## Aukosh Jagannath : Simple statistical tasks can be hard on average.

- Colloquium ( 231 Views )Consider the problem of recovering a rank 1 tensor of order k that has been subject to Gaussian noise. We will begin by reviewing results surrounding the statistical limits of maximum likelihood estimation for this problem and discuss an geometric analogue of the well-known BBP phase transition from the matrix setting. We then discuss recent analyses of the behavior of this problem from an optimization perspective. While the threshold for estimation occurs at a finite signal-to-noise ratio, it is expected that one needs a polynomially diverging signal-to-noise ratio to be able to do so efficiently. We present a recent study of the thresholds for efficient recovery for a simple family of algorithms, Langevin dynamics and gradient descent, to better understand the mechanism for this diverging statistical-to-computational gap. I will report on recent works with Ben Arousā??Gheissari on the algorithmic threshold and Lopatto-Miolane on the statistical threshold.

## Jonah Blasiak : Kronecker coefficients for one hook shape

- Colloquium ( 208 Views )The Kronecker coefficient $g_{\lambda \mu \nu}$ is the multiplicity of an irreducible $\mathcal{S}_n$-module $M_\nu$ in the tensor product $M_\lambda \otimes M_\mu$. A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. We give such a formula in the case that one of the partitions is a hook shape. Our main tool is Haiman's mixed insertion, which is a generalization of Schensted insertion to colored words. Prior familiarity with combinatorics of words and tableaux will not be assumed.

## Nicholas Cook : Large deviations for sparse random graphs

- Colloquium ( 202 Views )Let $G=G(N,p)$ be an Erd\H{o}s--R\'enyi graph on $N$ vertices (where each pair is connected by an edge independently with probability $p$). We view $N$ as going to infinity, with $p$ possibly going to zero with $N$. What is the probability that $G$ contains twice as many triangles (triples of vertices with all three pairs connected) as we would expect? I will discuss recent progress on this ``infamous upper tail" problem, and more generally on upper tail estimates for counts of any fixed subgraph. These problems serve as a test bed for the emerging theory of \emph{nonlinear large deviations}, and also connect with issues in extending the theory of \emph{graph limits} to handle sparse graphs. In particular, I will discuss our approach to the upper tail problems via new versions of the classic regularity and counting lemmas from extremal combinatorics, specially tailored to the study of random graphs in the large deviations regime. This talk is based on joint work with Amir Dembo.

## Joseph Teran : A second order virtual node algorithm for Poisson Interface Problems on Irregular Domains

- Colloquium ( 199 Views )I will present a second order accurate, geometrically flexible and easy to implement method for solving the variable coefficient Poisson equation with interfacial discontinuities on an irregular domain. We discretize the equations using an embedded approach on a uniform Cartesian grid employing virtual nodes at interfaces and boundaries. A variational method is used to define numerical stencils near these special virtual nodes and a Lagrange multiplier approach is used to enforce jump conditions and Dirichlet boundary conditions. Our combination of these two aspects yields a symmetric positive definite discretization. In the general case, we obtain the standard 5-point stencil away from the interface. For the specific case of interface problems with continuous coefficients, we present a discontinuity removal technique that admits use of the standard 5-point finite difference stencil everywhere in the domain. Numerical experiments indicate second order accuracy in L-infinity.

## Sarah Koch : Exploring moduli spaces in complex dynamics

- Colloquium ( 197 Views )A major goal in complex dynamics is to understand "dynamical moduli spaces"; that is, conformal conjugacy classes of holomorphic dynamical systems. One of the great successes in this regard is the study of the moduli space of quadratic polynomials; it is isomorphic to $\mathbb C$. This moduli space contains the famous Mandelbrot set, which has been extensively studied over the past 40 years. Understanding other dynamical moduli spaces to the same extent tends to be more challenging as they are often higher-dimensional. In this talk, we will begin with an overview of complex dynamics, focusing on the moduli space of quadratic rational maps, which is isomorphic to $\mathbb C^2$. We will explore this space, finding many interesting objects along the way. Note: special tea at 2:45.

## Ken Ono : CanĀ?t you just feel the Moonshine?

- Colloquium ( 188 Views )Richard Borcherds won the Fields medal in 1998 for his proof of the Monstrous Moonshine Conjecture. Loosely speaking, the conjecture asserts that the representation theory of the Monster, the largest sporadic finite simple group, is dictated by the Fourier expansions of a distinguished set of modular functions. This conjecture arose from astonishing coincidences noticed by finite group theorists and arithmetic geometers in the 1970s. Recently, mathematical physicists have revisited moonshine, and they discovered evidence of undiscovered moonshine which some believe will have applications to string theory and 3d quantum gravity. The speaker and his collaborators have been developing the mathematical facets of this theory, and have proved the conjectures which have been formulated. These results include a proof of the Umbral Moonshine Conjecture, and Moonshine for the first sporadic finite simple group which does not occur as a subgroup or subquotient of the Monster. The most recent Moonshine (announced here) yields unexpected applications to the arithmetic elliptic curves thanks to theorems related to the Birch and Swinnerton-Dyer Conjecture and the Main Conjectures of Iwasawa theory for modular forms. This is joint work with John Duncan, Michael Griffin and Michael Mertens.

## Li-Cheng Tsai : When particle systems meet PDEs

- Colloquium ( 187 Views )Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.

## Jianfeng Lu : Multiscale analysis of solid materials: From electronic structure models to continuum theories

- Colloquium ( 187 Views )Modern material sciences focus on studies on the microscopic scale. This calls for mathematical understanding of electronic structure and atomistic models, and also their connections to continuum theories. In this talk, we will discuss some recent works where we develop and generalize ideas and tools from mathematical analysis of continuum theories to these microscopic models. We will focus on macroscopic limit and microstructure pattern formation of electronic structure models.

## Subhabrata Sen : Random graphs, Optimization, and Spin glasses

- Colloquium ( 184 Views )Combinatorial optimization problems are ubiquitous in diverse mathemati- cal applications. The desire to understand their "typical" behavior motivates a study of these problems on random instances. In spite of a long and rich history, many natural questions in this domain are still intractable to rigorous mathematical analysis. Graph cut problems such as Max-Cut and Min-bisection are canonical examples in this class. On the other hand, physicists study these questions using the non-rigorous "replica" and "cavity" methods, and predict complex, intriguing features. In this talk, I will describe some recent progress in our understanding of their typical properties on random graphs, obtained via connections to the theory of mean-field spin glasses. The new techniques are broadly applicable, and lead to novel algorithmic and statistical consequences.