Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, January 31, 2017.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 31, 2017

11:00 am in 345 Altgeld Hall,Tuesday, January 31, 2017

#### Complex analytic elliptic cohomology and Looijenga line bundles

###### Charles Rezk (Illinois)

Abstract: I'll explain how, by taking the cohomology of suitable spaces and messing around a bit, you can get things like: the moduli stack of (analytic) curves, the universal curve, and Looijenga line bundles over these. This seems to have some relevance for the construction of complex analytic elliptic cohomology.

1:00 pm in 345 Altgeld Hall,Tuesday, January 31, 2017

#### Uncountable Categoricity in Continuous Logic

###### Victoria Noquez (UIC Math)

Abstract: In recent years, some progress has been made towards understanding uncountable categoricity in the continuous setting, particularly in the context of classes of Banach spaces. Currently, it is unknown if the Baldwin-Lachlan characterization of uncountable categoricity holds in continuous logic. Namely, is it the case a continuous theory T is $\kappa$-categorical for some uncountable cardinal $\kappa$ if and only if T is $\omega$-stable and has no Vaughtian pairs?
In order to address this question, we provide the necessary continuous characterization of Vaughtian pairs, and in the process, prove Vaught's two-cardinal theorem, as well as a partial converse of the theorem in the continuous setting. This allows us to prove the forward direction of the Baldwin-Lachlan characterization.
Trying to prove the reverse direction leads us to an attempt to characterize strong minimality in continuous logic. We propose a notion of strong minimality, and show that it has many of the properties of its classical analogue. Unfortunately, we see that this does not provide the machinery required to show that $\omega$-stability and the absence of Vaughtian pairs are sufficient conditions for uncountable categoricity. We provide some examples towards understanding this failure.

3:00 pm in 243 Altgeld Hall,Tuesday, January 31, 2017

#### Kirwan surjectivity for quiver varieties

###### Tom Nevins (UIUC)

Abstract: Many interesting hyperkahler, or more generally holomorphic symplectic, manifolds are constructed via hyperkahler/holomorphic symplectic reduction. For such a manifold there is a “hyperkahler Kirwan map,” from the equivariant cohomology of the original manifold to the reduced space. It is a long-standing question when this map is surjective (in the Kahler rather than hyperkahler case, this has been known for decades thanks to work of Atiyah-Bott and Kirwan). I’ll describe a resolution of the question (joint work with K. McGerty) for Nakajima quiver varieties: their cohomology is generated by Chern classes of “tautological bundles.” If there is time, I will explain that this is a particular instance of a general story in noncommutative geometry. The talk will not assume prior familiarity with any of the notions above.

3:00 pm in 241 Altgeld Hall,Tuesday, January 31, 2017

#### DP-colorings of graphs with high chromatic number

###### Anton Bernshteyn (Illinois Math)

Abstract: Let $G$ be an $n$-vertex graph with chromatic number $\chi(G)$. Ohba's conjecture (now a celebrated theorem due to Noel, Reed, and Wu) claims that whenever $\chi(G) \geq (n-1)/2$, the list chromatic number of $G$ equals $\chi(G)$. DP-coloring is a generalization of list coloring introduced recently by Dvořák and Postle. We establish an analog of Ohba's conjecture for the DP-chromatic number; namely we show that the DP-chromatic number of $G$ also equals $\chi(G)$ as long as $\chi(G)$ is sufficiently large''. In contrast to the list coloring case, large'' here means $\chi(G)\geq n-O(\sqrt{n})$, and we also show that this lower bound is best possible (up to the constant factor in front of $\sqrt{n}$). This is joint work with Alexandr Kostochka and Xuding Zhu.

4:00 pm in 131 English Building,Tuesday, January 31, 2017

#### Well-posedness for the "Good" Boussinesq on the Half Line

###### Erin Compaan   [email] (Erin Compaan)

Abstract: I'll present some recent results on well-posedness for the "good" Boussinesq equation on the half line at low regularities. The method is one introduced by Erdogan and Tzirakis, which involves extending the problem to the full line and solving it there with Bourgain space methods. A forcing term ensures that the boundary condition is enforced. I'll introduce the method and talk about the estimates required to close the argument. This is joint work with N. Tzirakis.