# 2017–2018 Mathematics Seminar

**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, c_{i}. Lastly, an edge connects two vertices if gcd(c_{i},c_{j})>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(D_{2n}×D_{2m}) and to study the completeness of Γ(S_{n}×S_{m}). 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 R^{d} whose vertices have integer coordinates. Then for any positive integer t, one can ask to compute the number f_{P(t)} of points in the lattice Z^{d} that lie within the t-th dilate of P. By a theorem of Ehrhart, the function f_{P(t)} is always a polynomial. If the vertices of P are rational (i.e. in Q^{d} instead of Z^{d}), then the function f_{P(t)} is no longer necessarily polynomial but it is a quasi-polynomial: there is a number m and polynomials g_{1}, ..., g_{m} such that f_{P(t)} = g_{i(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 R^{d} 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 2^{2n} + 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 x^{e} 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.