# 2018–2019 Mathematics Seminar

**Date**: Friday, September 21 at 3:30pm

**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 *p ^{n}*

**Abstract**: The Fibonacci sequence mod

*p*, where

^{n}*p*is prime, is periodic. Therefore, it is natural to ask what proportion of Fibonacci residues is attained modulo

*p*. As

^{n}*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.