# 2016–2017 Mathematics Seminar

**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 p^{α} 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.