Mathematics

On sizes of 1-cross intersecting set pair systems

Mina Nahvi (UIUC)

Abstract: Let $\{(A_i,B_i)\}_{i=1}^m$ be a set pair system. Furedi, Gyarfas and Kiraly called it $1$-cross intersecting if the size of intersection of $A_i$ and $B_j$ is $1$ when $i$ and $j$ are distinct, and $0$ if $i=j$. They studied such systems and their generalizations, and in particular considered $m(a,b,1)$, the maximum size of a $1$-cross intersecting set pair system in which $|A_i|\leq a$ and $|B_i|\leq b$ for all $i$.

Answering one of their questions, Holzman recently proved that if $a,b\geq 2$, then $m(a,b,1)\leq (29/30)((a+b) \text{ choose } a)$. He also conjectured that the factor $29/30$ in his bound can be replaced by $5/6$. The goal of this talk is to sketch the proof of this improved bound.

This is joint work with Grace Mc.Court and Alexandr V Kostochka.

For Zoom information, please contact Sean at SEnglish (at) illinois (dot) edu.

Towards optimal gradient bounds for torsion functions in the plane

Jeremy Hoskins (University of Chicago)

Abstract: In this talk we consider a natural question going back to Saint Venant in 1856: if a beam of constant cross-section is twisted, how large is the maximum shear stress? As it turns out, this problem can be formulated as a basic question about elliptic PDEs which arises in a number of settings, including electrostatics, constrained maximization of the lifetime of Brownian motion started close to the boundary, and optimal Hermite-Hadamard inequalities for subharmonic functions on convex domains. In this talk we will present upper and lower bounds for the largest possible shear stress for convex domains, and discuss extremal shapes.

Spring 2021 Department Faculty Meeting

(UIUC Department of Mathematics)

Abstract: The Spring 2021 Department Faculty Meeting will be held virtually on Tuesday, May 4, 2021, at 4 p.m. CT. Advance registration will be required: https://illinois.zoom.us/meeting/register/tZcsdeyhpjMvHdQSDeyJv0zk34B2Z-lzD-wZ