UIUC Dept. of Mathematics Seminar Calendar

    September 2019          October 2019          November 2019    
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Tuesday, October 8, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, October 8, 2019

Logic Seminar

A Categorical Semantics for Linear (and Quantum) Dependent Type Theory

Kohei Kishida (UIUC Philosophy)

Abstract: Desirable features of a quantum programming language include: treating both quantum resources and classical control; being a functional language admitting semantics and other formal methods; and dependent types. The type theory of such a language must be linear to reflect the linearity of quantum processes, and we want it to involve parameters (values that are known at circuit generation time) and states (values known at circuit execution time) to describe and generate quantum circuits. The goal of this talk is to provide a general semantic structure for linear dependent type theory of that sort. We review categorical models of quantum processes and the sets-and-functions model of classical dependent type theory, and show how they can be integrated to model linear dependent type theory of classical parameters and quantum states. This is joint work with Frank Fu, Julien Ross, and Peter Selinger.

Submitted by phierony

2:00 pm in 243 Altgeld Hall,Tuesday, October 8, 2019

Graph Theory and Combinatorics Seminar

Problems and results on $1$-cross intersecting set pair systems

Zolt�n F�redi (R�nyi Institute, Budapest, Hungary)

Abstract: The notion of cross intersecting set pair system of size $m$, $(\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m)$ with $A_i \cap B_i = \emptyset$ and $A_i \cap B_j \ne \emptyset$, was introduced by Bollob�s and it became an important tool of extremal combinatorics. His classical result states that $m \le {{a+b} \choose a}$ if $|A_i|\le a$ and $|B_i| \le b$ for each $i$.

Our central problem is to see how this bound changes with the additional condition $|A_i \cap B_j|= 1$ for $i \ne j$. Such a system is called $1$-cross intersecting. We show that the maximum size of a $1$-cross intersecting set pair system is

at least $5^{n/2}$ for $n$ even, $a=b=n$,
equal to $(\lfloor \frac n2 \rfloor + 1)(\lceil \frac n2\rceil + 1)$ if $a=2$, $b=n \ge 4$,
at most $|\bigcup_{i=1}^m A_i|$,
asymptotically $n^2$ if $\{A_i\}$ is a linear hypergraph ($|A_i\cap A_j| \le 1$ for $i \ne j$),
asymptotically $\frac12 n^2$ if $\{A_i\}$, $\{B_i\}$ are both linear.

Submitted by mlavrov

3:00 pm in 243 Altgeld Hall,Tuesday, October 8, 2019

Algebraic Geometry Seminar

Character stacks and shtukas in the topological setting

Nick Rozenblyum (U Chicago)

Abstract: I will describe a general categorical framework leading to shtukas (in the sense of Drinfeld) and excursion operators (in the sense of V. Lafforgue) on moduli spaces. In particular, I will give a concrete description of the space of functions on (derived) character varieties. I will explain how this leads to the spectral action in the context of Betti geometric Langlands and (conjecturally) to the spectral decomposition in geometric Langlands over finite fields via a categorification of Grothendieck's function-sheaf correspondence. This is joint work with Gaitsgory, Kazhdan, and Varshavsky.

Submitted by aronah

Mathematics