Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, October 1, 2019.

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Tuesday, October 1, 2019

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

Continuity of Functions on the Powerset of the First Uncountable Cardinal

William Chan (University of North Texas)

Abstract: Assume the axiom of determinacy. In this talk, we will discuss an almost everywhere uniformization for club subsets of $\omega_1$. Using this uniformization, we will show that every function $\Phi : [\omega_1]^{\omega_1} \rightarrow \omega_1$ has a club $C \subseteq \omega_1$ so that $\Phi | [C]^{\omega_1}$ is a continuous function. This continuity result can be used to make the following cardinality distinction, $|[\omega_1]^{<\omega_1}| < |[\omega_1]^{\omega_1}|$. This is joint work with Stephen Jackson.

2:00 pm in 347 Altgeld Hall,Tuesday, October 1, 2019

The genus of generalized sparse random graphs

Yifan Jing (UIUC)

Abstract: Determining the genus of graphs is one of the central problems in topological graph theory. In particular, it was proved by Thomassen that determining the genus of any given graph is NP-Complete. Recently, we provided a polynomial-time approximation scheme for the genus of dense graphs. One of the key ingredients there is that we approximate the genus of dense multipartite quasirandom graphs. Motivated by that project, we study the genus of generalized sparse random graphs, that is graphs can be decomposed by finite sum of sparse random graphs. This is joint work with Bojan Mohar.

2:00 pm in 243 Altgeld Hall,Tuesday, October 1, 2019

Long monochromatic paths and cycles in $2$-edge-colored graphs with large minimum degree

Mikhail Lavrov (Illinois Math)

Abstract: Our main result is a proof for sufficiently large $n$ of the conjecture by Benevides, Łuczak, Scott, Skokan and White that for every positive integer $n$ of the form $n=3t+r$ where $r \in \{0,1,2\}$ and every $n$-vertex graph $G$ with $\delta(G) \ge 3n/4$, in each $2$-edge-coloring of $G$ there exists a monochromatic cycle of length at least $2t+r$.

Our result also implies the conjecture of Schelp that for every sufficiently large $n$, every $(3n-1)$-vertex graph $G$ with minimum degree larger than $3|V(G)|/4$, in every $2$-edge-coloring of $G$, there is a monochromatic path with $2n$ vertices. Joint work with József Balogh, Alexandr Kostochka and Xujun Liu.

3:00 pm in 243 Altgeld Hall,Tuesday, October 1, 2019

Standard Conjecture D for matrix factorizations

Michael Brown (UW-Madison)

Abstract: In 1968, Grothendieck posed a family of conjectures concerning algebraic cycles called the Standard Conjectures. The conjectures have been proven in some special cases, but they remain open in general. In 2011, Marcolli-Tabuada realized two of these conjectures as special cases of more general conjectures, involving differential graded categories, which they call Noncommutative Standard Conjectures C and D. The goal of this talk is to discuss a proof, joint with Mark Walker, of Noncommutative Standard Conjecture D in a special case which does not fall under the purview of Grothendieck's original conjectures: namely, in the setting of matrix factorizations.

4:00 pm in 245Altgeld Hall,Tuesday, October 1, 2019

Limitations on All Known (and Some Unknown) Approaches to Matrix Multiplication

Virginia Vassilevska-Williams   [email] (Massachusetts Institute of Technology)

Abstract: In this talk we will consider the known techniques for designing Matrix Multiplication algorithms. The two main approaches are the Laser method of Strassen, and the Group theoretic approach of Cohn and Umans. We define generalizations that subsume these two approaches: the Galactic and the Universal method; the latter is the most general method there is. We then design a suite of techniques for proving lower bounds on the value of $\omega$, the exponent of matrix multiplication, which can be achieved by algorithms using many tensors $T$ and the Galactic method. Some of our techniques exploit `local' properties of $T$, like finding a sub-tensor of $T$ which is so `weak' that $T$ itself couldn't be used to achieve a good bound on $\omega$, while others exploit `global' properties, like $T$ being a monomial degeneration of the structural tensor of a group algebra. The main result is that there is a universal constant $\ell>2$ such that a large class of tensors generalizing the Coppersmith-Winograd tensor $CW_q$ cannot be used within the Galactic method to show a bound on $\omega$ better than $\ell$, for any $q$. We give evidence that previous lower-bounding techniques were not strong enough to show this. The talk is based on joint work with Josh Alman, which appeared in FOCS 2018. More recently, Alman showed how to extend our techniques so that they apply to the Universal method as well. In particular, Alman shows that the Coppersmith-Winograd tensor cannot yield a better bound on $\omega$ than 2.16805 even using the Universal method.