# 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**