I'll explain the notion of a p-adic L-function, try to
motivate why one might care about such a gadget, and give some history of their construction and applications. At the end of the talk I'll discuss a recent joint work with John Bergdall in which (among other things) we construct canonical p-adic L-functions associated with modular elliptic curves over totally real number fields.

Title: Uncertainty Quantification of Physics-constrained Problems – Data Assimilation and Parameter Estimation

Date: 01/09/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Yoonsang Lee, Lawrence Berkeley National Laboratory

Observation data along with mathematical models play a crucial role in improving prediction skills in science and engineering. In this talk we focus on the recent development of uncertainty quantification methods, data assimilation and parameter estimation, for Physics-constrained problems that are often described by partial differential equations. We discuss the similarities shared by the two methods and their differences in mathematical and computational points of view and future research topics. As applications, numerical weather prediction for geophysical flows and parameter estimation of kinetic reaction rates in the hydrogen-oxygen combustion are provided.

Title: Quantifying congruences between Eisenstein series and cusp forms

Date: 01/10/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Preston Wake, UCLA

Consider the following two problems in algebraic number theory:
1. For which prime numbers p can we easily show that the Fermat equation x^p + y^p =z^p has no non-trivial integer solutions?
2. Given an elliptic curve E over the rational numbers, what can be said about the group of rational points of finite order on E?
These seem like very different problems, but, surprisingly, they share a common theme: they are both related to the existence of congruences between two types of modular forms, Eisenstein series and cusp forms. We will explain these examples, and discuss a new technique for giving quantitative information about these congruences (for example, counting the number of cusp forms congruent to an Eisenstein series). We will explain how this can give finer arithmetic information than simply knowing the existence of a congruence. This is joint work with Carl Wang-Erickson.

Title: From Hilbert's Nullstellensatz to quotient categories

Date: 01/12/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: John Calabrese, Rice University

A common theme in algebraic geometry is the interplay between algebra and geometry. In this talk I will discuss a few "reconstruction theorems", in which the algebra determines the geometry.

For a filtered topological space, its persistent homology is a multi-set of half open real intervals known as barcode. Each bar represents the lifespan of a homology class. A fundamental principle is that the length of such a bar determines the significance of the corresponding class. In 2011, V. de Silva et al studied the relationships between (persistent) absolute homology, absolute cohomology, relative homology and relative cohomology. This talk will be a theoretical overview of that study.

I will discuss the connection between Sasaki-Einstein metrics and algebraic geometry in the guise of K-stability. In particular, I will give a differential geometric perspective on K-stability which arises from the Sasakian view point, and use K-stability to find infinitely many non-isometric Sasaki-Einstein metrics on the 5-sphere. This is joint work with G. Szekelyhidi.

Dr. Robert Caldwell, MSU's Ombudsperson, will attend this meeting. This will be an opportunity to ask him questions regarding challenging scenarios many of us have encountered during exam proctoring, grading of tests and projects.

Title: Investigating Subtleties of the Multiplication Principle

Date: 01/23/2018

Time: 1:15 PM - 2:45 PM

Place: 252 EH

Speaker: Elise Lockwood, Oregon State University

Central to introductory probability, and a primary feature of most discrete mathematics courses, the Multiplication Principle is fundamental to combinatorics, underpinning many standard formulas and providing justification for counting strategies. Given its importance, the ways it is presented in textbooks are surprisingly varied. In this talk, I identify key elements of the principle and present a categorization of statement types that emerged from a textbook analysis. I also incorporate excerpts from a reinvention study that sheds light on how students reason through key elements of the principle. Findings from both the textbook analysis and the reinvention study reveal surprisingly subtle aspects of the multiplication principle that can be made concrete for students through carefully chosen examples. I conclude with a number of potential mathematical and pedagogical implications of the categorization.

Title: An Introduction to Stanley's Theory of P-Partitions. I

Date: 01/23/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Bruce Sagan, MSU

Richard Stanley developed a powerful generalization of the theory of integer partitions where the parts of the partition are arranged on any labeled poset P. In this first lecture we will develop some intuition by computing the generating functions for various families of ordinary integer partitions. This will motivate Stanley's generalization which will be discussed in the second lecture. No background will be assumed.

Title: Dualities in Persistent (co)Homology-Part II

Date: 01/24/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

Speaker: Hitesh Gakhar, MSU

For a filtered topological space, its persistent homology is a multi-set of half open real intervals known as barcode. Each bar represents the lifespan of a homology class. A fundamental principle is that the length of such a bar determines the significance of the corresponding class. In 2011, V. de Silva et al studied the relationships between (persistent) absolute homology, absolute cohomology, relative homology and relative cohomology. This talk will be a theoretical overview of that study.

Title: Probabilistic scattering for the 4D energy-critical defocusing nonlinear wave equation

Date: 01/25/2018

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

Speaker: Jonas Lührmann

We consider the Cauchy problem for the energy-critical defocusing
nonlinear wave equation in four space dimensions. It is known that for
initial data at energy regularity, the solutions exist globally in time
and scatter to free waves. However, the problem is ill-posed for initial
data at super-critical regularity, i.e. for regularities below the
energy regularity.
In this talk we study the super-critical data regime for this Cauchy
problem from a probabilistic point of view, using a randomization
procedure that is based on a unit-scale decomposition of frequency
space. We will present an almost sure global existence and scattering
result for randomized radially symmetric initial data of super-critical
regularity. This is the first almost sure scattering result for an
energy-critical dispersive or hyperbolic equation for scaling
super-critical initial data.
The main novelties of our proof are the introduction of an approximate
Morawetz estimate to the random data setting and new large deviation
estimates for the free wave evolution of randomized radially symmetric data.
This is joint work with Ben Dodson and Dana Mendelson.

Title: The Extended Bogomolny Equations and Teichmuller space

Date: 01/25/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Speaker: Siqi He, Caltech

We will discuss Witten’s gauge theory approach to Jones polynomial and Khovanov homology by counting solutions to some gauge theory equations with singular boundary conditions. When we reduce these equations to 3-dimensional, we call them the extended Bogomolny equations. We will discuss a Donaldson-Ulenbeck-Yau type correspondence of the moduli space of the singular solutions to the Extended Bogomolny equations and Teichmuller space. If time permits, we will also discuss the relationship of the singular solutions moduli space with higher Teichmuller theory. This is joint work with Rafe Mazzeo.

Title: An invitation to large scale sojourn properties of Brownian motion

Date: 01/25/2018

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

Speaker: Xiaochuan Yang, MSU

For a one dimensional Brownian motion, we consider the sets of times where Brownian motion stays inside some moving boundaries. The boundaries considered are power functions with the power in [0, 1/2]. Since the usual scaling for Brownian motion at time t is square root of t, the sojourn sets we considered describe the recurrence of a Brownian motion around zero. We give large scale geometric properties of these sets using macroscopic dimensions introduced by Barlow and Taylor in the late 80's. The audience of 881/882 might find this talk interesting.

Title: Invariant and characteristic random subgroups and their applications

Date: 01/30/2018

Time: 10:20 AM - 11:10 AM

Place: C304 Wells Hall

Speaker: Rostyslav Kravchenko, Northwestern University

The invariant random subgroups (IRS) were implicitly used by Stuck and Zimmer in 1994 and defined explicitly by Abert, Glasner and Virag in 2012. We recall the definition of IRS and discuss their properties. We also define the notion of characteristic random subgroups (CRS) which are a natural analog of IRSs for the case of the group of all automorphisms. We determine CRS for free abelian groups and for free groups of finite rank. Using our results on CRS of free groups we show that for some groups of geometrical nature there are infinitely many continuous ergodic IRS.

Title: Amenability of discrete groups and their actions

Date: 01/30/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Kate Juschenko, Northwestern University

The subject of amenability essentially begins in 1900's with Lebesgue. He asked whether the properties of his integral are really fundamental and follow from more familiar integral axioms. This led to the study of positive, finitely additive and translation invariant measure on reals as well as on other spaces. In particular the study of isometry-invariant measure led to the Banach-Tarski decomposition theorem in 1924. The class of amenable groups was introduced by von Neumann in 1929, who explained why the paradox appeared only in dimensions greater or equal to three, and does not happen when we would like to decompose the two-dimensional ball. In 1940's, M. Day formally defined a class of elementary amenable groups as the largest class of groups amenability of which was known to von Naumann. He asked whether there are other groups then that. Currently there are many groups that answer von Neumann-Day's question. However, in each particular case it is algebraically difficult to show that the group is not elementary amenable, and analytically difficult to show that it is amenable. The talk is aimed to discuss recent developments and approaches in the field. In particular, it will be shown how to prove amenability of all known non-elementary amenable groups using only one single approach. We will also discuss techniques coming from random walks of groups.

Title: An Introduction to Stanley's Theory of P-Partitions, II

Date: 01/30/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

Speaker: Bruce Sagan, MSU

In this second lecture we will describe how Stanley associated to any labelled poset P a set of partitions having a rational generating function. Its denominator only depends on the number of elements of P and the numerator can be computed using an associated set of permutations and the major index statistic. If one bounds the size of the parts, then the major index is replaced by the number of descents.

Speaker: Arie Israel, University of Texas at Austin

The Brudnyi-Shvartsman finiteness principle is a foundational result in the study of Whitney-type extension problems. This result provides an answer to the following question: How can we tell whether there exists a Hölder-smooth function that takes prescribed values on a given (arbitrary) subset of Euclidean space? In this talk I will describe new machinery for answering this question based on the notion of the “local complexity” of a set at a given position and scale. To complete the main induction argument we must prove that the complexity of an arbitrary set is bounded uniformly by an absolute constant. This is accomplished through an elementary lemma on the stabilization of the dynamics of a 1-parameter family of non-isotropic dilations acting on the space of positive-definite matrices. We conjecture an improvement to the constants in the stabilization lemma which would result in an improvement to the best-known constants in the finiteness principle. This is joint work with A. Frei-Pearson and B. Klartag.

Reaction-diffusion equations describe a variety of physical and biological phenomena. In this talk, I begin by presenting the classical Fisher-KPP equation and its significance to ecology. I then describe recent results on other PDEs of reaction-diffusion type, including non-local equations arising in evolutionary ecology, as well as ones that model tumor growth (joint with Inwon Kim). I will highlight the mathematical challenges and techniques that arise in the analysis of these PDEs.