This talk will be about a pair of related matrix integrals, the Harish-Chandra/Itzykson-Zuber integral and the Brezin-Gross-Witten integral, which play an important role in random matrix theory, representation theory, and mathematical physics. While these integrals cannot be exactly evaluated, an old conjecture says that they admit asymptotic expansions whose coefficients are themselves generating functions for some unspecified combinatorial invariants of compact Riemann surfaces (or smooth projective curves).

When $G$ is a Polish group, one way of knowing that it has nice
dynamics is to show that $M(G)$, the universal minimal flow of $G$, is
metrizable. For non-Polish groups, this is not the relevant dividing
line: the universal minimal flow of the symmetric group of a set of
cardinality $\kappa$ is the space of linear orders on $\kappa$---not
a metrizable space, but still nice---, for example.
In this talk, we present a set of equivalent properties of topological
groups which characterize having nice dynamics. We show that the class
of groups satisfying such properties is closed under some topological
operations and use this to compute the universal minimal flows of some
concrete groups, like $\mathrm{Homeo}(\omega_{1})$.
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This is joint work with Andy Zucker.

We investigate some structural properties of C*-algebras generated by quasi-regular representations of stabilizers of boundary actions of discrete groups G. Our main tool is the notion of (noncommutative) boundary maps, namely G-equivariant unital positive maps from G-C*algebras to C(B), where B is the Furstenberg boundary of G. We completely describe the tracial structure and characterize the simplicity of these C*-algebras. As an application, we show that the C*-algebra generated by the quasi-regular representation associated to Thompson's groups $F < T$ does not admit traces and is simple.
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This is joint work with Eduardo Scarparo.

Levenberg-Marquardt (LM) algorithms are a class of methods that add a regularization term to a Gauss-Newton method to promote better convergence properties. This talk presents three works on this class of methods. The first discusses a new method that simultaneously achieves all types of state of the art convergence guarantees for unconstrained problems. Stochastic LM is discussed next, which is an algorithm to handle noisy data. An example is presented on data assimilation. Finally, a LM method is presented to handle equality constraints, with examples from inverse problems in PDEs.

The fundamental group is one of the most powerful invariants to distinguish closed three-manifolds. One measure of the non-triviality of a three-manifold is the existence of non-trivial SU(2)-representations. In this talk I will show that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)-representations.
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This is joint work with Tye Lidman and Raphael Zentner.

Real Grassmannians' uses in geometry are manifold, but in general, their integral (co)homology groups were unknown. Until now. I won't Stiefel myself any longer, and I will (co)change your views on this class of manifolds. At the end of this talk, you should be able to differentiate Grassmannian manifolds and feel right at (co)home with them, K? This talk will Blow Up your mind, Link some ideas you may not have heard of, and you'll take an Exit-Path out with your mind (po)set straight.

In this talk, we consider Riemannian manifolds with almost non-negative scalar curvature and Perelman entropy. We establish an $\epsilon$-regularity theorem showing that such a space must be close to Euclidean space in a suitable sense. Interestingly, such a result is false with respect to the Gromov-Hausdorff and Intrinsic Flat distances, and more generally the metric space structure is not controlled under entropy and scalar lower bounds. Instead, we introduce the notion of the $d_p$ distance between (in particular) Riemannian manifolds, which measures the distance between $W^{1,p}$ Sobolev spaces, and it is with respect to this distance that the $\epsilon$ regularity theorem holds. We will discuss various applications to manifolds with scalar curvature and entropy lower bounds, including a compactness and limit structure theorem for sequences, a uniform $L^\infty$ Sobolev embedding, and a priori $L^p$ scalar curvature bounds for $p<1$.
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This is joint work with Man-Chun Lee and Aaron Naber.

We study the benign overfitting phenomenon induced by simple optimization algorithms in deep learning.
Oftentimes the neural network is over-parameterized in the sense that the number of parameters exceeds the
training data size, but the obtained solution generalizes well to unseen data. The generalization stems from
an implicit regularization of the optimization algorithm. We present the recent theoretical development of
over-parameterization for linear/non-linear models, together with some numerical experiments.

We briefly overview the current developments of rigorous constructions in "stochastic quantization” - an active field linking quantum field theory with stochastic PDE.
We then focus on stochastic quantization of the Yang-Mills model in 2 and 3 space dimensions.
This includes constructing the Langevin dynamic for the formal Yang-Mills measure, defining the state space of gauge orbits, proving gauge equivariance of the dynamic, and making sense of Wilson loop observables in this context. We will also discuss some future directions.
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The talk is based on several works mostly joint with A.Chandra, I.Chevyrev, and M.Hairer.

Sadly, I won't have time to prove the Riemann hypothesis in
this talk. However, I do hope to outline recent work in a record
partial-verification of RH.
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This is joint work with Dave Platt, in
Bristol, UK.

Epistasis is a situation when the effect of one mutation changes as other mutations are introduced into the genome. Epistasis is used in genetics to probe functional relationships between genes, and it also plays an important role in evolution. However, there is no theory to understand how functional relationships at the molecular level translate into epistasis at the level of whole-organism phenotypes, such as fitness. I will present a simple model of a hierarchical metabolic network with first-order kinetics which helps us gain some intuition in this problem. I will derive two rules for how epistasis between mutations with small effects propagates from lower- to higher-level phenotypes and how such epistasis depends on the topologyof the network. Most importantly, weak epistasis at a lower level may be distorted as it propagates to higher levels. These results suggest that pairwise inter-gene epistasis should be common and it should generically depend on the genetic background and environment. Furthermore, the epistasis coefficients measured for high-level phenotypes may not be sufficient to fully infer the underlying functional relationships.

The effectiveness of a graduate school admissions ultimately
rests on the quality of the information collected and the decision
making process that is used to arrive at a decision. Admissions relies
on faculty judgment combining a variety of tools including grades, test
scores, letters of recommendation, interviews, and student essays to
identify the best candidates. Unfortunately, current practice often
falls short of well established best practices leading to lower quality
decisions and the possible introduction of bias. In this talk, I will
make the case that improvement is urgently needed and then lay out both
a short and long term place for modernizing graduate school admissions.

I will introduce and discuss an invariant of
hypersurface singularities, Saito's minimal exponent (a.k.a. Arnold
exponent in the case of isolated singularities). This can be considered
as a refinement of the log canonical threshold, which is interesting in
the case of rational singularities. I will focus on recent work on this
invariant and remaining open problems.