Week of November 27, 2022

Mon Nov 28, 2022
12:00pm - zoom - Probability and Analysis Webinar
Emanuel Milman - (Technion - Israel Institute of Technology)
Multi-Bubble Isoperimetric Problems - Old and New

The classical isoperimetric inequality in Euclidean space R^n states that among all sets ("bubbles") of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems for more general metric-measure spaces, such as on the n-sphere S^n and on n-dimensional Gaussian space G^n (i.e. R^n endowed with the standard Gaussian measure). Furthermore, one may consider the "multi-bubble" isoperimetric problem, in which one prescribes the volume of p ≥ 2 bubbles (possibly disconnected) and minimizes their total surface area -- as any mutual interface will only be counted once, the bubbles are now incentivized to clump together. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to p=1; the case p=2 is called the double-bubble problem, and so on. 

In 2000, Hutchings, Morgan, Ritoré and Ros resolved the double-bubble conjecture in Euclidean space R^3 (and this was subsequently resolved in R^n as well) --  the boundary of a minimizing double-bubble is given by three spherical caps meeting at 120-degree angles. A more general conjecture of J. Sullivan from the 1990's asserts that when p ≤ n+1, the optimal multi-bubble in R^n (as well as in S^n) is obtained by taking the Voronoi cells of p+1 equidistant points in S^n and applying appropriate stereographic projections to R^n (and backwards).

In 2018, together with Joe Neeman, we resolved the analogous multi-bubble conjecture for p ≤ n bubbles in Gaussian space G^n -- the unique partition which minimizes the total Gaussian surface area is given by the Voronoi cells of (appropriately translated) p+1 equidistant points. In the talk, I will describe our approach in that work, as well as recent progress we have made on the multi-bubble problem on R^n and S^n. In particular, we show that minimizing bubbles in R^n and S^n are always spherical when p ≤ n, and we resolve the latter conjectures when in addition p ≤ 5 (e.g. the triple-bubble conjectures when n ≥ 3 and the quadruple-bubble conjectures when n ≥ 4). 


4:00pm to 5:30pm - RH 440 R - Logic Set Theory
Jouko Vaananen - (University of Helsinki)
An inner model from stationary logic

Godel's constructible universe uses first order logic to build a model of ZFC containing all of the ordinals and where the GCH holds.  This talk describes a method of building analogous models using Stationary Logic. These are well-founded inner models of stronger axioms than ZFC that retain elements of fine structure. 

Stationary Logic is a stronger logic than first order logic, but retains desirable model theoretic aspects. 

Tue Nov 29, 2022
4:00pm to 5:00pm - ISEB 1200 - Differential Geometry
Richard Wentworth - (Maryland)
A Donaldson-Uhlenbeck-Yau theorem for normal projective varieties

The correspondence between polystable reflexive sheaves on compact Kaehler manifolds and the existence of suitably singular Hermitian-Einstein metrics can be extended to normal projective varieties that are smooth in codimension two. A particular application is a characterization of those sheaves which saturate the Bogomolov-Gieseker inequality. This talk will present some of the key details of this result, which is joint work with Xuemiao Chen.

Wed Nov 30, 2022
2:00pm to 3:00pm - 510R Rowland Hall - Combinatorics and Probability
Jiapeng Zhang - (USC)
Detecting Hidden Communities by Power Iterations with Connections to Vanilla Spectral Algorithms

Community detection in the stochastic block model is one of the central problems of graph clustering. In this setup, spectral algorithms have been one of the most widely used frameworks for the design of clustering algorithms. However, despite the long history of study, there are still unsolved challenges. One of the main open problems is the design and analysis of ``simple'' spectral algorithms, especially when the number of communities is large. 

In this talk, I will discuss two algorithms. The first one is based on the power-iteration method. Our algorithm performs optimally (up to logarithmic factors) compared to the best known bounds in the dense graph regime by Van Vu (Combinatorics Probability and Computing, 2018). 

Then based on a connection between the powered adjacency matrix and eigenvectors, we provide a ``vanilla'' spectral algorithm for large number of communities in the balanced case. Our spectral algorithm is as simple as PCA (principal component analysis).

This talk is based on joint works with Chandra Sekhar Mukherjee. (https://arxiv.org/abs/2211.03939)

Thu Dec 1, 2022
9:00am to 9:50am - Zoom - Inverse Problems
María Ángeles García-Ferrero - (Universitat de Barcelona)
The Calderón problem for directionally antilocal operators


10:00am - Zoom ID: 99342387189 - Harmonic Analysis
Christos Saroglou - (University of Ioannina )
On a j-Santaló conjecture

Let $k\geq 2$ be an integer. In the spirit of Kolesnikov-Werner, for each $j\in\{2,\ldots,k\}$, we conjecture a sharp Santaló type inequality (we call it $j$-Santal\'{o} conjecture)  for many sets (or more generally for many functions), which we are able to confirm in some cases, including the case $j=k$ and the unconditional case. Interestingly, the extremals of this family of inequalities are tuples of the $l_j^n$-ball. 
Our results also strengthen one of the main results of Kolesnikov-Werner, which corresponds to the case $j=2$. All members of the family of our conjectured inequalities can be interpreted as generalizations of the classical Blaschke-Santaló inequality.
Related, we discuss an analogue of a conjecture due to K. Ball in the multi-entry setting and establish a connection to the $j$-Santaló conjecture.

1:00pm to 2:00pm - RH 306 - Algebra
Mohamed Omar - (Harvey Mudd)
How many cards avoid a SET?

SET is a popular real-time card game where players search for special triples of cards among a table of cards that are face-up. A common issue when playing the game is not having a SET among the face-up cards. What is the maximum number of cards that can be face-up while avoiding a SET? Surprisingly, this question is at the heart of a decades old central problem in extremal combinatorics and additive number theory that had a major breakthrough in 2017. In this talk, we describe the breakthrough, and how the presenter used ideas in its development to make headway on a range of disparate problems in combinatorics.

3:00pm to 4:00pm - RH 306 - Number Theory
Siegfred Baluyot - (AIM)
Twisted $2k$th moments of primitive Dirichlet $L$-functions: beyond the diagonal

In this joint work with Caroline Turnage-Butterbaugh, we study the family of Dirichlet $L$-functions of all even primitive characters of conductor at most $Q$, where $Q$ is a parameter tending to infinity. We approximate the twisted $2k$th moment of this family using Dirichlet polynomials of length between $Q$ and $Q^2$. Assuming the Generalized Lindelof Hypothesis, we prove an asymptotic formula for these approximations. Our result agrees with the prediction of Conrey, Farmer, Keating, Rubinstein, and Snaith, and provides the first rigorous evidence beyond the diagonal terms for their conjectured asymptotic formula for the general $2k$th moment of this family. The main device we use in our proof is the asymptotic large sieve developed by Conrey, Iwaniec, and Soundararajan. 

4:00pm - RH 192 - Colloquium
Thomas Hou - (Caltech)
On Finite Time Blowup of the 3D Euler Equations and Related Models Using Computer-Assisted Proofs