In 2010, Dabrowski showed that a von Neumann algebra generated by self-adjoint operators is a factor when they admit a conjugate system. We extend this to the operator-valued case by defining an operator valued partial derivative and conjugate systems with respect to completely positive maps. We show that the center of the von Neumann algebra generated by B and its relative commutant is the center of B.

Given a representation $W$ of a group $G$, polarization is a technique to obtain polynomial invariants for the diagonal action of $G$ on $W^{\oplus r+1}$ from invariants of $W^{\oplus r}$. Weyl's theorem on polarization tells us when one can obtain all polynomial invariants of $W^{\oplus r+1}$ via this process. I will survey some results on polarization in the positive characteristic setting from the last three decades and explain how this can be used to obtain negative answers to some noetherian problems in infinite-dimensional/noncommutative algebra.

Connes Rigidity Conjecture (1980) states that any ICC (infinite conjugacy class) property (T) group is W*-superrigid, meaning the group can be completely recognized from its group von Neumann algebra. The first examples of groups satisfying the conjecture, wreath-like product groups, were constructed in the work of Chifan-Ioana-Osin-Sun (2021). Building on these groups, we investigate the reconstruction of groups with infinite center from their group von Neumann algebras. We introduce the first examples of groups with infinite center whose direct product structure and ICC part are completely recognizable, and the first examples of property (T) W*-superrigid groups with infinite center.  This is based on joint work with Ionuţ Chifan and Adriana Fernández Quero, and upcoming joint work with Ionuţ Chifan, Adriana Fernández Quero and Denis Osin.

This talk will concern the three dimensional half-wave maps equation (HWM), a nonlocal geometric equation arising in the study of integrable spin systems. In high dimensions, n≥4, the equation is known to admit global solutions for suitably small initial data, however the extension of these results to three dimensions presents significant difficulties due to the loss of a key Strichartz estimate. In this talk I will introduce the half-wave maps equation and discuss a global wellposedness result for the three dimensional problem under the assumption that the initial data has angular regularity. The proof combines techniques from the study of wave maps with new microlocal arguments involving commuting vector fields and improved Strichartz estimates.

This work concerns the zeroth-order global minimization of continuous nonconvex functions with a unique global minimizer and possibly multiple local minimizers. We formulate a theoretical framework for inexact proximal point (IPP) methods for global optimization, establishing convergence guarantees under mild assumptions when either deterministic or stochastic estimates of proximal operators are used. The quadratic regularization in the proximal operator and the scaling effect of a positive parameter create a concentrated landscape of an associated Gibbs measure that is practically effective for sampling. The convergence of the expectation under the Gibbs measure is established, providing a theoretical foundation for evaluating proximal operators inexactly using sampling-based methods such as Monte Carlo (MC) integration. In addition, we propose a new approach based on tensor train (TT) approximation. This approach employs a randomized TT cross algorithm to efficiently construct a low-rank TT approximation of a discretized function using a small number of function evaluations, and we provide an error analysis for the TT-based estimation. We then propose two practical IPP algorithms, TT-IPP and MC-IPP. The TT-IPP algorithm leverages TT estimates of the proximal operators, while the MC-IPP algorithm employs MC integration to estimate the proximal operators. Both algorithms are designed to adaptively balance efficiency and accuracy in inexact evaluations of proximal operators. The effectiveness of the two algorithms is demonstrated through experiments on diverse benchmark functions and various applications.

Consider a variety $X$ in the space of matrices, stable under the action of a product of general linear groups by row and column operations. How does its coordinate ring decompose as a direct sum of irreducible representations? We argue that this question is effectively studied by imposing a crystal graph structure on the standard monomials of the defining ideal of $X$ (with respect to some term order). For the standard monomials of "bicrystalline" ideals, we obtain such a crystal structure from the crystal graph on monomials introduced by Danilov–Koshevoi and van Leeuwen. This yields an explicit combinatorial rule we call "filtered RSK" for their irreducible representation multiplicities. In this talk, we will explain our rule and show that Schubert determinantal ideals (among others) are bicrystalline. Based on joint work with Abigail Price and Alexander Yong, https://arxiv.org/abs/2403.09938.

A central problem in cryptanalysis involves computing the set of solutions within a bounded region to systems of modular multivariate polynomials. Typical approaches to this problem involve identifying shift polynomials, or polynomial combinations of input polynomials, with good computational properties. In particular, we care about the size of the support of the shift polynomials, the degree of each monomial in the support, and the magnitude of coefficients. While shift polynomials for systems of a single modular univariate polynomial have been well understood since Coppersmith's original 1996 work, multivariate systems have been more difficult to analyze. Most analyses of shift polynomials only apply to specific problem instances, and it has long been a goal to find a general method for constructing shift polynomials for any system of modular multivariate polynomials. In recent work, we have made progress toward this goal by applying Groebner bases, graph optimization algorithms, and Ehrhart's theory of polytopes to this problem. This talk focuses on these mathematical aspects as they relate to our work, as well as open conjectures about the asymptotic performance of our strategies.

[pre-talk at 3:00PM]

TBA

I will review a general framework for developing numerical methods working with non-parametrically defined surfaces for various problems involving. The main idea is to formulate appropriate extensions of a given problem defined on a surface to ones in the narrow band of the surface in the embedding space. The extensions are arranged so that the solutions to the extended problems are equivalent, in a strong sense, to the surface problems that we set out to solve. Such extension approaches allow us to analyze the well-posedness of the resulting system, develop, systematically and in a unified fashion, numerical schemes for treating a wide range of problems involving differential and integral operators, and deal with similar problems in which only point clouds sampling the surfaces are given.

We prove a homotopy formula which yields almost sharp estimates in all (positive-indexed) Sobolev and Hölder-Zygmund spaces for the $\overline \partial$ equation on finite type domains in $\mathbb C^2$, extending the earlier results of Fefferman-Kohn (1988), Range (1990), and Chang-Nagel-Stein (1992). The main novelty of our proof is the construction of holomorphic support functions that admit precise estimates when the parameter variable lies in a thin shell outside the domain, which generalizes Range's method.

Nonstandard analysis involves the direction manipulation of infinite and infinitesimal quantities to circumvent many uses of epsilons and deltas in analytical arguments. These techniques provide a new perspective on analysis, and they can make rigorous many intuitively appealing arguments that are difficult to formalize with the standard approach. In this talk, we will build the logical foundation for nonstandard analysis and the ever-important transfer principle, and discuss some applications. For instance, we’ll reinterpret the df/dx notation for derivatives as a genuine quotient of infinitesimals.