Calendar
- 01March 1, 2022
The Rank of Syzygies
I will explain a notion of rank for the relations amongst the equations of a projective variety, generalizing the classical notion of rank of a quadric. I will then explain a result telling us that, for a general canonical curve, all syzygies are linear combinations of syzygies of minimal rank. This is a sweeping generalization of old results of Andreotti-Mayer, Harris-Arbarello and Green, which tell us that canonical curves are defined by quadrics of rank four. As a special case, this perspective gives us a new, and simpler, proof of Green’s conjecture for general curves.
https://harvard.zoom.us/j/91082896381?pwd=TmJCanRPVlM4VGR2L1RzZGVGbHRVQT09
- 01March 1, 2022
Joint Harvard-CUHK-YMSC Differential Geometry: Tropical Lagrangian multi-sections and locally free sheaves
The SYZ proposal suggests that mirror symmetry is T-duality. It is a folklore that locally free sheaves are mirror to a Lagrangian multi-section of the SYZ fibration. In this talk, I will introduce the notion of tropical Lagrangian multi-sections and discuss how to obtain from such object to a class of locally free sheaves on the log Calabi-Yau spaces that Gross-Siebert have considered. I will also discuss a joint work with Kwokwai Chan and Ziming Ma, where we proved the smoothability of a class of locally free sheaves on some log Calabi-Yau surfaces by using combinatorial data obtained from tropical Lagrangian multi-sections.
Zoom Link: https://cuhk.zoom.us/j/95804207557
Meeting ID: 958 0420 7557; Passcode: 20220302
- 01March 1, 2022
CMSA Combinatorics, Physics and Probability: Rational Polypols
Eugene Wachspress introduced polypols as real bounded semialgebraic sets in the plane that generalize polygons. He aimed to generalize barycentric coordinates from triangles to arbitrary polygons and further to polypols. For this, he defined the adjoint curve of a rational polypol. In the study of scattering amplitudes in physics, positive geometries are real semialgebraic sets together with a rational canonical form. We combine these two worlds by providing an explicit formula for the canonical form of a rational polypol in terms of defining equations of the adjoint curve and the facets of the polypol. For the special case of polygons, we show that the adjoint curve is hyperbolic and provide an explicit description of its nested ovals. Finally, we discuss the map that associates the adjoint curve to a given rational polypol, in particular the cases where this map is finite. For instance, using monodromy we find that a general quartic curve is the adjoint of 864 heptagons.This talk is based on joint work with R. Piene, K. Ranestad, F. Rydell, B. Shapiro, R. Sinn, M.-S. Sorea, and S. Telen.https://harvard.zoom.us/j/91799784675?pwd=MS9LV25DWk9RcmJoRVM0K3RGWkFRdz09
- 01March 1, 2022
Rigorous results about entropy in QFT
We will present some rigorous results about entropy and relative entropy in QFT, motivated in part by recent physicists’ work which however depends on heuristic arguments such as introducing cut off and using path integrals.
https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09
- 01March 1, 2022
CMSA Algebraic Geometry in String Theory: Virtual localization for Artin stacks
This is a report about work in progress with: Adeel Khan, Aloysha Latyntsev, Hyeonjun Park and Charanya Ravi. We will describe a virtual Atiyah-Bott formula for Artin stacks. In the Deligne-Mumford case our methods allow us to remove the global resolution hypothesis for the virtual normal bundle.
https://harvard.zoom.us/j/97335783449?pwd=S3U0eVdyODFEdzNaRXVEUTF3R3NwZz09