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for events the day of Friday, February 11, 2022.

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Friday, February 11, 2022

1:00 pm in Altgeld Hall 147,Friday, February 11, 2022

Einstein metrics with prescribed conformal infinity

Xinran Yu (UIUC)

Abstract: Given a conformal class of metrics on the boundary of a manifold, one can ask for the existence of an Einstein metric whose conformal infinity satisfies the boundary condition. In 1991, Graham and Lee studied this boundary problem on the hyperbolic ball. They proved the existence of metrics sufficiently close to the round metric on a sphere by constructing approximate solutions to a quasilinear elliptic system. In his monograph (2006), Lee discussed the boundary problem on a smooth, compact manifold-with-boundary. Using a similar construction, he proved the existence and regularity results for metrics sufficiently close to a given asymptotically hyperbolic Einstein metric. The proof is based on a linear theory for Laplacian and the inverse function theorem.

3:00 pm in 341 Altgeld Hall,Friday, February 11, 2022

An Introduction to Exodromy

Brian Shin (UIUC Math)

Abstract: One of the first major topics we learn about in algebraic topology is the classification of locally constant sheaves of sets (i.e. covering spaces) of a sufficiently nice topological space in terms of its fundamental group. This classification is mediated by an equivalence of categories known as the monodromy equivalence. An insight of Kan was that, in order to classify locally constant sheaves of more interesting objects, one must pass from fundamental groups to fundamental infinity-groupoid. In this expository talk, I'd like to talk about work of Barwick-Glasman-Haine pushing this circle ideas further into the realm of stratified spaces. The main result is the exodromy equivalence, which classifies constructible sheaves on a stratified space in terms of its profinite stratified shape.

4:00 pm in 347 Altgeld Hall,Friday, February 11, 2022

Surfaces in Knot Complements

Brannon Basilio (UIUC)

Abstract: We shall survey the role that surfaces play in knot complements. Many surfaces can be embedded in the complement of a knot in S^3. However, we will only concern ourselves with certain surfaces called incompressible surfaces. These surfaces play a major role in studying knots. For example, it turns out that the existence of certain surfaces tells us what kind of geometry the knot complement admits. As this is a survey talk, we will only go over definitions, main ideas, foundational results, new results, and open questions. We will not assume previous knowledge of knots nor will we explicitly prove any theorems, but rather give the overall main ideas and the tools used in the proofs.