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for events the day of Monday, March 1, 2021.

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Monday, March 1, 2021

3:00 pm in Zoom,Monday, March 1, 2021

Generalized Gauge Theory: Where Logic Meets Homotopy Theory

Joseph Rennie (UIUC)

Abstract: Higher categories admit a notion of internal groupoids which Nikolaus et. al have shown yield a nice theory of principle bundles in any higher topos. An example of the practical use of this can be seen work of Freed-Hopkins where they define a higher topos of “generalized spaces” which then admits a universal bundle with connection. In an attempt to extend the results of Nikolaus to more kinds of categories, we inevitably end up working with the same kinds of structures as logicians. Namely, with pretoposes and logical functors, as opposed to the more homotopy theoretic grothendieck toposes and geometric morphisms. The goal of this talk will be to demystify this deep connection between model theory (in the logician’s sense) and homotopy theory. This talk will mostly operate at a conceptual level to more insightfully navigate the fact that many results that we want don’t yet have analogs proven in the higher-categorical setting, and the fact that the lower setting doesn’t quite have as nice of a picture. I assume no background in logic, and a vague awareness of the use (conceptually) of toposes in homotopy theory. Please email vb8 at illinois dot edu for the zoom details.

3:00 pm in zoom,Monday, March 1, 2021

Poisson manifolds of strong compact type over 2-tori

Luka Zwann (UIUC)

Abstract: An integrable Poisson manifold is said to be of strong compact type if the source 1-connected groupoid integrating it is compact. A trivial class of such manifolds is that of compact symplectic manifolds with finite fundamental group, but beyond that finding examples is difficult. The first non-trivial example was given by D. Martínez Torres in 2014. The construction there is inspired by D. Kotschick’s construction of a free symplectic circle action with contractible orbits. In this talk I will go over the original construction, recalling the relevant results on Poisson manifolds of compact types as well as the geometry of the moduli spaces of K3 surfaces, and then modify the construction to obtain more examples. In the end, we will have for every strongly integral affine 2-torus (i.e. integral affine 2-torus with integral translational part) a Poisson manifold of strong compact type having said torus as its leaf space.

5:00 pm in Altgeld Hall,Monday, March 1, 2021

Subriemanian Geodesic Flow

Advith Agovind

Abstract: Part 2 subriemanian geometry