Buchberger introduced in 1965 the concept of a Groebner basis for a polynomial ideal over a field and gave an algorithm to compute it. Since the 1980s, this concept has been extensively studied and generalized; it has found many applications in diverse areas of mathematics and computer science. The talk will integrate the concepts of a universal Groebner basis which serves as a Groebner basis for all admissible term orderings with a parametric (more popularly called comprehensive) Groebner basis which serves as a Groebner basis for all possible specializations of parameters. This integration defines a mega Groebner basis that works for every admissible ordering as well as for any specialization of parameters. Algorithms for constructing comprehensive Groebner bases, their canonicity, and generalization to universal comprensive Groebner bases will be presented.

In this talk we discuss qualitative behavior of certain solutions to linear and nonlinear dispersive partial differential equations such as Schrodinger and Korteweg-de Vries equations. In particular, we will present results on the fractal dimension of the solution graph and the dependence of solution profile on the algebraic properties of time.

The main problem in phylogenetics is to reconstruct evolutionary relationships between collections of species, typically represented by a phylogenetic tree. In the statistical approach to phylogenetics, a probabilistic model of mutation is used to reconstruct the tree that best explains the data (the data consisting of DNA sequences from homologous genes of the extant species). In algebraic statistics, we interpret these statistical models of evolution as geometric objects in a high-dimensional probability simplex. This connection arises because the functions that parametrize these models are polynomials, and hence we can consider statistical models as algebraic varieties. The goal of the talk is to introduce this connection and explain how the algebraic perspective leads to new theoretical advances in phylogenetics, and also provides new research directions in algebraic geometry. The talk material will be kept at an introductory level, with background on phylogenetics and algebraic geometry.

Bio: Seth Sullivant received his PhD in 2005 from the University of California, Berkeley. After a Junior Fellowship in Harvard's Society of Fellows, he joined the department of mathematics at North Carolina State University in 2008 as an assistant professor. He was promoted to full professor in 2014 and distinguished professor in 2018. Sullivant's work has been honored with a Packard Foundation Fellowship and an NSF CAREER award and he was selected as a Fellow of the American Mathematical Society. He helped to found the SIAM activity group in Algebraic Geometry where he has served as both secretary and chair. Sullivant's current research interests include algebraic statistics, mathematical phylogenetics, applied algebraic geometry, and combinatorics.

The Local Langlands program and its variants have lead to the study of smooth, admissible representations of p-adic algebraic groups. The degree to which these are understood depends on the field over which the representations are being taken. Over a field of characteristic p, the usual dual in the category of smooth representations gives less information: in most cases of interest, it is 0! Kohlhaase has defined candidates-the higher duality functors-for a useful replacement. We will go over their properties, and some examples in rank one where they can be computed.

Biochemical oscillation is one of the most important way in living systems to track the information of time, or to communicate with population members. A good clock needs to be function accurately in the presence of noise and at the same time respond sensitively to external signals. Low fluctuation and high sensitivity are incompatible in equilibrium systems due to the fluctuation-dissipation theorem (FDT). In biology, biochemical oscillators are fueled by dissipative processes such as ATP hydrolysis, which is inherently nonequilibrium and the FDT is broken. In our recent work, we show that for a single oscillator, the lower bound of oscillation phase fluctuation, and the upper bound of phase sensitivity are determined by the free energy dissipation. Real biological clocks are composed of multiple oscillators and synchronization is necessary to drive their collective dynamics. Inspired by the cyanobacterial circadian clock, we proposed a model of coupled oscillators. We find that synchronization of oscillators cost free energy even though the coupling is conservative. By analytical solving the model, we show that the many-body system goes through a nonequilibrium phase transition driven by energy dissipation.

A fundamental tool in graph theory is Szemeredi's regularity lemma which asserts that any dense graph can be partitioned into finitely many parts so that almost all edges are contained in the union of bipartite subgraphs between pairs of the parts and these bipartite subgraphs are random-like under the notion of $\epsilon$-regular.

\medskip

Here, we consider a variation of the regularity lemma for graphs with a nontrivial
clustering coefficient. The clustering coefficient is the ratio of the number triangles and the number of paths of length $2$ in a graph. Note many real-world graphs have large clustering coefficients and such clustering effect is one of the main characteristics of the so-called ``small world phenomenon''.

\medskip

In this talk, We give a regularity lemma for clustering graphs without any restriction on edge density. We also discuss several generalizations of the regularity lemma and mention some related problems.

Given a collection of input rankings provided as permutations $\pi_i : [n] \rightarrow [n]$ for $1 \leq i \leq m$, the rank aggregation problem seeks to find another permutation $\sigma: [n] \rightarrow [n]$ that minimizes $\sum_{i=1}^m K(\sigma, \pi_i)$ where $K$ is the Kendall distance between the two permutations. In this talk, we will discuss motivation for the problem and some existing Markov chain based algorithms along with an investigation of their performance guarantees. Necessary background information will also be provided.