Department of

# Mathematics

Seminar Calendar
for events the day of Wednesday, October 4, 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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Wednesday, October 4, 2017

3:00 pm in 243 Altgeld Hall,Wednesday, October 4, 2017

#### Instability of Steep Ocean Waves and Whitecapping

###### Sergey Dyachenko (UIUC)

Abstract: Wave breaking in deep oceans is a challenge that still defies complete scientific understanding. Sailors know that at wind speeds of approximately 5m/sec, the random looking windblown surface begins to develop patches of white foam (’whitecaps’) near sharply angled wave crests. We idealize such a sea locally by a family of close to maximum amplitude Stokes waves and show, using highly accurate simulation algorithms based on a conformal map representation, that perturbed Stokes waves develop the universal feature of an overturning plunging jet. We analyze both the cases when surface tension is absent and present. In the latter case, we show the plunging jet is regularized by capillary waves which rapidly become nonlinear Crapper waves in whose trough pockets whitecaps may be spawned.

4:00 pm in 245 Altgeld Hall,Wednesday, October 4, 2017

#### Modular forms, physics, and topology

###### Dan Berwick-Evans (Illinois)

Abstract: Modular forms appear in a wide variety of contexts in physics and mathematics. For example, they arise in two dimensional quantum field theories as certain observables. In algebraic topology, they emerge in the study of invariants called elliptic cohomology theories. A long-standing conjecture suggests that these two appearances of modular forms are intimately related. After explaining the ingredients, I’ll describe some recent progress.

4:00 pm in 343 Altgeld Hall,Wednesday, October 4, 2017

#### Enumerative Geometry and the Schubert Problem

###### Anna Weigandt (UIUC)

Abstract: How many lines meet four fixed lines in three space? This question has its roots in classical enumerative geometry. We will discuss the answer to the Schubert problem from the perspective of geometry, symmetric function theory, and combinatorics. This talk will be accessible for beginning graduate students.

4:00 pm in Altgeld Hall 141,Wednesday, October 4, 2017

#### Parabolic Higgs bundles

###### Georgios Kydonakis (Illinois Math)

Abstract: The Narasimhan and Seshadri theorem, one of the seminal first results in the study of the moduli space of vector bundles over a Riemann surface, relates degree zero, stable vector bundles on a compact Riemann surface $X$ with unitary representations of ${{\pi }_{1}}\left( X \right)$. One direction to generalize this theorem is by allowing punctures in the Riemann surface and the correspondence, which now involves parabolic bundles, was carried out by Mehta and Seshadri. The version for fundamental group representations of the punctured Riemann surface into Lie groups other than $G=\text{U}\left( n \right)$ entails introducing the notion of parabolic Higgs bundles. We will describe these holomorphic objects and see examples of those corresponding to Fuchsian representations of the fundamental group of the punctured Riemann surface.

4:00 pm in 314 Altgeld Hall,Wednesday, October 4, 2017

#### Applying to Graduate School in Mathematics

###### Lee DeVille and Karen Mortensen (Math Graduate Office, UIUC)

Abstract: Undergraduates who are interested in applying to graduate school in mathematics, whether this year or in the future, are invited to attend. You'll learn general information and tips about the process.