Jarod Alper: Colloquium: Evolution of Moduli
In the rich landscape of algebraic varieties, moduli spaces stand out as some of the most enchanting varieties, capturing the imagination of algebraic geometers with their profound elegance and deep connections to other branches of mathematics. Moduli, the plural of modulus, is a term coined by Riemann to describe a space whose points afford an alternative description as certain classes of geometric objects. We will trace the origins of moduli spaces through the discoveries of Riemann, Hilbert, Grothendieck, Mumford, and Deligne, as a means to explain many of the fundamental concepts of moduli spaces. We will then survey how the foundations of moduli theory have further evolved over the last 50 years.
Rachel Pries: Four perspectives on rational points on one Shimura curve
In this talk, I describe the points on a special Shimura curve from four perspectives: a family of cyclic covers of the projective line, a hyperbolic triangle, quadratic forms, and a unitary similitude group. By leveraging these four perspectives, we generalize a result of Elkies about supersingular reduction of elliptic curves to the case of genus four curves having an automorphism of order five. This is joint work with Wanlin Li, Elena Mantovan, and Yunqing Tang.
Dan Kaplan: TBA
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Phil Engel: TBA
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Sheel Ganatra: TBA
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Kristin DeVleming: TBA
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Joshua Mundinger: Hochschild homology of algebraic varieties in characteristic p
Hochschild homology is an invariant of an associative algebra. When the associative ring is commutative, the Hochschild-Kostant-Rosenberg theorem gives a formula for Hochschild homology in terms of differential forms. This formula extends to the Hochschild-Kostant-Rosenberg decomposition for complex algebraic varieties. In this talk, we quantitatively explain the failure of this decomposition in positive characteristic.
Xinwen Zhu: TBA
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