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Title: TBA Speaker: Steve Wang, Carleton University Canada Time: 3:00pm‐4:00pm Place: CMC 130

TBA

Title: A deformation for the Kadomtsevâ€“Petviashvili (KP) hierarchy Speaker: Baofeng Feng, University of Texas Rio Grande Valley Time: 3:00pm‐4:00pm Place: CMC 130

It is observed that some bilinear equations to soliton equations, such as the Camassa-Holm (CH), modified CH equations and even the nonlinear Schrödinger equation and Sasa-Satsuma equation in deriving dark soliton solutions, cannot be obtained within the framework of the KP theory developed by Kyoto school. In this talk, by introducing one or more nonzero constant in pseudo-differential operators including the dressing operator, we attempt to give a modification of the KP theory. We will give the Lax equation, the Sato equation and the corresponding tau functions. In addition, we will develop a family of bilinear equations which include the onese for the CH and mCH equations.

Title: TBA Speaker: Diego Ricciotti Time: 3:00pm‐4:00pm Place: CMC 130

Title: Object Oriented Data Analysis Speaker: J. S. Marron, Statistics and Operations Research University of North Carolina Time: 3:00pm‐4:00pm Place: CMC 130

The rapid change in computational capabilities has made Big Data a major modern statistical challenge. Less well understood is the rise of Complex Data as a perhaps greater challenge. Object Oriented Data Analysis (OODA) is a framework for addressing this, in particular providing a general approach to the definition, representation, visualization and analysis of Complex Data. The notion of OODA generally guides data analysis, through providing a useful terminology for interdisciplinary discussion of the many choices typically needed in modern complex data analyses. The main ideas are illustrated via a survey of a number of approaches which integrate differential geometry and Bayesian statistics, yielding powerful image segmentations.

Due to COVID-19, this presentation has been CANCELLED.

Title: Gauges, Loops, and Polynomials for Partition Functions of Graphical Models Speaker: Michael Chertkov, University of Arizona (Tucson) Time: 3:00pm‐4:00pm Place: CMC 130

Graphical models (GM) represent multivariate and generally not normalized probability distributions. Computing the normalization factor, called the partition function (PF), is the main inference challenge relevant to many statistical and optimization applications. The problem has exponential complexity with respect to the number of variables. In this talk, aimed at approximating the PF, we consider Multi-Graph Models (MGMs) where binary variables and multivariable factors are associated with edges and nodes, respectively, of an undirected multi-graph. We suggest a new methodology for analysis and computations that combines the Gauge Function (GF) technique with techniques from the field of real stable polynomials. We show that the GF, representing a single-out term in a finite sum expression for the PF which achieves extremum at the so-called Belief-Propagation (BP) gauge, has a natural polynomial representation in terms of gauges/variables associated with edges of the multi-graph. Moreover, GF can be used to recover the PF through a sequence of transformations with algebraic and graphical interpretations. Even though the complexity of computing factors in the sequence of derived MGMs and respective GFs is exponential in the number of eliminated edges, polynomials associated with the new factors remain bi-stable if the original factors have this property.

Title: Geometric methods in representation theory Speaker: Mohammed Bouhada, Sherbrooke University, Canada Time: 3:00pm‐4:00pm Place: CMC 130

This talk will introduce the basics of quiver theory and its representations. In general, the goal of representation theory of algebras is to understand the category of modules over a given associative unital \(k\)-algebra \(A\), where \(k\) is a commutative ring.

In this talk, we will discuss the case of finite dimensional algebras, and we will focus on the geometrical aspects of representations of quivers.

Title: What is Yang-Baxter homology and how do you compute it? Speaker: Jozef H. Przytycki, George Washington University Time: 3:00pm‐4:00pm Place: CMC 130

I will describe the path from Fox 3-colorings of links to Yang-Baxter operator with many stops on the way (like quandles). I will show how quandle homology can be generalized to Yang-Baxter homology. Finally, I will demonstrate how to compute the second Yang-Baxter homology for the Yang Baxter operators giving Jones and HOMFLYPT polynomials. I will speculate on relations with Khovanov homology.

Title: Curves over finite fields and polynomial problems Speaker: Daniele Bartoli, University of Perugia Italy Time: 3:00pm‐4:00pm Place: CMC 130

Algebraic curves over finite fields are not only interesting objects from a theoretical point of view, but they also have deep connections with different areas of mathematics and combinatorics. In fact, they are important tools when dealing with, for instance, permutation polynomials, APN functions, planar functions, exceptional polynomials, scattered polynomials. In this talk I will present some applications of algebraic curves to the above mentioned objects.

Title: Random Unitary Matrices and Matrix Models for the Circular Beta Ensemble Speaker: Mihai Stoiciu, Williams College Time: 4:00pm‐5:00pm Place: CMC 130

We consider various families of random unitary band matrices (CMV and Joye) and study their spectral properties. In particular, we investigate the distribution of the eigenvalues of these matrices and use them to construct matrix models for the circular beta ensemble. We also study spectral properties of these matrix models for beta approaching zero and infinity.

Title: Integrability and limit cycles in polynomial systems of ODEs Speaker: Valery Romanovsky, University of Maribor Slovenia Time: 3:00pm‐4:00pm Place: CMC 130

We discuss two problems related to the theory of polynomial plane differential systems, that is, systems of the form \[ \tag{1} \frac{dx}{dt}=P_{n}(x,y), \ \ \ \frac{dy}{dt}=Q_{n}(x,y), \] where \(P_{n}(x,y), Q_{n}(x,y)\) are polynomials of degree \(n\), \(x\) and \(y\) are real unknown functions.

The first one is the problem of local integrability, that is, the problem of finding local analytic integrals in a neighborhood of singular points of system (1). We present a computational approach to find integrable systems within given parametric families of systems and describe some mechanisms of integrability.

The second problem is called the cyclicity problem, or the local 16th Hilbert problem, and is related to the estimation of the number of limit cycles arising in system (1) after perturbations of integrable systems. The approach is algorithmic and is based on algorithms of computational commutative algebra relying on the Groebner bases theory.

Title: Images of arboreal Galois representations Speaker: Andrea Ferraguti, ICMAT Time: 3:00pm‐4:00pm Place: CMC 130

Arboreal Galois representations are a central topic in modern arithmetic dynamics. In this talk, we will briefly review their construction, and then we will focus on arboreal representations attached to quadratic polynomials. First, we will explain a new simple but extremely powerful lemma that relates the dynamic of the polynomial to the image of the associated arboreal representation. Next, we will give an overview of several applications of this lemma: how to recover known results with one-line proofs, how to construct representations with maximal image, and how to prove that many subgroups of automorphisms of the infinite binary tree cannot arise as images of arboreal representations over global fields. This is joint work with Carlo Pagano and Daniele Casazza.

Title: Approximation theory in several complex variables Speaker: Myrto Manolaki, University College Dublin Time: 2:00pm‐3:00pm Place: CMC 108

In one complex variable, approximation theory is well developed. For example, the celebrated theorem of Mergelyan states that if \(K\) is a compact subset of the complex plane with connected complement, then every continuous function on \(K\) which is holomorphic on its interior can be uniformly approximated on \(K\) by polynomials. In this talk, we will discuss what happens in several complex variables, where the situation is far from being understood. In particular, I will present some new Mergelyan-type theorems for products of planar compact sets and graphs of functions. (Joint work with J. Falco, P. Gauthier and V. Nestoridis.