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for events the day of Friday, March 25, 2022.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, March 25, 2022

1:00 pm in 147 Altgeld Hall,Friday, March 25, 2022

Index theorems and boundary conditions

Gayana Jayasinghe (UIUC)

Abstract: The study of the Fredholm Index of operators is of great interest in analysis. The 60's saw the development of the theory of the index of elliptic operators as a topological invariant on compact manifolds, a vast generalization of the Gauss Bonnet theorem. Now there are generalizations of all sorts of spaces including spaces with boundary, singularities and in non commutative geometry. I will begin by introducing Index theory, motivating the key ideas with the example of the Gauss Bonnet theorem, on manifolds with boundary. The index in this example has a topological contribution as well as a contribution coming from the asymmetry of the boundary, which can be described in terms of geodesic curvature. I will then describe how for general operators, this asymmetry can be expressed using boundary conditions for the operator, and in the case of spectral boundary conditions, one deals with the spectral asymmetry. I won't assume any prior knowledge on Index theory, and it should be accessible for analysts

3:00 pm in 341 Altgeld Hall,Friday, March 25, 2022

An introduction to 2-categories

Yigal Kamel (UIUC Math)

Abstract: This talk will be a survey of some of the basic notions and facts about 2-categories. After introducing 2-categories and various types of functors between them, we will consider constructions that map 2-categories to more familiar objects. On one hand, viewing a 2-category as a “poor” higher category, we can “reduce” it to an ordinary category, via a homotopy construction. On the other hand, viewing a 2-category as a “rich” ordinary category, we can “uplift” it to a simplicial set, via the duskin nerve. I will talk about these and related ideas, indicate some quirks of the theory, and provide examples along the way.

4:00 pm in 347 Altgeld Hall,Friday, March 25, 2022

Modular Forms in Geometry and Physics

Saaber Pourmotabbed

Abstract: Modular forms appear as generating functions of curve counts and BPS states, partition functions of conformal field theories, in moonshine phenomena, sections of line bundles, spectrums of cohomology theories, elliptic genera of K3 surfaces, and many other interesting cases. In this talk we will look at some examples of where modular forms occur in physics and geometry, why they occur, and their interactions between these different fields.