Georgia Institute of Technology College of Sciences

The Reissner–Nordström spacetime models a spherically symmetric and time-independent charged black hole in general relativity. The Cauchy horizon in the interior of such a black hole is subject to an infinite blueshift instability. In 1989, Poisson and Israel discovered a nonlinear manifestation of this instability in the spherically symmetric setting called "mass inflation," where the Hawking mass becomes identically infinite at the Cauchy horizon. 

We complete the first proof of mass inflation for a wave-type matter model, namely the spherically symmetric Einstein–Maxwell–scalar field system. This result follows from a large-data decay result for the scalar field in the black hole exterior combined with works of Dafermos, Luk–Oh, and Luk–Oh–Shlapentokh-Rothman.

I'll present various methods, some old, some new,  leading to estimates on the variance of $f(X_1, X_2, \dots, X_n)$ where  

$X_1, X_2, \dots, X_n$ are independent random variables.  These methods will be illustrated with various examples.

Let us discuss how to use generative AI to help with math and coding.

My presentation features two scenarios:

Coding in LaTeX. Suppose you have a raw draft of what potentially could be a math paper. We will consider and apply simple AI tools that may help realizing the potential.

Coding in CAS. Suppose you have a raw idea for a package in a Computer Algebra System; your raw idea may be limited to a rough description of the input/output of a method you would like to implement. How far can an AI assistant take you? Can it autonomously code a working software package?   

Let $LC_n$ be the length of the longest common subsequences of two independent random words whose letters are taken  in a finite alphabet and when the alphabet is totally ordered and let $LCI_n$ be the length of the longest common and increasing subsequences of the words.   Results on the asymptotic means, variances and limiting laws of these well-known random objects will be described and compared.

Abstract:
This talk explores the Uzawa method, tracing its development from early applications in partial differential equations (PDEs) to modern advancements in optimization, image processing, and scientific computing. We will examine recent refinements for developing GPU-adaptive solvers for huge-scale linear programming and its extension to semidefinite programming arising in quantum information science. The discussion will also highlight the method's integration with deep learning and unrolling techniques for optimal control problems of PDEs, as well as its applications in industry.

Bio:

Xiaoming Yuan is a Professor in the Department of Mathematics at The University of Hong Kong. His research spans optimization, optimal control, scientific machine computing, and artificial intelligence. He is well recognized for his fundamental contributions to first-order optimization algorithms, including the Alternating Direction Method of Multipliers (ADMM), primal-dual methods, and proximal point algorithms. He also collaborates extensively with the AI and cloud computing industries. He led the development of the first automatic bandwidth allocation system for the cloud computing sector. His team was honored as a Franz Edelman Award Finalist in 2023.

In a highly influential paper from 1978, Lovász used topological methods to determine the chromatic number of the Kneser graph of the set of k-element subsets of a set with n elements. In this talk, we will discuss the Kneser graph of the set of triangulations of a convex n-gon and a recent proof that the chromatic number of this graph is n-2. The geometry of the associahedron will play a particularly important role in the argument. Based on a joint work with Anton Molnar, Cosmin Pohoata and Daniel Zhu.

The local spacing of zeros of orthogonal polynomials is studied using scaling limits of Christoffel--Darboux kernels. Different limit kernels are associated with different universality classes, e.g. sine kernel with bulk universality and locally asymptotically uniform zero spacing. In recent years, new results have been obtained by using the de Branges theory of canonical systems. This includes necessary and sufficient conditions for a family of scaling limits corresponding to homogeneous de Branges spaces; this family includes bulk universality, hard edge universality, jump discontinuities in the weight, and other notable universality classes. It also includes local behaviors beyond scaling limits. The talk is based on joint works with Benjamin Eichinger, Brian Simanek, Harald Woracek, Peter Yuditskii.

We present a random construction proving that the extreme off-diagonal Ramsey numbers satisfy $R(3,k)\ge  \left(\frac12+o(1)\right)\frac{k^2}{\log{k}}$ (conjectured to be asymptotically tight), improving the previously best bound $R(3,k)\ge  \left(\frac13+o(1)\right)\frac{k^2}{\log{k}}$. In contrast to all previous constructions achieving the correct order of magnitude, we do not use a nibble argument.

Beyond the paper, we will explore a bit further how the approach can be used for other problems.