairing can exist, independent of any algorithm. Lastly, we analyze the run time of the algorithm when each of the mens preference lists has order type ω + 1.

### 2016–2017

**Date**: September 28, 2016, 11:20 a.m.**Room**: Roosevelt 213**Speaker**: Kira Adaricheva, Hofstra University**Title:** Representation of finite convex geometries by circles on the plane**Abstract:** Convex geometries are closure systems satisfying the anti-exchange axiom that models the behavior of convex hull operator in Euclidean space. Other than geometrical models appear in semilattices, graphs, logic and theory of human learning.

The possibility to represent every convex geometry by convex sets of finite point configuration in an n-dimensional space was an open problem until a result of K. Kashiwabara, M. Nakamura and Y. Okamoto (2005). Allowing circles rather than points, as was suggested by G.Czedli (2014), may presumably reduce the dimension for representation.

In this paper we introduce a property, the Weak 2x3-Carousel rule, which is satisfied by all convex geometries of circles on a plane but not by all finite convex geometries. This raises a number of representation problems for convex geometries which may allow us to better understand the properties of Euclidean space related to its dimension.

This work is coauthored by M. Bolat (currently a fourth-year math major at Nazarbayev University, Kazakhstan), and the paper is available on the arXiv here. The results were first presented at the 2016 SIAM Discrete Mathematics conference.

**Date**: October 21, 2016, 3:30 p.m.**Room**: Roosevelt 213**Speaker**: Quinn Culver, Fordham University**Title:** Algorithmically Random Tango**Abstract:** Algorithmic randomness was originally defined on the Cantor space of infinite sequences of 0s and 1s. Since then, other objects have been defined to be algorithmically random by coding via sequences or by adapting the definition directly to those objects' spaces. It turns out that it doesn't really matter which approach is taken; the same random objects are achieved. This is due to a general theorem, called preservation of randomness (PoR) and its converse, no randomness ex nihilo (NREN). This theorem also allows one to prove results about algorithmically random objects by doing probability and simply observing that the map in question is computable.

In this talk, the algorithmically random objects we'll focus on are algorithmically random sequences, algorithmically random closed subsets (RCSs) of Cantor space, algorithmically random continuous functions (RCFs) from Cantor space to Cantor space, and algorithmically random Borel probability measures on Cantor space. We are particularly interested in the interplay between them. It's fun to let random objects play together! For example, every RCS contains a random sequence and every random sequence is contained in some RCS. The zero sets (that is, the preimages of the sequence of all 0s) of RCFs are exactly the RCSs. If time permits, we'll discuss some other facts that don't necessarily use PoR or NREN but still exhibit what can happen when random objects dance.

**Date**: October 26, 2016, 11:20 a.m.**Room**: Roosevelt 213**Speaker**: Eric Rowland, Hofstra University**Title:** Unanswered questions about the Fibonacci numbers**Abstract:** Leonardo of Pisa wrote about the sequence 1, 1, 2, 3, 5, 8, 13, ... in the year 1202. But 800 years later, there are still basic questions about the Fibonacci numbers whose answers we don't know. The sequence obtained by reducing every Fibonacci number modulo m is periodic; but a general expression for the period length depends on whether or not special prime numbers, known as Wall–Sun–Sun primes, exist. In 1966, D. D. Wall conjectured that they don't, but a heuristic argument suggests there are infinitely many! Another question concerns the density of residues attained by the Fibonacci sequence modulo pα as α→∞. This question should be more tractable, and I'm hoping to find students interested in working on it.

**Date**: November 11, 2016, 3:30 p.m.**Room**: Roosevelt 213**Speaker**: Henry Towsner, University of Pennsylvania**Title:** Why (and when) is there only one way to random?**Abstract:** A large finite graph (in the sense of combinatorics) is called quasirandom if it "resembles" the graph we would get by choosing each edge randomly based on a coin flip. "Resembles" sounds like a vague notion, but it turns out that lots of very different ways of saying that a graph resembles a random graph end up all being equivalent. To explain why this happens, we'll have to replace large but finite graphs with properly infinite graphs, so that we can use abstract probability theory to identify what a "kind of randomness" looks like.

Time permitting, we'll touch on hypergraphs—like graphs, except that instead of having edges which connect two vertices, they have "*k*-hyperedges" which connect *k* vertices at once—and explain why there are multiple ways for hypergraphs to be random, and how we can tell that we've found them all.

**Date**: March 17, 3:30 p.m.**Room**: Roosevelt 213**Speaker**: Linda Brown Westrick, University of Connecticut**Title:** Computation and information in sofic shifts**Abstract:** Any two-dimensional sofic shift can be described of as the set of infinite tilings from a fixed tileset, in which some of the distinctions between the tiles have subsequently been erased. Classically, there are tilesets whose infinite tilings perform arbitrary computations, so in a sofic shift these computations can be hidden, even as they control what is visible. By contrast, in an effectively closed shift, the restrictions on what patterns occur are enumerated by an algorithm that does not have to share physical space with the patterns it controls. The sofic shifts are a proper subclass of the effectively closed shifts, but exactly what the limitations of the sofic computations are is not well understood. Towards one direction of this problem, we construct new examples of "computationally-intensive" sofic shifts.

**Date**: March 31, 4:00 p.m.**Room**: Roosevelt 213**Speaker**: Victor Donnay, Bryn Mawr College**Title:** Connecting Math and Sustainability**Abstract:** How can we better inspire our students to study and succeed in mathematics? Victor Donnay will discuss his experiences in using issues of civic engagement, particularly environmental sustainability, as a motivator. He will present a variety of ways to incorporate issues of sustainability into math and science classes ranging from easy to adapt extensions of standard homework problems to more elaborate service learning projects. He will share some of the educational resources that he helped collect as chair of the planning committee for Mathematics Awareness Month 2013 - The Mathematics of Sustainability as well as his TED-Ed video on Tipping Points and Climate Change. He has used these approaches in a variety of courses including Calculus, Differential Equations (chosen as a SENCER model course), Mathematical Modeling and Senior Seminar.

**Date**: April 14, 3:30 p.m.**Room**: Roosevelt 213**Speaker**: Richard Myers, Hofstra University**Title:** Randomness: A Computable Story**Abstract:** What is a random string of ones and zeros? This can be a difficult thing to describe, since, could we describe a random string directly, we would be hard pressed to call it truly random. Instead, we describe the complementary set of nonrandom strings. However, there is not just one natural description for the nonrandom strings, and some of these descriptions are nonequivalent, giving rise to distinct sets of random strings. We will discuss some of these descriptions and prove the equivalence of two such classes, Martin-Löf and *r*-Hempstead randomness.

**Date**: April 21, 3:30 p.m.**Room**: Roosevelt 213**Speaker**: Suresh Eswarathasan, Cardiff University**Title:** Overview of Some Problems in Eigenfunction Asymptotics**Abstract:** In this lecture, I will give an overview of some classical and recent results concerning the spectral asymptotics for eigenfunctions of the Laplace-Beltrami operator on a compact boundaryless Riemannian manifold (M,g). In particular, I will cover results regarding semiclassical measures (which quantify the asymptotic profile of the eigenfunctions, in some sense) and L^p norms (which provide various measures of their size). If time permits, I will present some joint work in progress with Malabika Pramanik (U. British Columbia) regarding these eigenfunctions and fractal sets on M.

**Date**: April 26, 11:30 a.m.**Room**: Roosevelt 213**Speaker**: Stephanie Nagel, Hofstra University**Title:** A New Method to Generate Uniformly Distributed Random Variables on the D-Dimensional Unit Spherical Shell**Abstract:** Generating uniformly distributed random variables on the unit spherical shell is extremely important. For example, generating these random variables is a crucial step in the simulation of the multivariate normal distribution. There already exist methods to generate uniformly on the unit spherical shell, including methods involving scaling of normal vectors and spherical coordinates. The new method presented in this paper involves recursively writing the joint density of the vector of random variables as the product of the marginal distribution of one component and the corresponding conditional distribution for the rest of the components. After this decomposition, the re-scaling turns out to correspond to the density of another uniformly distributed random variable on a unit spherical shell of lower dimension. In this research, we detail each algorithm that can be used to generate on the unit spherical shell, and we conduct tests on each method to determine which one is quicker and more efficient.

**Date**: April 26, 12:10 p.m.**Room**: Roosevelt 213**Speaker**: Stephanie Nagel, Hofstra University**Title:** My Research Experience at Frontline Education**Abstract:** For the past three months, I have worked as a data analyst intern at Frontline Education. Through my internship, I have gained invaluable work experience. Also, my passion for statistics has grown tremendously. Adapting what I have learned in my statistics classes at Hofstra to the workplace has been extremely useful. In this presentation, I will speak about the projects that I worked on during the internship, what I have learned, and the role of mathematics in the internship.

**Date**: April 28, 3:30 p.m.**Room**: Roosevelt 213**Speaker**: Waseet Kazmi, Hofstra University**Title:** Computing in Logspace**Abstract:** Most of combinatorial and computational group theory focuses on computing efficiently in finitely generated groups. In this talk, we will consider the class of groups which have logspace computable normal form over some finite generating set. We will show some basic properties of logspace computable functions and various closure properties satisfied by the class of groups with logspace computable normal form. Finally, we will show that the free group has logspace conjugacy problem and that the property of having logspace conjugacy problem is closed under direct product.

**Date**: May 10, 11:30 a.m.**Room**: Roosevelt 213**Speaker**: Tim McNicholl, Iowa State University**Title:** What computers can't do**Abstract:** In 1936, British mathematician A.M. Turing proved that there are mathematical problems that cannot be solved by any discrete computing device such as a digital computer. His insights founded the mathematical discipline of computability theory, which is the study of the limits and potentialities of computing machines. It is now known that every discipline in mathematics contains incomputable problems that are fundamental and natural. We will discuss some of these problems, some standard methods for showing that a problem is incomputable, and some of the frontiers in current computability research.

### 2015–2016

**Date**: October 30, 2015, 3:30 p.m.**Speaker**: Steve Costenoble, Hofstra University**Title:** Calculations in Equivariant Ordinary Cohomology**Abstract:** Ordinary cohomology is one of the calculational workhorses of algebraic topology. However, when we add in the consideration of symmetries, to get equivariant algebraic topology, the analogous cohomology theory is poorly understood and calculations are few and far between. Stefan Waner and I have worked to try to rectify these problems. I'll describe equivariant ordinary cohomology and the version we've developed with "extended grading," some of the calculations that have been done, and the hope for the future.

**Date**: November 18, 2015, 11:30 a.m.**Speaker**: Barbara Gonzalez, Hofstra University**Title:** Students as Partners in Curricular Design**Abstract:** In order to increase student interest in mathematics and its connections to real applications, Roosevelt University began incorporating semester-long projects into its Calculus II course. Different project topics are used each semester. Project creation has lead to opportunities for student involvement, including work as embedded tutors, undergraduate research projects, and opportunities for students to present posters and talks to a broader audience. The student project designers are often mathematically early in their careers, and so this provides them with an opportunity to create and explore new mathematics while giving faculty the ability to involve students of all levels in research projects.

Evidence was gathered from interviews, surveys, and observation of student research work and its implementation in the classroom. We found that embedded tutors reported more confidence in their knowledge of calculus and insights into teaching it, and project designers experienced similar benefits to those of a traditional research experience.

(joint work with S. Cohen and M. Pivarski)

**Date**: March 18, 2016, 4:00 p.m.**Speaker**: Patrick Dragon, Bard College at Simon's Rock**Title:** The Grandmama De Bruijn Sequence for Binary Strings**Abstract:** A de Bruijn sequence is a binary string of length 2*n* which, when viewed cyclically, contains every binary string of length *n* exactly once as a substring. For example, 00010111 suffices for *n*=3. Knuth refers to the lexicographically least de Bruijn sequence for each *n* as the "Grandaddy" sequence due to its venerable origin. Martin originally constructed these sequences greedily and later it was shown by Fredericksen et al. that the Grandaddy sequences can also be constructed by concatenating the aperiodic prefixes of the binary necklaces of length *n* in lexicographic order. It was recently proven that the Grandaddy has a lexicographic partner. The "Grandmama" sequence is constructed by concatenating the aperiodic prefixes of necklaces in co-lexicographic order. We will discuss the construction and some interesting properties of both the Grandaddy and Grandmama de Bruijn sequences.

Based off a similarly-titled paper by Williams, Hernandez, and Dragon (November 2015).

**Date**: April 13, 2016, 11:30 a.m.**Speaker**: Scott Jeffreys, Hofstra University**Title:** Computational Finance, Applications of Calculus, and R Programming**Abstract:** Computational finance forges together applied mathematics, economics, computer science, and finance to solve problems in algorithmic and high-frequency trading, quantitative investing, and portfolio management. Financial engineers, or "quants," create models against which markets can be measured for returns, hedging opportunities, or risk controls.

During the Spring 2016 term following 18 months of research, the Hofstra University School of Engineering and Applied Sciences launched our first course in computational finance using algorithms developed in R to solve otherwise untenable problems. In this seminar, we will look at three complex problems from finance and offer elegant solutions drawn from our applied mathematics and computer science tools:

[1] How are coupon bonds efficiently priced?

[2] How are options priced using the Black-Scholes model?

[3] Why do heavy tails warn us about portfolio risk?

**Date**: April 22, 2016, 4:00 p.m.**Speaker**: Mutiara Sondjaja, New York University**Title:** A New Proof of Tucker's Lemma with a Volume Argument**Abstract:** Tucker's lemma is a combinatorial theorem about labeled triangulations of the d-dimensional sphere. It states that if the vertices of such triangulations receive a label from {±1, ±2, ..., ±d} with the labels of antipodal vertices summing to zero, then there must exists a pair of adjacent vertices whose labels sum to zero. The lemma is known to be equivalent to the Borsuk-Ulam theorem in topology.

In this talk, we discuss an application of Tucker's lemma in constructing a solution to a fair division problem (Su and Simmons (2002)). Then we present a new proof of Tucker's lemma based on a volume argument. This is joint work with F. E. Su and undergraduates at the 2015 MSRI Undergraduate Program (B. Kutture, O. Leong, C. Loa).

**Date**: April 27, 2016, 11:30 a.m.**Speaker**: Stephanie Nagel, Hofstra University**Title:** Generating on the N-Sphere: An Undergrad Research Experience**Abstract:** Before taking my departmental honors research course, I had some knowledge of what math research entailed, but I have learned so much through this experience. We will discuss what it is like to do research, and how doing research is different from taking a regular class. We will also talk about the research that I have been doing with Dr. Gonzalez, which is on the methods to generate uniform random variables on the d-dimensional unit spherical shell. These methods include scaling vectors of normal random variables and an algorithm involving spherical coordinates. We will look at the code for these methods and discuss further research that I plan to do with these methods.