I will present some recent work with Debmalya Basak and Alexandru Zaharescu on potential improvements to the Siegel—Walfisz upper bound on the greatest real zero of a Dirichlet $L$-function.
I will present on recent work - joint with John Green, Terence Harris, Kevin Ren, and Yumeng Ou - towards proving lower bounds for the dimensions of Furstenberg sets of circles and sine curves in the plane. A circular $(u,v)$-Furstenberg set is a set that contains a $u$-dimensional subset of each circle from a $v$-dimensional family of circles. One can approach the circular Furstenberg problem by proving estimates for the number of incidences between families of $\delta$-disks and $\delta$-annuli that satisfy certain dimension conditions. For different values of $u$ and $v$, we prove incidence estimates using local smoothing and using trilinear restriction estimates for the cone in $\mathbb{R}^3$. As time permits, I will discuss work relevant to proving dimension estimates for Furstenberg sets of sine curves (which satisfy all of the bounds we prove for circular Furstenberg sets) and/or work for Furstenberg sets of curves that satisfy a more general cinematic curvature condition.
The quest of blowup solutions to the NSE remains a challenging topic. We will discuss a recent construction that illustrates a blowup phenomenon unexplored previously for the NSE. In particular, the constructed solutions start from smooth initial data and exhibit an instantaneous blowup saturating Type-I blowup rate at a finite time. Moreover, there are infinitely many such blowup solutions; and the non-uniqueness occurs in borderline spaces of known criteria which ensure uniqueness. This is joint work with Alexey Cheskidov and Stan Palasek.
This talk provides an elementary introduction to skein algebras of surfaces, which serve as quantizations of $SL_n$-character varieties. For surfaces with boundary, we extend this framework to stated skein algebras, demonstrating how they provide simple and transparent geometric interpretations of various quantum group structures.
Specifically, we present a geometric realization of the dual canonical basis of $\mathscr{O}_q(\mathfrak{sl}_n)$ using skeins for $n=2$ and $n=3$. If time permits, we will also show how the skein algebra framework can be used to recover the Shapiro–Schrader embedding of the quantized enveloping algebra into a quantum torus algebra.
In work in progress with Ryan Stees, we show that every closed, oriented 3-manifold can be obtained by 0-surgery on a link. Since the 0-surgery of a link can capture the data of many of the typical isotopy and concordance invariants of a link, particularly in the pairwise linking number 0 case, this result gives us a nice lens through which to study both 3-manifolds and links. However, 0-surgery on a link is certainly not a complete link invariant, and we also give multiple constructions for non-isotopic (and even non-concordant) links with homeomorphic 0-surgeries.
I will review a general framework for developing numerical methods working with non-parametrically defined surfaces for various problems. In this talk, I will focus on boundary integral equations. 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. At the end of this talk, I will mention our work in developing multilevel neural network methods for inverting dense and large matrices that arise from boundary integral equations.
A fundamental problem in algebraic geometry is to construct compact moduli spaces parametrizing algebraic shapes. I will discuss a new approach to this problem in the case of Calabi--Yau pairs (X,D) for which D is ample. Such pairs arise from many well-studied algebraic varieties such as plane curves, K3 surfaces, and del Pezzo surfaces. In the case of Calabi-Yau pairs of dimension two, this approach outputs a projective moduli space on which the Hodge line bundle is ample. This is based on joint work with Yuchen Liu that builds on earlier work with Ascher, Bejleri, DeVleming, Inchiostro, Liu, and Wang.
The capillary energy functional is used to model the equilibrium shape of a liquid drop meeting a substrate at a prescribed interior contact angle. We will discuss a rigidity theorem for volume-preserving critical points of the capillary energy in the half-space: among all sets of finite perimeter, every such critical configuration corresponding to a prescribed contact angle between 90 degrees and 120 degrees must be a finite union of spheres and spherical caps with the correct contact angle. Assuming that the tangential part of the capillary boundary is $\mathcal{H}^n$-null, this rigidity extends to the full hydrophobic range of contact angles between 90 degrees and 180 degrees. We will also present an anisotropic counterpart, establishing rigidity under suitable lower density assumptions. This talk is based on joint work with A. De Rosa and R. Resende.
A well known result in graph theory states that a graph is connected if and only if the second eigenvalue of its Laplacian matrix is positive. In fact, the larger the second eigenvalue, the more connected the graph is. By varying the weights on edges, one can in general increase the second eigenvalue which in turn affects many graph properties such as expansion, mixing times of random walks etc.
In this talk, I will introduce conformally rigid graphs, which are those unweighted undirected graphs in which one cannot increase the second eigenvalue or decrease the largest eigenvalue by changing
weights. This notion turns out to be deeply connected to graph embeddings, semidefinite programming and other ideas in geometry, optimization and combinatorics.
Joint work with Joao Gouveia and Stefan Steinerberger
