Algebra and Number Theory Seminar Winter 2019
Friday January 11th, 2019
Ryan Drury, University of California Santa Cruz
Steenrod operations on algebraic De Rham cohomology
Let X be a smooth projective scheme over a field k.In the case k = C, Atiyah and Hirzebruch used Steenrod operations to find the first counter examples to the Integral Hodge Conjecture. For char(k) = p,we construct Steenrod operations on the algebraic De Rham cohomology of X. It is likely one can also define compatible Steenrod operations on the E_1 page of the Hodge to De Rham spectral sequence.
Friday January 18th, 2019
TBA
Friday January 25th, 2019
TBA
Friday February 1st, 2019
TBA
Friday February 8th, 2019
TBA
Friday February 15th, 2019
Daniel Bragg, University of California Berkeley
Derived equivalences of twisted supersingular K3 surfaces
Two smooth projective varieties with equivalent derived categories of coherent sheaves are said to be Fourier-Mukai partners. In general, it is a difficult and interesting question to determine the set of Fourier-Mukai partners of a given variety, or to tell whether two given varieties are Fourier-Mukai partners. For K3 surfaces over the complex numbers, a satisfactory answer to both questions was found by Orlov, who proved a derived Torelli theorem characterizing derived equivalences in terms of Hodge theory. Much less is known in positive characteristic. We will focus our attention on supersingular K3 surfaces, which are a special class of K3 surfaces in positive characteristic. Using crystalline cohomology, we will formulate a derived Torelli theorem for (twisted) supersingular K3 surfaces, giving a positive characteristic analog of Orlov's result.
Friday February 22nd, 2019
Theo Johnson-Freyd, Perimeter Institute
Galois actions on VOA gauge anomalies
Symmetries of a physical system can be "anomalous"; when they are, there is a "gauge anomaly" living in the cohomology of the group of symmetries. I will explain the definition of this anomaly in the case of quantum mechanics (also called Azumaya algebra) and holomorphic conformal field theory (also called holomorphic vertex operator algebra). I will then explain an anomaly of the VOA case: if a finite group $G$ acts on a holomorphic VOA $V$ over $\CC$, then the anomaly lives in $H^3(G; \CC^\times)$, but a Galois automorphism $\gamma$ does not act simply on the coefficients by $\alpha \mapsto \gamma(\alpha)$, but rather by $\alpha \mapsto \gamma^2(\alpha)$. This explains many appearances of the number 24 in moonshine and suggests many questions relating VOAs to K-theory.
Friday March 1st, 2019
TBA
Friday March 8th, 2019
TBA
