Isotropic Quot scheme is a projective scheme that compactifies the moduli space of maps (of a given degree) from a smooth projective algebraic curve to an Isotropic Grassmannian. Isotropic quot schemes are often not smooth, but we construct a virtual fundamental class which makes it 'virtually' smooth. We use this to understand intersection theory of this space and find a Vafa-Intriligator type formula for the intersection of certain cohomology classes.

By Ratner's famous equidistribution theorem, we know that unipotent orbits in finite volume quotients of Lie groups equidistribute in their closures. Often, in applications, one needs to know more: specifically, at what rate does the orbit equidistribute? We call a statement that includes a quantitative error term effective. In this talk, I will present an effective equidistribution theorem, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is joint work with Nattalie Tamam.

In a series of papers, Voiculescu generalized the notions of entropy and Fisher information to the free probability setting. In particular, the notions of free entropy have several applications in the theory of von Neumann algebras and free probability such as demonstrating certain von Neumann algebras do not have property Gamma, demonstrating the absence of atoms in the distributions of polynomials of random matrices, and the construction of free monotone transport. With the recent bi-free extension of free probability being sufficiently developed, it is natural to ask whether there are bi-free extensions of Voiculescu's notions of free entropy. In this talk, we will provide an introduction to a few notions of bi-free entropy and discuss the difficulties and peculiarities that occur. This is joint work with Ian Charlesworth.

This talk discusses the saddle point problem of polynomials. We give an algorithm for computing saddle points, based on Lasserre's hierarchy of Moment-SOS relaxations. Under some genericity assumptions, we show that: i) if there exists a saddle point, the algorithm can get one by solving a finite number of relaxations; ii) if there is no saddle point, the algorithm can detect its nonexistence.

In 2010, Hedden showed that the Ozsvath-Szabo concordance invariant tau detects whether a fibered knot induces a tight contact structure on the three-sphere. In 2017, Ozsvath, Stipsicz, and Szabo constructed a one-parameter family of concordance invariants Upsilon, which recovers tau as a special case. I will discuss a sufficient condition using Upsilon for the monodromy of the open book decomposition of a fibered knot to be right-veering. I will also discuss a generalization of a result of Baker on ribbon concordances between fibered knots. This is joint work with Dongtai He and Diana Hubbard.

A mean field game is usually denoted as a game with the number of
players tending to infinity, which leads us to consider population
densities of players and their resulting ``Nash equilibrium'' strategies.
Mean field games have been used to model financial markets, population
dynamics, and more, but solving the mean field games ends up requiring us
to solve a high-dimensional PDE. To avoid the curse of dimensionality, we
can instead use the method of characteristics to turn one of the PDEs in
the mean field game into a system of ODEs and then solve the mean field
game by parameterizing the dynamics of the ODE with a neural network.

Theta functions are certain quasi-periodic functions in several complex variables. ``Non-abelian'' analogues of theta functions can be defined via moduli of stable bundles over a curve. There exist surprising symmetries in the dimension formulas, called Verlinde numbers, of spaces of non-abelian theta functions. We will talk about the strange duality which is a geometric explanation for the symmetry between spaces of non-abelian theta functions.

The fractional obstacle problem in $\mathbb R^n$ with obstacle $\varphi\in C^\infty(\mathbb{R}^n)$ can be written as \[ \min\{(-\Delta)^s u , u-\varphi\} = 0,\quad\textrm{in }\quad\mathbb{R}^n. \] The set $\{u = \varphi\} \subset \mathbb{R}^n$ is called the contact set, and its boundary is the free boundary, an unknown of the problem. The free boundary for the fractional obstacle problem can be divided between two subsets: regular points (around which the free boundary is smooth, and is $n-1$ dimensional) and degenerate points. The set of degenerate points, even for smooth obstacles, can be very large (for example, with infinite $\mathcal{H}^{n-1}$ measure). In a joint work with X. Ros-Oton we show, however, that generically solutions to the fractional obstacle problem have a lower dimensional degenerate set. That is, for almost every solution (in an appropriate sense), the set of degenerate points is lower dimensional.

I will present MCMC algorithms as optimization over the KL-divergence in the space of probabilities. By incorporating a momentum variable, I will discuss an algorithm which performs accelerated gradient descent over the KL-divergence. Using optimization-like ideas, a suitable Lyapunov function is constructed to prove that an accelerated convergence rate is obtained. I will then discuss how MCMC algorithms compare against variational inference methods in parameterizing the gradient flows in the space of probabilities and how it applies to sequential decision making problems.

Right and left eigenvectors of non-Hermitian matrices form a bi-orthogonal system to which one can associate homogeneous quantities known as overlaps. The matrix of overlaps quantifies the stability of the spectrum and characterizes the joint eigenvalues increments under Dyson-type dynamics. Overlaps first appeared in the physics literature: Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results have been obtained in other integrable settings, namely quaternionic Gaussian matrices, as well as matrices from the spherical and truncated unitary ensembles.

Outside of laboratories, microbial communities (biofilms and other types)
often exist in relatively stable environments where, on average, resource quality and quantity
are predictable. In these conditions, these communities are able to organize
into tuned chemical factories, efficiently turning resources into biomass and
waste byproducts. To do so, physical, chemical, and biological constraints
must be accomodated. In this seminar, techniques to model this
organization will be discussed. In particular, the importance of coupling microscale
metabolic information to community scale transport processes will be emphasized.

Modeling Malle's Conjecture with Random Groups
Abstract: We construct a random group with a local structure that models
the behavior of the absolute Galois group ${\rm Gal}(\overline{K}/K)$,
and prove that this random group satisfies Malle's conjecture for
counting number fields ordered by discriminant with probability 1. This
work is motivated by the use of random groups to model class group
statistics in families of number fields (and generalizations). We take
care to address the known counter-examples to Malle's conjecture and how
these may be incorporated into the random group.

One approach to computing integrals over Hilbert schemes of
points on surfaces (and other moduli spaces of sheaves on surfaces) is
to reduce to the special case when the surface in question is $C^2$.
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I'll explain how to use the (K-theoretic) Donaldson-Thomas theory of
threefolds to deduce identities for holomorphic Euler characteristics
of tautological bundles over the Hilbert scheme of points on $C^2$. I'll
also explain how these identities control the behavior of such Euler
characteristics over Hilbert schemes of points on general surfaces.

It is known both scientifically and anecdotally that typical large-scale
social networks exhibit short paths between pairs of nodes, hence
the common phrase ``six degrees of separation''. In this talk, mathematical
background for this phenomenon is given, together with a study of the
algorithmic question of how to search such large-scale networks using only
local information. In particular, one could imagine that each
person in the network is allowed to pass a message to one of their
acquaintances, until a particular target person in the network receives the
message. Remarkably, for many networks with $n$ nodes, there is a simple
algorithm which does this extremely efficiently -- a ``decentralized
search'' algorithm which runs in time polylogarithmic in the number of
nodes. The mathematics in this talk involves only elementary combinatorics
and probability.