Department of

Mathematics


Seminar Calendar
for events the day of Friday, February 1, 2019.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, February 1, 2019

2:00 pm in 141 Altgeld Hall,Friday, February 1, 2019

A Heat Trace Anomaly on Polygons

Hadrian Quan (Illinois Math)

Abstract: Given a planar domain with smooth boundary, one can associate its heat kernel, a time dependent operator whose trace admits an asymptotic expansion in t. The coefficients in this expansion turn out to all be geometric/topological invariants of the domain. However, by considering a smooth family of domains converging to a polygon, one can conclude that these heat trace coefficients are not continuous under such domain deformation. In this talk Iíll describe work of Mazzeo-Rowlett which recasts this apparent anomaly using renormalized invariants. Iíll also use it as an excuse to talk about uncommon but useful techniques in the study of linear PDEs e.g.: domain blow-ups, polyhomogeneous expansions, and more.

4:00 pm in 145 Altgeld Hall,Friday, February 1, 2019

Vector fields on Spheres

Brian Shin (UIUC)

Abstract: In this talk, I would like to tell the story of one of the classical problems in topology: how many pointwise linearly independent vector fields can you put on a sphere of dimension $n$. The famous Hairy Ball Theorem tells us that there are none if $n$ is even. On the other hand, if $n$ is one of 1, 3, or 7, we can construct $n$ such vector fields using the normed divison $\mathbb{R}$-algebra structures on complex numbers, quaternions, and octonions. In this talk, we'll discuss the complete resolution of this problem by Adams, using methods of geometry, algebra, and homotopy theory along the way.

4:00 pm in 245 Altgeld Hall,Friday, February 1, 2019

Nice rack: The evolution of deer antlers and other mating displays

Dr. Sara Clifton   [email] (UIUC Math)

Abstract: Species spanning the animal kingdom have evolved extravagant and costly ornaments to attract mating partners. Zahavi's handicap principle offers an elegant explanation for this: ornaments signal individual quality and must be costly to ensure honest signaling, making mate selection more efficient. Here, we incorporate the assumptions of the handicap principle into a mathematical model and show that they are sufficient to explain the heretofore puzzling observation of bimodally distributed ornament sizes in a variety of species.