In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation T, one takes averages of a given integrable function over the intervals {x, T(x), T2(x),..., Tn(x)} in front of the point x. We prove a "backward" ergodic theorem for a countable-to-one pmp T, where the averages are taken over subtrees of the graph of T that are rooted at x and lie behind x (in the direction of T-1). Surprisingly, this theorem yields forward ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank, where the averages are taken along subtrees of the standard Cayley graph rooted at the identity. This strengthens Bufetov's theorem from 2000, which was the most general result in this vein.
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This is joint work with Anush Tserunyan.

Logarithmic Sobolev inequalities (LSI) were first introduced by
Gross in the 70's, and later found rich connections to geometry,
probability, graph theory, optimal transport as well as information
theory. In recent years, logarithmic Sobolev inequalities for quantum
Markov semigroups have attracted a lot of attention and found
applications in quantum information theory and quantum many-body
system. For classical Markov semigroup on a probability space, an
important advantage of log-Sobolev inequalities is the tensorization
property that if two semigroups satisfies LSI, so does their tensor
product semigroup. Nevertheless, tensoraization property fails for LSI
in the quantum cases. In this talk, I'll present some recent progress
on tensor stable log-Sobolev inequalities for quantum Markov
semigroups.
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This talk is based on joint works with Michael Brannan,
Marius Junge, Nicholas LaRacuente, Haojian Li and Cambyse Rouze.

One well-known strategy for distinguishing smooth structures on closed 4-manifolds is to produce a knot $K$ in $S^3$ which is (smoothly) slice in one smooth filling $W$ of $S^3$ but not slice in some homeomorphic smooth filling $W’$. There are many techniques for distinguishing smooth structures on complicated closed 4-manifolds, but this strategy stands out for it’s potential to work for 4-manifolds $W$ with very little algebraic topology. However, this strategy had never actually been used in practice, even for complicated $W$. I’ll discuss joint work with Manolescu and Marengon which gives the first application of this strategy. I’ll also discuss joint work with Manolescu which gives a systematic approach towards using this strategy to produce exotic definite closed 4-manifolds.

Algebraic Geometry is known for its abstract nonsense and its towers of abstraction (one might even say skyscraper sheaves of abstraction). But for this talk, we'll forget about all of that -- we'll explore toric varieties, an extremely concrete way of constructing examples of algebraic varieties. We'll even see examples that are *gasp* not complex manifolds.

In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of Kaehler metrics with good geometric properties. By definition, this class is invariant under biholomorphism. It also includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller space. Analytic problems are also tractable for this class, in particular we show that compactness of the dbar-Neumann operator on (0,q)-forms is equivalent to a growth condition of the Bergman metric. This generalizes an old result of Fu-Straube for convex domains.

Stochastic optimization algorithms have become indispensable in modern machine learning. An important question in this area is the difference between with-replacement sampling and without-replacement sampling -- does the latter have superior convergence rate compared to the former? A paper of Recht and Re reduces the problem to a noncommutative analogue of the arithmetic-geometric mean inequality where n positive numbers are replaced by n positive definite matrices. If this inequality holds for all n, then without-replacement sampling (also known as random reshuffling) indeed outperforms with-replacement sampling in some important optimization problems. In this talk, We will explain basic ideas and techniques in polynomial optimization and the theory of noncommutative Positivstellensatz, which allows us to reduce the conjectured inequality to a semidefinite program and the validity of the conjecture to certain bounds for the optimum values. Finally, we show that Recht--Re conjecture is false as soon as $n = 5$.
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This is a joint work with Lek-Heng Lim.

I will report on my joint work with Samit Dasgupta on the
Brumer-Stark conjecture proving existence of the Brumer-Stark units and
on a conjecture of Dasgupta giving a p-adic analytic formula for these
units. I will present a sketch of our proof of the Brumer-Stark
conjecture and also mention applications to Hilbert's 12th problem, or
explicit class field theory.

Despite breakthrough performance, modern learning models are known to be highly vulnerable to small adversarial perturbations in their inputs. While a wide variety of recent adversarial training methods have been effective at improving robustness to perturbed inputs (robust accuracy), often this benefit is accompanied by a decrease in accuracy on benign inputs (standard accuracy), leading to a tradeoff between often competing objectives. Complicating matters further, recent empirical evidence suggests that a variety of other factors (size and quality of training data, model size, etc.) affect this tradeoff in somewhat surprising ways. In this talk we will provide a precise and comprehensive understanding of the role of adversarial training in the context of linear regression with Gaussian features and binary classification in a mixture model. We precisely characterize the standard/robust accuracy and the corresponding tradeoff achieved by a contemporary mini-max adversarial training approach in a high-dimensional regime where the number of data points and the parameters of the model grow in proportion to each other. Our theory for adversarial training algorithms also facilitates the rigorous study of how a variety of factors (size and quality of training data, model overparametrization etc.) affect the tradeoff between these two competing accuracies.

Virtual counting of coherent sheaves has seen recently a large
development in complex dimension four, where it was defined for
Calabi--Yau fourfolds by Borisov--Joyce and Oh--Thomas. I will focus
on invariants for Hilbert schemes of points as they have not been well
understood before. The only known result expressed integrals of top
Chern classes of tautological vector bundles associated to smooth
divisors in terms of the MacMahon function and Cao--Kool conjectured
this holds for any line bundle. To address these questions I discuss
the conjectural wall-crossing formulae of Joyce and discuss how to
relate them to the conjectures on Hilbert schemes. On the other hand,
Arbesfeld--Johnson--Lim--Oprea--Pandharipande studied Quot-schemes on
surfaces and their virtual integrals giving explicit expressions for
their generating series. Interestingly, these satisfy similar
wall-crossing formulae as Hilbert schemes in the fourfold case when
the curve class is zero. As a consequence their general invariants
share a large similarity. Computing explicitly virtual fundamental
classes and integrals on both, we can firstly recover the results in
the five author paper from a small piece of data. Moreover, we obtain
a universal transformation comparing integrals on Hilbert schemes on
fourfolds and elliptic surfaces.

Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations to other parts of geometric and algebraic combinatorics. These polytopes were recently related to (multiplex) juggling sequences of Butler, Graham and Chung. The Chan-Robbins-Yuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which Mészáros gave an interpretation as the volume of a collection of flow polytopes. In this talk we will talk about the connection between juggling and flow polytopes and introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes.
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The first part of the talk is based on joint work with Benedetti, Hanusa, Harris and Simpson and the second part is based on joint work with William Shi.