Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, December 3, 2019.

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Tuesday, December 3, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, December 3, 2019

Steklov Spectral Asymptotics for Polygons

David Sher   [email] (DePaul Mathematics)

Abstract: We consider the Steklov eigenvalue problem on curvilinear polygons in the plane, with all interior angles measuring less than pi. In this setting, we formulate and prove precise spectral asymptotics, with error converging to zero as the spectral parameter increases. These asymptotics have a surprising dependence on arithmetic properties of the angles. Moreover, the problem turns out to have an interesting relationship to a scattering-type eigenvalue problem on the one-dimensional boundary of the polygon, viewed as a quantum graph.

2:00 pm in 243 Altgeld Hall,Tuesday, December 3, 2019

Monochromatic cycles of given length in $2$-edge-colored graphs with large minimum degree

Xujun Liu (Illinois Math)

Abstract: We prove that if $n$ is sufficiently large and has the form $n=3t+r$ where $r \in \{0,1,2\}$, then for every $n$-vertex graph $G$ with $\delta(G) \ge (3n-1)/4$, in each $2$-edge-coloring of $G$, either there is a monochromatic subgraph containing cycles of every length $\{3, 4, 5, \dots, 2t+r\}$, or there is a monochromatic subgraph containing cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$. This is tight and slightly sharpens for sufficiently large $n$ a conjecture by Benevides, Łuczak, Scott, Skokan and White that for every graph $G$ of order $n=3t+r$ where $r \in \{0,1,2\}$ with $\delta(G) \ge 3n/4$ and every $2$-edge-coloring of $G$, there exists a monochromatic cycle of length at least $2t+r$. Joint work with József Balogh, Alexandr Kostochka and Mikhail Lavrov.

2:00 pm in 347 Altgeld Hall,Tuesday, December 3, 2019

Path independence for additive functionals of stochastic equations with jumps

Huijie Qiao (Southeast University and UIUC Math)

Abstract: In the talk, I will first introduce the definition of path independence for additive functionals of stochastic equations with jumps. And then for three types of stochastic equations with jumps, I state concrete path independence for their additive functionals. Three types of stochastic equations with jumps are stochastic differential equations with jumps, stochastic evolution equations with jumps and McKean-Vlaso stochastic differential equation with jumps.

3:00 pm in 243 Altgeld Hall,Tuesday, December 3, 2019

The Beauville-Voisin conjecture for Hilb(K3) and the Virasoro algebra

Andrei Negut (MIT)

Abstract: We give a geometric representation theory proof of a mild version of the Beauville-Voisin Conjecture for Hilbert schemes of K3 surfaces, namely the injectivity of the cycle map restricted to the subring of Chow generated by tautological classes. Our approach involves lifting formulas of Lehn and Li-Qin-Wang from cohomology to Chow groups, and using them to solve the problem by invoking the irreducibility criteria of Virasoro algebra modules, due to Feigin-Fuchs. Joint work with Davesh Maulik.

4:00 pm in 245 Altgeld Hall,Tuesday, December 3, 2019

On the Duffin-Schaeffer conjecture

Dimitris Koukoulopoulos   [email] (University of Montreal)

Abstract: Given any real number $\alpha$, Dirichlet proved that there are infinitely many reduced fractions $a/q$ such that $|\alpha-a/q|\le 1/q^2$. Can we get closer to $\alpha$ than that? For certain "quadratic irrationals" such as $\alpha=\sqrt{2}$ the answer is no. However, Khinchin proved that if we exclude such thin sets of numbers, then we can do much better. More precisely, let $(\Delta_q)_{q=1}^\infty$ be a sequence of error terms such that $q^2\Delta_q$ decreases. Khinchin showed that if the series $\sum_{q=1}^\infty q\Delta_q$ diverges, then almost all $\alpha$ (in the Lebesgue sense) admit infinitely many reduced rational approximations $a/q$ such that $|\alpha-a/q|\le \Delta_q$. Conversely, if the series $\sum_{q=1}^\infty q\Delta_q$ converges, then almost no real number is well-approximable with the above constraints. In 1941, Duffin and Schaeffer set out to understand what is the most general Khinchin-type theorem that is true, i.e., what happens if we remove the assumption that $q^2\Delta_q$ decreases. In particular, they were interested in choosing sequences $(\Delta_q)_{q=1}^\infty$ supported on sparse sets of integers. They came up with a general and simple criterion for the solubility of the inequality $|\alpha-a/q|\le\Delta_q$. In this talk, I will explain the conjecture of Duffin-Schaeffer as well as the key ideas in recent joint work with James Maynard that settles it.