Mathematics

Hypersurfaces with Central Convex Cross-Sections

Alper Gur (Indiana U Math)

Abstract: The compact transverse cross-sections of a cylinder over a central ovaloid in \(\bf R^n\), \(n \geq 3\), with hyperplanes are central ovaloids. A similar result holds for quadrics (level sets of quadratic polynomials in \( \bf R^n\), \(n \geq 3\) ). Their compact transverse cross-sections with hyperplanes are ellipsoids, which are central ovaloids.
\(\quad\)In \( \bf R^3\), Blaschke, Brunn, and Olovjanischnikoff found results for compact convex surfaces that motivated B. Solomon to prove that these two kinds of examples provide the only complete, connected, smooth surfaces in \( \bf R^3\), whose ovaloid crosssections are central. We generalize that result to all higher dimensions, proving: If \( M^{n-1} \subseteq \bf R^n\), \(n \geq 4\), is a complete, connected, smooth hypersurface, which intersects at least one hyperplane transversally along an ovaloid, and every such ovaloid on \(M\) is central, then \(M\) is either a cylinder over a central ovaloid or a quadric.

Robinson forcing in $C^*$-algebras

Thomas Sinclair (Purdue University Math)

Abstract: Several long-standing open problems in the theory of $C^*$-algebras reduce to whether for a given class of $C^*$-algebras there is a locally universal one among them with certain nice properties. I will discuss how techniques from model theory, in particular model-theoretical forcing, can be used to shed light on these problems. This is joint work with Isaac Goldbring.

On Sharpness of the Kato Smoothing Property of Dispersive Wave Equations

Bingyu Zhang (University of Cincinnati)

Abstract: Constantin and Saut showed in 1988 that solutions of the Cauchy problem for general dispersive equations $$ w_t +iP(D)w=0,\quad w(x,0)=q (x), \quad x\in \mathbb{R}^n, \ t\in \mathbb{R} , $$ enjoy the local smoothing property $$ q\in H^s (\Bbb R ^n) \implies w\in L^2 \Big (-T,T; H^{s+\frac{m-1}{2}}_{loc} \left (\Bbb R^n\right )\Big ) , $$ where $m$ is the order of the pseudo-differential operator $P(D)$. This property, now called local Kato smoothing, was first discovered by Kato for the KdV equation and implicitly shown later by Sj\"olin for the linear Schr\"odinger equation. In this talk, we show that the local Kato smoothing property possessed by solutions general dispersive equations in the 1D case is sharp, meaning that there exist initial data $q\in H^s \left (\Bbb R \right )$ such that the corresponding solution $w$ does not belong to the space $ L^2 \Big (-T,T; H^{s+\frac{m-1}{2} +\epsilon}_{loc} \left (\Bbb R\right )\Big )$ for any $\epsilon >0$.

The functional equation of the Dedekind zeta function of a number field

Ravi Donepudi (UIUC )

Abstract: Riemann’s seminal paper of 1859 establishes (among other things) the analytic continuation and functional equation of the Riemann zeta function. These results have since been generalized to several other similarly defined functions. Indeed, Wikipedia lists at least 33 "zeta functions" defined in diverse areas of mathematics. In this talk, we will discuss one such class of objects: the Dedekind zeta function attached to a number field. Erich Hecke was the first to prove a functional equation for these functions. We will describe Hecke’s proof in detail, highlighting it’s similarities to Riemann’s original proof. As a corollary we will be able to better see why important invariants of a number field, like the discriminant and regulator, mysteriously appear in the class number formula. People of both algebraic and analytic persuasions should find something interesting in this talk. If I use fancy words like Adeles or Ideles in the talk, you can shoot me.

Star decompositions of random regular graphs

Michelle Delcourt (Illinois Math)

Abstract: In 2006, Barat and Thomassen conjectured that the edges of every planar 4-regular 4-edge-connected graph can be decomposed into claws. Shortly afterward, Lai constructed a counterexample to this conjecture. Using the small subgraph conditioning method of Robinson and Wormald, Luke Postle and I showed that a.a.s. a random 4-regular graph has a decomposition into claws, provided the number of vertices is divisible by 3. I will also discuss more recent results decomposing regular graphs into stars; this is joint work with Bernard Lidicky and Luke Postle.

Applications of Numerical Algebraic Geometry: Bertini and Blood Coagulation

Francesco Pancaldi (University of Notre Dame)