2018-19 Colloquia talks

Knotting in DNA Replication

Abstract: Replication is when a single DNA molecule is reproduced to form two new identical DNA molecules. In the middle of replication a more complex structure is formed. When a circular piece of DNA is replicated, the intermediate structure is that of a theta-graph. A theta-graph has two vertices and three edges between them. I study embedded graphs and graph embeddings. This talk will focus on my work on unknotting numbers of theta-graphs and understanding DNA replication.

Gröbner Bases, Smooth Centralisers and the Lefschetz Principle

Abstract: In a paper in the now defunct LMS Journal of Computation I used GAP to compute the Lie-theoretic centralisers in the exceptional groups of elements in their minimal modules in all characteristics, establishing when the centralisers in the groups were smooth. Non-smoothness was only found in very small characteristics, even where there are infinitely many orbits. This led to a question on when all centralisers of elements in a Z-defined representation would be smooth if the characteristic were large enough. With my co-authors we managed to prove this using the Lefschetz principle from model theory applied to Gröbner bases, which gave rise to what we call 'd-bounded Hopf quadruples'. I'll explain some of this.

This is joint with Ben Martin and Lewis Topley.

The Complement of a Linklessly Embeddable Graph with Thirteen Vertices is Intrinsically Linked

Abstract: A simple graph is called intrinsically linked if every embedding of it into R³ contains a non-trivial link. Campbell et al. showed that a graph on n ≥ 6 vertices and at least 4n-9 edges is intrinsically linked because it contains a K₆; this implies that for a graph G with n ≥ 15 vertices, either G or its complement is intrinsically linked. We improve this result by proving that given any simple non-oriented graph G with at least thirteen vertices either G or its complement is intrinsically linked. In the process, we discuss the de Verdière invariant, an algebraic construction which yields surprising results about the topological properties of a graph.

Facilitating Collaborative Learning in Mathematics Courses

Abstract: With support from SGA and the Provost's Office, in spring 2018 MSPB 360 was converted into a collaborative learning classroom. In this talk, we will describe several of the features of the classroom that can be used to enhance the student experience and give you a more engaged classroom. In addition to suggesting simple techniques that you may incorporate with your existing teaching style, we will give an overview of how to use Team-Based Learning in mathematics, based upon our collaboration with our university's TeamUSA Quality Enhancement Program.

Some Conjectures about the Colored Jones Polynomial

Abstract: We will discuss some old and new conjectures about the colored Jones polynomial. These include the volume conjecture, AJ conjecture, slope conjecture, and strong slope conjecture. The volume conjecture of Kashaev-Murakami-Murakami relates the colored Jones polynomial of a knot and the hyperbolic volume of the knot complement in S³. The AJ conjecture of Garoufalidis relates the A-polynomial and the colored Jones polynomial of a knot. The A-polynomial was introduced by Cooper et al. in 1994 and has been fundamental in geometric topology. A similar conjecture to the AJ conjecture was also proposed by Gukov from the viewpoint of the Chern-Simons theory. The slope conjecture of Garoufalidis and the strong slope conjecture of Kalfagianni-Tran assert that certain boundary slopes and Euler characteristics of essential surfaces in a knot complement can be read off from the degree of the colored Jones polynomial.

Nilpotent Orbits and Tilting Modules for the General Linear Group

Abstract: This talk is about a certain class of finite-dimensional representations (called "tilting modules") for the group GL_n(k), where k is an algebraically closed field of positive characteristic. Here are some things one can do with these modules: (1) Classify the tensor ideals of tilting modules (this makes sense because the tilting modules are closed under taking tensor products). (2) Compute their support varieties, which are closures of nilpotent orbits in the Lie algebra of GL_n. I will explain what is known about these questions, and I will discuss a conjectural link between them, with some concrete examples. This is joint work with W. Hardesty and S. Riche.

Pairwise and Variance Balanced n-ary Block Designs

Abstract: We give methods to construct n-ary block designs which are pairwise balanced (PB) and also variance balanced (VB). While pairwise balance can be seen as a combinatorial property, the variance balance is a statistical property. We construct the designs with k=4,5. Two series of designs are constructed for a general value of k. Apart from two types of balances, it is shown that these designs are also E-optimal and MV-optimal in a general class of designs.

The Dollar Game: An Introduction to Sandpiles and Chip-Firing

Abstract: We introduce the Dollar Game, a game in which we trade dollars across edges of a graph with the goal of eliminating debt. This talk is intended to motivate the theory of abelian sandpiles and chip-firing, topics that we will explore in a seminar this semester.

Rainbow Spanning Trees in Edge-Colored Complete Graphs

Abstract: A spanning tree of an edge-colored graph is rainbow provided that each of its edges receives a distinct color. In 1996, Brualdi and Hollingsworth conjectured that if K_2m is properly (2m − 1)-edge-colored, then the edges of K_2m can be partitioned into m rainbow spanning trees, except when m = 2. In this talk, we’ll look at the history and recent results concerning this conjecture and consider the extremal question of maximizing and minimizing the number of rainbow spanning trees in K_n, given a special type of (n − 1)-edge-coloring which is surjective and rainbow cycle free, called a JL-coloring.

L-Moments of the Inverse Gaussian Distribution

Abstract: L-moments are robust alternatives to the moments for the skewed distributions with heavy tails or one with high variability. The inverse Gaussian distribution is one such distribution. We give L-moments formulae and methods for parameter estimation for the inverse Gaussian distribution. Simulation studies are conducted to compare the performance of the L-moments estimates with the maximum likelihood estimates of the distribution parameters. These L-moments are used to estimate the critical point of the hazard function. Further we obtain the L-moments estimates and admissible method estimates of the critical point of the hazard function and then compare this with two other existing methods. To illustrate these, a practical application will be carried out at the end.

Jensen-Pólya Criterion for the Riemann Hypothesis and Related Problems

Abstract: In this talk, I will summarize forthcoming work with Griffin, Ono, and Zagier. In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann's Xi-function. This hyperbolicity has been proved for degrees less than or equal to 3. We obtain an arbitrary precision asymptotic formula for the derivatives of Xi, which allows us to prove the hyperbolicity of 100% of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials. In the case of Riemann's Xi-function, this proves the GUE random matrix model prediction for the distribution of zeros in derivative aspect. This general condition also confirms a conjecture of Chen, Jia, and Wang on the partition function.

Escher Squares and Lattice Links

Abstract: A lattice link is a link embedded in the 3-dimensional integer lattice. Allardice and Bloch showed that a lattice diagram in the 2-dimensional integer lattice is the projection of a lattice link if and only if it does not contain a Celtic configuration. We give a new proof of this result, by considering a height function on a digraph associated to the lattice diagram.

This is joint work with Ramin Naimi and Andrei Pavelescu.

Rainbow Cycles on Complete Graphs

Abstract: It is well known that a complete graph on n vertices can be edge colored with n-1 colors in order to avoid rainbow cycles. No such coloring exists using n colors. A certain encoding of full binary trees produces edge colorings using this maximum number of colors, n-1, in order to avoid rainbow cycles. Interestingly, all such colorings can be formed using this encoding. A few years later a similar result was found to hold for complete bipartite graphs and, subsequently, complete multipartite graphs. Most recently, an analogous theorem was found for all general connected graphs. First, we will look at the connection between these edge colorings and full binary trees; we then will highlight some of the important ideas used in order to prove the general case.

Prediction Methods for Semi-Continuous Data with Applications in Climate Science

Abstract: Semi-continuous random variables have discrete and continuous components with support on a set of discrete points and a subset on the real line. Daily precipitation (rainfall) data is an example of such a random variable with a point mass at 0 and an absolutely continuous distribution function on the positive real line. When the Probability of observing a 0 is assumed to be independent of the parameters of the continuous part, the density of the random variable takes the form a Two-Part model. A popular form that enforces a dependency is the standard Tobit model. We briefly review some inferential aspects of the semi-continuous distributions and present several methods of prediction and derivation of predictive densities, motivated by applications of spatio-temporal models in Climate Science. This is joint work with Sai Kumar Popuri, UMBC, and Dr. Amita Mehta of Joint Center for Earth Systems Technology.

Fifth Satya Mishra Memorial Lecture

The Statistical Modeling of Self-Reported and Proxy Observations in Longitudinal Studies

Abstract: In gerontological studies when the patients become unable to provide responses by themselves due to advancing severity of their conditions, proxy responses by a relative or a caregiver “proxy” are used. The resulting database contains both self- reported as well as proxy observations for the same subject at different time points. Some statistical models are being investigated that can analyze self-reported and proxy observations together so that relevant parameters and their standard errors can be estimated in a single framework. This is joint work with Mina Hosseini, UMBC, and Dr. Ann Gruber-Baldini, School of Public Health, University of Maryland, Baltimore.

Wiener’s Lemma and Beyond

Abstract: The classical Wiener’s lemma states that if f(x) is a function with an absolutely convergent Fourier series, which nowhere vanishes for real arguments, 1/f(x) has an absolutely convergent Fourier series. In this talk, I will discuss various aspects of Wiener’s lemma from reciprocal of periodic functions to inverse of matrices and to inverse functions, and their applications to spectral invariance of integral operators and signal processing on spatially distributed networks.

A New Procedure to Detect Differentially Expressed Genes from NGS-seq Data

Abstract: Gene expressions profiled using next generation sequencing techniques are highly discretized and have a lot of zeros. This poses difficulties to modeling the NGS gene expression data. Based on a two-component measurement error model, we propose to model the NGS gene expression data using finite mixture models and fit the distributions using an EM-algorithm via data binning. The applications of the proposed methods will be illustrated through a real TCGA lung cancer dataset and benchmarked with existing methods in CRAN R package edgeR.

Stability and Triviality of Plamenevskaya’s Transverse Invariant from Khovanov Homology

Abstract: We prove a stability property of Plamenevskaya’s transverse invariant from Khovanov homology, and use it to give several families of examples of 3-6 braids for which the invariant may be used to detect non quasi-positivity and non right-veeringness. We also construct an infinite family of pretzel knots for which Plamenevskaya’s invariant vanishes for every single transverse link representative from a braid representative. This is joint work with Diana Hubbard.