Colloquia 2021-2022

On Lie's Third Theorem for Leibniz Algebras

Leibniz algebras generalize Lie algebras in the sense that the bracket is not necessarily antisymmetric. We will describe in our talk attempts to integrate Leibniz algebras into Lie racks. Here a rack is an algebraic structure more general than the notion of a group; in fact, one retains from the notion of a group only the conjugation map. M. Kinyon showed in 2007 that the tangent space of a Lie rack carries naturally the structure of a Leibniz algebra. S. Covez showed in 2010 that Leibniz algebras integrate into local Lie racks. Many other integration procedures have been proposed since then. We will focus on the integration procedure of Bordemann-Wagemann (2017), where a general Leibniz algebra is integrated into a Lie rack which is an affine bundle over a Lie group such that in case the Leibniz algebra is a Lie algebra, one obtains the standard integration of Lie algebras. The drawback of this procedure is that it is not functorial.

This talk is directed especially toward undergraduate and graduate students in mathematics or related subjects.

Counting the Uncountable, or, How I Learned to Stop Worrying and Love Elementary Submodels

Consider the following game. Let W be some subset of the reals. During the initial round, Alice chooses -∞ and Bob chooses +∞. Then each subsequent move of either player is a real number strictly between the two most recently chosen numbers. After ω-many rounds, Alice wins this game provided that some number in W is both greater than all her choices and less than all Bob's choices.

Bob has a winning strategy in this game if and only if W is countable. To see this, we will demonstrate two proofs, one utilizing countable elementary submodels, and the other involving limited information strategies.

This talk is based on work done in collaboration with Will Brian (UNC Charlotte).

Multi-Step Batch Testing for Infectious Diseases

We propose a mathematical model based on probability theory to optimize COVID-19 testing by a multi-step batch testing approach with variable batch sizes. This model and simulation tool dramatically increase the efficiency and efficacy of the tests in a large population at a low cost, particularly when the infection rate is low. The proposed method combines statistical modeling with numerical methods to solve nonlinear equations and obtain optimal batch sizes at each step of tests, with the flexibility to incorporate geographic and demographic information. In theory, this method substantially improves the false positive rate and positive predictive value as well. We also conducted a Monte Carlo simulation to verify this theory. Our simulation results show that our method significantly reduces the false negative rate. More accurate assessment can be made if the dilution effect or other practical factors are taken into consideration. The proposed method will be particularly useful for the early detection of infectious diseases and prevention of future pandemics. The proposed work will have broader impacts on medical testing for contagious diseases in general.

Bifurcation Phenomena in the Family of Cubic Polynomials

This talk will begin with a survey of complex dynamics and known results regarding bifurcation in the family, {f_a,c}_c∈C, of cubic polynomials with critical points ±a; similar phenomena in another family will be presented. I will also discuss a variety of attempts to explain resonance phenomena in these bifurcations.

Imagining (or Reimagining) the Finite Subgroups of SU(2)

First of all, the talk is aimed at upper level math/stat majors and graduate students. The prior knowledge I am assuming is that the listener has an understanding of the definition of a group, that of a subgroup, and the collection of cosets of the subgroup. The groups that I'll be talking about are the so-called dicyclic groups, the binary tetrahedral group (which is also known as the group of (3-by-3)-matrices of determinant 1 with entries in Z-mod 3 [SL₂(Z/3)]), the binary octahedral group, and the binary dodecahedral group [SL₂(Z/5)]. All of these groups can be thought of as sitting inside the 3-dimensional sphere. I'll demonstrate a method for representing the group elements as strings with beads, or quipu. The beads compose by means of modular (clock) arithmetic and slide through crossings between pairs of strings.

The talk will be replete with pictures and ad hoc animations. Some techniques for obtaining similar pictures will be presented.

On Random Polynomials Generated by a High-Order Recurrence Relation

We consider a class of random banded Hessenberg matrices with independent entries having identical distributions along diagonals. The distributions may be different for entries belonging to different diagonals. For a sequence of n x n matrices in the class considered, we investigate the asymptotic behavior of their empirical spectral distribution as n tends to infinity.

Numerical Analysis meets Algebraic Topology

One of the fundamental tools in numerical analysis and PDE is the finite element method (FEM).  A main ingredient in FEM are splines: piecewise polynomial functions on a mesh. Even for a fixed mesh in the plane, there are many open questions about splines: for a triangular mesh T and smoothness order one, the dimension of the vector space C₃(T) of splines of polynomial degree at most three is unknown. In 1973, Gil Strang conjectured a formula for the dimension of the space C₂(T) of splines of polynomial degree at most two in terms of the combinatorics and geometry of the mesh T, and in 1987 Lou Billera used algebraic topology to prove the conjecture (and win the Fulkerson prize). I'll describe recent progress on the study of spline spaces, including a quick and self contained introduction to some basic but quite useful tools from topology.

The talk will be accessible to undergraduates.

Artistic Mathematics: Truth and Beauty

I'll talk about my work in mathematical visualization: making accurate, effective, and beautiful pictures, models, and experiences of mathematical concepts. I'll discuss what it is that makes a visualization compelling, and show many examples in the medium of 3D printing, as well as some work in virtual reality and spherical video. I'll also discuss my experiences in teaching a project-based class on 3D printing for mathematics students.

There are too many Knotted Graphs!

The Graph Minor Theorem of Robertson and Seymour can be thought of as a powerful generalization of Kuratowski's Theorem. It asserts that, for any graph property, whatsoever, there is an associated finite list of graphs that are minor minimal for that property. For example, Kuratowski's Theorem states that K₅ and K_3,3 are the two minor minimal non-planar graphs. As another example, Robertson, Seymour, and Thomas showed that there are seven minor minimal intrinsically linked graphs.

We'll report on the lack of progress in resolving the analogous question for intrinsic knotting. We know the number of minor minimal intrinsically knotted graphs must be finite and there are at least 260 of them. However, it seems that new ideas are needed to get an upper bound or otherwise move the theory forward.

This talk is directed especially toward undergraduate and graduate students in mathematics or related subjects.

Cutting Up a Pizza and Related Topics

Suppose you are cutting up a pizza and you are trying to maximize the number of pieces of pizza you can get with only straight-line cuts. Can you find a formula for this maximum number just in terms of the number of cuts? What if you can only use straight-line cuts and you are trying to maximize the number of pieces of pizza that don't have any crust? What if you are super lucky and have to answer the same questions for a three-dimensional pizza - maybe a large calzone? These are some of the questions you might start with if you are studying a mathematical object called a hyperplane arrangement. It turns out that hyperplane arrangements are not only useful for cutting up pizza, but also for motion planning, studying singularities, and lots of other things! We'll start with the pizza question and see how far we get.

This talk is directed especially toward undergraduate and graduate students in mathematics or related subjects.

Maximal Linklessly Embeddable Graphs

This talk is about graphs embedded in 3-dimensional space. We study graphs which are linklessly embeddable and, in particular, graphs which are maximal with this property (maxnil graphs). Linklessly embeddable graphs can be embedded in space in such a way that no two cycles link non-trivially. We learn how to construct large maxnil graphs from smaller maxnil graphs, and we ask how many edges can a maxnil graph with N vertices have.

Here is a warm up question. A graph is maximal planar if it can be drawn in the plane without edge intersections, while the addition of any one edge obstructs this kind of drawing. How many edges can a maximal planar graph with 5 vertices have? 6 vertices? 7 vertices? Can you find a pattern?

This talk is based on joint work with Ramin Naimi and Andrei Pavelescu. 

This talk is directed especially toward undergraduate and graduate students in mathematics or related subjects.

The Shape of the Universe

Topology studies the shapes of objects that are unchanged by stretching and shifting. In low-dimensional topology, we study the shapes of 3- and 4-dimensional objects that model the physical world that we live in. In this talk, I will consider the question: "What is the shape of the universe?" I will introduce the mathematical context for manifolds, knots, and surfaces for making sense of the question, discuss how geometric topology helps explore the range of possibilities, and relate current physical evidence supporting the likelihood of different answers.

This talk is directed especially toward undergraduate and graduate students in mathematics or related subjects.

Rabbits and Airplanes and Dust, Oh My!

Join me on a tour of the famous Mandelbrot set! See functions that eat their own outputs. Chase complex numbers through these functions and figure out how far they need to go to escape. Color by number to reveal pictures of monstrous beasts or cute rabbits. Fall down rabbit holes of self-symmetry. By the end, will you be brave enough to tackle new, unexplored realms beyond the Mandelbrot set?

Likelihood-Based Finite Sample Inference for Synthetic Data from the Pareto Model

Statistical agencies often publish microdata or synthetic data to protect confidentiality of survey respondents. This is more prevalent in case of income data. We have developed likelihood-based finite sample inferential methods for singly imputed synthetic data using plug-in sampling and posterior predictive sampling techniques under the Pareto distribution, a well-known income distribution. The estimators are constructed based on sufficient statistics and the estimation methods possess desirable properties. For example, the estimators are unbiased and confidence intervals developed are exact. An extensive simulation study has been carried out to analyze the performance of the proposed methods.

This is joint work with Sandeep Barui

On the Humphreys-Verma Conjecture and Donkin's Tilting Module Conjecture

In 1973 Humphreys and Verma conjecture that the principal indecomposable modules of a restricted Lie algebra can be lifted to their ambient algebraic group. In 1990 Donkin conjectured that these liftings should in fact be tilting modules for the algebraic group. Donkin linked his tilting module conjecture to a second conjecture of his on the existence and relationship between certain filtrations for modules of the algebraic group. Donkin's second conjecture would imply a positive answer to a question raised by Jantzen in 1980.

In this talk I will discuss some recent developments on these conjectures and Jantzen's question. Most of the results presented are based on joint work with Dan Nakano, Chris Bendel and Paul Sobaje. All efforts will be made to keep the exposition accessible.

Graduate students are strongly encouraged to attend.

This talk is directed especially toward undergraduate and graduate students in mathematics or related subjects.

The Mathematics of Mario

While often derided as a mindless distraction, many video games actually contain deep mathematical ideas. In this talk, the presenter will demonstrate instances of how mathematics can be used to overcome challenges found within the classic video games Super Mario RPG (Super Nintendo) and Teenage Mutant Ninja Turtles: Fall of the Foot Clan (Game Boy).

At the end of the talk, upcoming extracurricular and honors opportunities in mathematics and statistics will be discussed.