I will briefly review the convergence theory for non-collapsed Einstein 4-manifolds developed by Anderson, Bando-Kasue-Nakajima and Tian around 1990. This was the main precursor for the more recent higher-dimensional theory of Cheeger-Colding-Naber. However, several difficult problems have remained open even in dimension 4. I will focus on the structure of the possible bubbles and bubble trees in the 4-dimensional theory. In particular, I will explain Kronheimer's classical work on gravitational instantons as well as a recent result of Biquard-H concerning the renormalized volume of a 4-dimensional Ricci-flat ALE space.

The f-invariant was introduced by Lewis Bowen in 2008 and is a real-valued isomorphism invariant that is defined for a large class of probability measure-preserving actions of finite-rank free groups. Most notably, the f-invariant provided the first classification up to isomorphism of Bernoulli shifts over finite-rank free groups. It is also quite useful for the study of finite state Markov chains with values indexed by a finite-rank free group. The f-invariant is conceptually similar to entropy, and it has a formal connection to sofic entropy. In this expository talk, I will introduce the f-invariant and discuss some of its basic properties.

In free probability theory, there is no direct analog of density for probability distributions, but there is something like a notion of ``log-density'' in the study of free Gibbs laws and free score functions. Non-commutative notions of smoothness are important for studying both these log-densities and the functions used for changes of variables (or transport of measure) in free probability. In the single-variable setting, we have a good understanding of the smoothness properties of a function $f: \mathbb{R} \to \mathbb{R}$ applied to self-adjoint operators thanks to the work of Peller, Aleksandrov, and Nazarov; see the recent paper of Evangelos Nikitopoulos. However, in the multivariable setting, much of the literature has used classes of functions that are either too restrictive (such as analytic functions) or very technical to define (such as Dabrowski, Guionnet, and Shlyakhtenko's smooth functions where Haagerup tensor norms were used for the derivatives). We will discuss a notion of tracial non-commutative smooth functions that is modeled on trace polynomials. These smooth functions have many desirable properties, such as a chain rule, good behavior under conditional expectations, and a natural way to incorporate the one-variable functional calculus. We will sketch current work about the smooth transport of measure for free Gibbs laws as well as future directions in relating these smooth functions to free SDE.

Detecting differences and building classifiers between distributions, given only finite samples, is an important task in data science applications. Optimal transport (OT) has evolved as the most natural concept to measure the distance between distributions, and has gained significant importance in machine learning in recent years. There are some drawbacks to OT: Computing OT is usually slow, and it often fails to exploit reduced complexity in case the family of distributions is generated by simple group actions.
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In this talk, we discuss how optimal transport embeddings can be used to deal with these issues, both on a theoretical and a computational level. In particular, we'll show how to embed the space of distributions into an $L^2$-space via OT, and how linear techniques can be used to classify families of distributions generated by simple group actions in any dimension. We'll also show conditions under which the $L^2$ distance in the embedding space between two distributions in arbitrary dimension is nearly isometric to Wasserstein-2 distance between those distributions. This is of significant computational benefit, as one must only solve N optimal transport problems to define the $N^2$ pairwise distances between N distributions. We'll present some applications in image classification and supervised learning.
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This is joint work with Alex Cloninger.

In this talk, I will introduce the Chow t-structure on the motivic stable homotopy category over a general base field. This t-structure is a generalization of the Chow-Novikov t-structure defined on a p-completed cellular motivic module category in work of Gheorghe--Wang--Xu.
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Moreover, we identify the heart of this t-structure with a purely algebraic category, and expand the results of Gheorghe-Wang-Xu to integral results on the entire motivic category over general base fields. This leads to computational applications on determining the Adams spectral sequences in the classical stable homotopy category, as well as that in the motivic stable homotopy category over C, R, and $F_p$. This is joint work with Tom Bachmann, Guozhen Wang and Zhouli Xu.

Once upon a time, in the land of Mathlandia, the people in the city of Algebraic Geometria had a brilliant vision: what if all math was algebra, every theorem was category theory, and every proof used cohomology? Armed with their categories and rings, they declared war on the other cities of Mathlandia. Number Theoryberg was the first to fall, succumbing to the glory of sheaves and schemes. Next was Representation Theory City, which fell soon after. But alas, despite their efforts the algebraists had not made any progress since... until now! Scholze and Clausen have started the Condensed Mathematics program, which vows to take over Functional Analysisberg and Differential Geometria. We will deal with the very first matter in this program, the story of locally compact abelian groups.

Hurwitz spaces parametrize finite covers of the projective line by curves of genus g. We show that the Chow rings of Hurwitz spaces of low degree covers exhibit interesting stabilization properties as g goes to infinity. We also consider applications to the moduli space of curves. This is joint work with Hannah Larson.

Often in Number Theory and Geometry, it is necessary to consider topological algebraic objects, like locally compact abelian groups, continuous group cohomology, and adic spaces. However, many of these topological structures don't have good categorical properties, at least when compared to their purely algebraic counterparts. We will explore Condensed Mathematics, a program of Clausen and Scholze to systematically add topological structure to algebra with nice categorical properties.

We will discuss classification of ancient solutions to geometric flows. We will focus especially on the Ricci flow.

Machine learning achieves tremendous success in image classification, speech recognition, and medical diagnosis. In the meantime, machine learning also brings some interesting mathematical questions: How to solve the resulting optimization more efficiently and robustly? How to apply the machine learning technique to tackle challenging problems in mathematics. In this talk, I will view the neural networks model from a nonlinear scientific computing point of view and present some recent work on developing a homotopy training algorithm to train neural networks ``layer-by-layer'' and ``node-by-node.'' I will also review the neural network from numerical algebraic geometry point view and provide a novel initialization for ReLU networks.

The classical convergence theory and singularity formation for non-collapsed Einstein 4-manifolds leaves some important issues open. Which singular Einstein metrics are actually limits of smooth ones? Can we describe the moduli space of Einstein metrics close to its boundary? We partially answer these by proving that any Einstein manifold sufficiently GH-close to an Einstein orbifold is the result of a gluing-perturbation procedure. This completely describes the singularity formation of compact Einstein metrics and moreover lets us show that the desingularization of Einstein metrics by Kronheimer's gravitational instantons is obstructed. The recent results of Biquard-Hein let us extend a weaker obstruction for general Ricci-flat ALE spaces.

The 2-spin spherical Sherrington Kirkpatrick (SSK) model was introduced by Kosterlitz, Thouless and Jones as a simplification of the usual Sherrington-Kirkpatrick model with Ising spins. The SSK model admits tractable formulas for many of its observables, allowing for a detailed analysis of its fluctuations using techniques from random matrix theory. We discuss recent results on the fluctuations of the SSK model with a magnetic field as well as at critical temperature. Based on joint work with P. Sosoe.

The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal and image processing. However, in many modern applications, the signal bandwidths have increased tremendously, while the acquisition capabilities have not scaled sufficiently fast. Consequently, conversion to digital has become a serious bottleneck. Furthermore, the resulting digital data requires storage, communication and processing at very high rates which is computationally expensive and requires large amounts of power. In the context of medical imaging sampling at high rates often translates to high radiation dosages, increased scanning times, bulky medical devices, and limited resolution.
In this talk, we present a framework for sampling and processing a large class of wideband analog signals at rates far below Nyquist in space, time and frequency, which allows to dramatically reduce the number of antennas, sampling rates and band occupancy. Our framework relies on exploiting signal structure and the processing task. We consider applications of these concepts to a variety of problems in communications, radar and ultrasound imaging and show several demos of real-time sub-Nyquist prototypes including a wireless ultrasound probe, sub-Nyquist MIMO radar, super-resolution in microscopy and ultrasound, cognitive radio, and joint radar and communication systems. We then discuss how the ideas of exploiting the task, structure and model can be used to develop interpretable model-based deep learning methods that can adapt to existing structure and are trained from small amounts of data. These networks achieve a more favorable trade-off between increase in parameters and data and improvement in performance, while remaining interpretable.

In this talk, we study the Chow group of the motive associated
to a tempered global L-packet $\pi$ of unitary groups of even rank with
respect to a CM extension, whose global root number is -1. We show that,
under some restrictions on the ramification of $\pi$, if the central
derivative $L'(1/2,\pi)$ is nonvanishing, then the $\pi$-nearly isotypic
localization of the Chow group of a certain unitary Shimura variety over
its reflex field does not vanish. This proves part of the
Beilinson--Bloch conjecture for Chow groups and L-functions. Moreover,
assuming the modularity of Kudla's generating functions of special
cycles, we explicitly construct elements in a certain $\pi$-nearly
isotypic subspace of the Chow group by arithmetic theta lifting, and
compute their heights in terms of the central derivative $L'(1/2,\pi)$ and
local doubling zeta integrals. This is a joint work with Chao Li.

Kuznetsov conjectured the existence of a correspondence
between different types of Fano threefolds which identifies a
distinguished semiorthogonal component of the derived category on each
side. I will explain joint work with Arend Bayer which resolves one of
the outstanding cases of this conjecture. This relies on the study of
the Hodge theory of certain K3 categories associated to the
semiorthogonal components.

Arithmetic statistics is a subfield of number theory which studies arithmetic objects in the context of statistical properties like distribution, expected value, etc. Work in this field involves both an understanding of the underlying arithmetic structure and the application of strong analytic tools. We will first explore a few types of questions that are studied in this field today, then we discuss a major analytic tool used by researchers called a Tauberian theorem.