In this talk we will explore one useful connection between graphs and permutation groups. With a little help from A. Cayley we'll see how many spanning trees a graph with n vertices has. Then we'll use this to find the number of minimal-length factorizations of a permutation into transpositions. The material should be accessible to anyone with a rudimentary knowledge of group theory. Undergraduates welcome!

We'll be talking about recent results on the problem of splitting Brauer classes by torsors for genus one curves. In its geometric form the question to be asked is: does every Severi--Brauer variety contain a smooth and projective genus one curve? Algebraically, this question is related to the existence of certain finite Galois modules inside the linear algebraic automorphism group of the Severi--Brauer variety. Our goal will be to motivate why this is an intuitive and interesting question, giving some new results along the way.

One of the most fundamental unsolved problems in representation theory is to classify the set of irreducible unitary representations of a semisimple Lie group. In this talk, I will define a class of such representations coming from filtered quantizations of certain graded Poisson varieties. The representations I construct are expected to form the ''building blocks'' of all unitary representations.

This is joint work with Srivatsav Kunnawalkam Elayavalli. I will discuss a new viewpoint we developed on the study of II_1 factors involving sequential commutation. This gives us new insights on the elementary equivalence problem, and also reveals a new natural spectral gap type property for II_1 factors strictly strengthening fullness. 

This research presents a framework for evaluating policies in a continuous-time setting, where the dynamics are unknown and represented by Lévy processes. Initially, we estimate the model using available trajectory data, followed by solving the associated PDE to conduct the policy evaluation. Our approach encompasses not only the conventional Brownian motion but also the non-Gaussian and heavy-tailed Lévy processes. We have developed an algorithm that demonstrates enhanced performance compared to existing techniques tailored for Brownian motion. Furthermore, we provide a theoretical guarantee regarding the error in policy evaluation given the model error. Experimental results involving both light-tailed and heavy-tailed data will be presented. This research provides a first step to continuous-time model-based reinforcement learning, particularly in scenarios characterized by irregular, heavy-tailed dynamics.

The famous Caccetta-Häggkvist conjecture states that for any $n$-vertex directed graph $D$, the directed girth of $D$ (the minimum length of a directed cycle in $D$) is at most $\lceil n/k \rceil$, where $k$ is the minimum out-degree of $D$. Aharoni raised a strengthening conjecture: for any $n$-vertex graph $G$ equipped with an edge coloring (not necessarily proper) using $n$ colors, the rainbow girth of $G$ (the minimum length of a cycle in $G$ with distinctly colored edges) is at most $\lceil n/k \rceil$, where $k$ is the minimum size of the color class. We will discuss some results in the non-uniform degrees and rainbow versions of the Caccetta-Häggkvist conjecture.

Based on joint work with Ron Aharoni, Eli Berger, Maria Chudnovsky, and Shira Zerbib.

The exploratory and interactive nature of modern data analysis often introduces selection bias and poses challenges to traditional statistical inference methods. To address selection bias, a common approach is to condition on the selection event. However, this often results in a conditional distribution that is intractable and requires Markov chain Monte Carlo (MCMC) sampling for inference. Notably, some of the most widely used selection algorithms yield selection events that can be characterized as polyhedra, such as the lasso for variable selection and the $\varepsilon$-greedy algorithm for multi-armed bandit problems. This talk will present a method that is tailored for conducting inference following polyhedral selection. The proposed method transforms the variables constrained within a polyhedron into variables within a unit cube, allowing for exact sampling. Compared to MCMC, this method offers superior speed and accuracy. Furthermore, it facilitates the computation of maximum likelihood estimators based on selection-adjusted likelihoods. Numerical results demonstrate the enhanced performance of the proposed method compared to alternative approaches for selective inference.

In this talk we will introduce big mapping class groups and compare them to classical mapping class groups. The goal of the talk is to convince you that big MCGs form an interesting class of Polish groups.

In this work, we investigate applications of no-collision transportation maps introduced by Nurbekyan et al. in 2020 in manifold learning for image data. Recently, there has been a surge in applying transportation-based distances and features for data representing motion-like or deformation-like phenomena. Indeed, comparing intensities at fixed locations often does not reveal the data structure. No-collision maps and distances developed in [Nurbekyan et al., 2020] are sensitive to geometric features similar to optimal transportation (OT) maps but much cheaper to compute due to the absence of optimization. In this work, we prove that no-collision distances provide an isometry between translations (respectively dilations) of a single probability measure and the translation (respectively dilation) vectors equipped with a Euclidean distance. Furthermore, we prove that no-collision transportation maps, as well as OT and linearized OT maps, do not in general provide an isometry for rotations.  The numerical experiments confirm our theoretical findings and show that no-collision distances achieve similar or better performance on several manifold learning tasks compared to other OT and Euclidean-based methods at a fraction of a computational cost.

The first non-round Einstein metrics on spheres were described in 1973 by Jensen in dimensions 4n+3 (n > 0). For the next 25 years it remained an open problem whether the same could be done in even dimensions. This question was settled in 1998 when C. Böhm constructed infinite families of Einstein metrics on all Spheres of dimension between 5 and 9, in particular on $S^6$ and $S^8$.

In the 25 years since then, all spheres of odd dimension (at least 5) have been shown to admit non-round Einstein metrics. However, there have been no new developments in even dimensions above 8, leaving open to speculation the question of whether, if the dimension is even, non-uniqueness of the round metric is a low-dimensional phenomenon or to be expected everywhere.

I will give an overview of the methods used to construct such Einstein metrics, which we recently used to construct the first examples of non-round Einstein metrics on $S^{10}$.

This is joint work with Matthias Wink.

The classical Spezialschar is the subspace of the space of holomorphic modular forms on  $\mathrm{Sp}_4(\mathbb{Z})$ whose Fourier coefficients satisfy a particular system of linear equations. An equivalent characterization of the Spezialschar can be obtained by combining work of Maass, Andrianov, and Zagier, whose work identifies the Spezialschar in terms of a theta-lift from $\widetilde{\mathrm{SL}_2}$. Inspired by work of Gan-Gross-Savin, Weissman and Pollack have developed a theory of modular forms on the split adjoint group of type D_4. In this setting we describe an analogue of the classical Spezialschar, in which Fourier coefficients are used to characterize those modular forms which arise as theta lifts from holomorphic forms on $\mathrm{Sp}_4(\mathbb{Z})$.

It is well-known that isoperimetric hypersurfaces in a smooth, compact (n+1)-manifold are smooth up to a closed set of codimension at least 7. We prove that the dimension estimate of singularities is sharp. In this talk, we will explore an example of an 8-dimensional closed smooth Riemannian manifold, whose unique isoperimetric region, with half the volume of the manifold, displays two isolated singularities on its boundary. Furthermore, for n > 7, we utilize similar methods to construct singular isoperimetric hypersurfaces in higher dimensions.
 

Rank aggregation from pairwise and multiway comparisons has drawn considerable attention in recent years and has a variety of applications, ranging from recommendation systems to sports rankings to social choice. The existing literature on the ranking problem mainly concerns parameter estimation and algorithm implementation. However, there has been little investigation on the statistical inference theory of ranks. In this talk, I will start with a novel inference framework for ranks based on a modified Plackett-Luce model for multiway ranking with only the top choice observed. Then I will present a new methodology to construct simultaneous confidence intervals for the corresponding ranks through a sophisticated maximum pairwise difference statistic based on the MLE. Practically a valid Gaussian multiplier bootstrap procedure is developed to approximate the distribution of the proposed statistic. With the constructed simultaneous confidence intervals, we are able to study various inference problems on ranks such as testing whether an item of interest is among the top-K ranking. Our inference framework for the ranks can be widely applicable in many other ranking problems.  

The main objects of study in algebraic geometry are varieties, which are geometric objects locally defined by polynomial equations, and one goal of the subject is to classify all algebraic varieties of a given type.  We approach this problem by constructing parameter spaces, called moduli spaces, whose points correspond to the geometric objects we aim to parameterize.  Depending on the type of variety, there are several different ways to construct a compact moduli space and in this talk we will survey these different moduli spaces and stability conditions (such as GIT stability, K-stability, KSB/KSBA-stability), discuss their relationships, and give several applications.

I will discuss joint work with Kenny Ascher, Dori Bejleri, Harold Blum, Giovanni Inchiostro, Yuchen Liu, and Xiaowei Wang on construction of moduli stacks and moduli spaces of boundary polarized log Calabi Yau pairs. Unlike moduli of canonically polarized varieties (respectively, Fano varieties) in which the moduli stack of KSB stable (respectively, K semistable) objects is bounded for fixed volume, dimension, the objects here form unbounded families. Despite this unbounded behavior, we define the notion of asymptotically good moduli space, and, in the case of plane curve pairs (P2, C), we construct a projective good moduli space parameterizing S-equivalence classes of such pairs. Time permitting, I will discuss applications to the classification of special degenerations of P2, the b-semiampleness conjecture of Shokurov and Prokhorov, and the Hassett-Keel program.