Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, October 13, 2015.

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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, October 13, 2015

11:00 am in 243 Altgeld Hall,Tuesday, October 13, 2015

#### Intermediate Hopf-Galois Extensions and the Nilpotence Theorem

###### Jon Beardsley (Johns Hopkins)

Abstract: We will define Hopf-Galois extensions of ring spectra and give some examples from chromatic homotopy theory. We also describe a method of producing intermediate extensions from normal-sub-Hopf-algebras, similarly to the way one produces intermediate Galois extensions from normal subgroups of the Galois group. We finally describe ongoing work to use this structure to streamline the proof of the Nilpotence Theorem of Devinatz, Hopkins and Smith.

12:00 pm in 345 Altgeld Hall,Tuesday, October 13, 2015

#### The topology of local commensurability graphs

###### Khalid Bou-Rabee (City College of CUNY)

Abstract: The p-local commensurability graph of a group has vertices consisting of all finite-index subgroups, where an edge is drawn between two subgroups if their commensurability index is a power of p. While commensurability is a fundamental notion, these graphs also have ties to subgroup growth, another natural invariant. What group-theoretic information can we draw from the topology of these graphs? To initiate the study of this question, we explore these graphs for a number of examples and share some of our findings. It turns out that any such graph for a group with all nilpotent finite quotients is complete. Further, this topological criteria characterizes such groups. In contrast, for any prime p, any large group (e.g. a nonabelian free group or a surface group of genus two or, more generally, any virtually special group) has geodesics in its p-local commensurability graph of arbitrarily long length. This talk covers joint work with Daniel Studenmund.

1:00 pm in 214 Altgeld Hall,Tuesday, October 13, 2015

#### Regularity of Monge-Ampere equations and Gromov-Hausdorff limits

###### Gabe La Nave (UIUC Math)

Abstract: I will discuss how to obtain regularity results for complex Monge-Ampere equations using structure theory of Gromov-Hausdorff limits.

2:00 pm in 347 Altgeld Hall,Tuesday, October 13, 2015

#### A drunk walk in a drunk world

###### Ivan Corwin (Columbia University, Clay Mathematics Institute)

Abstract: In a simple symmetric random walk on Z a particle jumps left or right with 50% chance independently at each time and space location. What if the jump probabilities are taken to be random themselves (e.g. uniformly distributed between 0% and 100%). In this talk we will describe the effect of this random environment on a random walk, in particular focusing on a new connection to the Kardar-Parisi-Zhang universality class and to the theory of quantum integrable systems. No prior knowledge or background will be expected.

3:00 pm in 241 Altgeld Hall,Tuesday, October 13, 2015

#### A refinement of two theorems on chorded cycles

###### Michael Santana   [email] (UIUC Math)

Abstract: In 1963, Corrádi and Hajnal verified a conjecture of Erdős showing that every graph $G$ on at least $3k$ vertices with $\delta(G) \ge 2k$ contains $k$ disjoint cycles. Independently, Enomoto and Wang (1998 and 1999, resp.) extended this result of Corrádi and Hajnal by replacing the minimum degree condition with an Ore-degree condition. Recently, Kierstead, Kostochka, Molla and Yeager completed a characterization of the graphs that just fail the Ore-degree condition of Enomoto and Wang and do not contain k disjoint cycles. In 2008, Finkel proved an analogue of Corrádi-Hajnal for chorded cycles, requiring G to have at least $4k$ vertices and $\delta(G) ≥ 3k$ in order to guarantee $k$ disjoint chorded cycles. This result was extended by Chiba et al. in 2010, who replace the minimum degree condition with an Ore-degree condition. In this talk, we will discuss our proof of a chorded cycle analogue to that of Kierstead et al., in which we characterize the graphs that just fail the Ore-degree condition of Chiba et al. and do not contain $k$ disjoint chorded cycles. Time permitting, we will also present current progress towards a result that will provide a transition from our result to that of Kierstead et al. This is joint work with Theodore Molla and Elyse Yeager.