Calendar

We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomology with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields. This is a joint work with J. Cao and R. Villaflor. (arXiv:2109.00587)

https://harvard.zoom.us/j/98781914555?pwd=bmVzZGdlRThyUDREMExab20ybmg1Zz09

Let $M(\beta,\chi)$ be the moduli space of one-dimensional semi-stable sheaves on a del Pezzo surface $S$, supported on an ample curve class $\beta$ and with Euler-characteristic $\chi$. Working over a non-archimedean local field $F$, we define a natural measure on the $F$-points of $M(\beta,\chi)$. We prove that the integral of a certain gerbe on $M(\beta,\chi)$ with respect to this measure is independent of $\chi$ if $S$ is toric. A recent result of Maulik-Shen then implies that these integrals compute the Donaldson-Thomas invariants of $M(\beta,\chi)$. A similar result holds for suitably twisted Higgs bundles. This is joint work with Francesca Carocci and Giulio Orecchia.

We will explain how to model the Hitchin integrable system on a certain Lagrangian upward flow as the spectrum of equivariant cohomology of a Grassmannian.

Zoom link: https://harvard.zoom.us/j/95767170359 (Password: cmsa)

For any quantum system invariant under gauging a higher-form global symmetry, we construct a non-invertible topological defect by gauging in only half of spacetime. This generalizes the Kramers-Wannier duality line in 1+1 dimensions to higher spacetime dimensions. We focus on the case of a one-form symmetry in 3+1 dimensions, and determine the fusion rule. From modular invariance and a direct analysis of one-form symmetry protected topological phases, we show that the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase. By further assuming time-reversal symmetry, we find that the presence of certain duality defects implies that the low-energy phase has to be gapless unless the one-form symmetry is spontaneously broken. We give an explicit realization of this duality defect in the free Maxwell theory where the duality defect is realized by a Chern-Simons coupling between the gauge fields from the two sides.

Recent technological developments now allow for the accurate, high resolution measurement of millions of human genomes. I will describe some of the challenges and opportunities of this postgenomic era, and how these technologies stand poised to revolutionize our understanding of human biology and human health.

Website: https://people.math.harvard.edu/~ana/ons/

Topological defects provide a modern perspective on symmetries in quantum field theory. They generalize the familiar invertible symmetries described by groups to non-invertible symmetries described by fusion categories. Such generalized symmetries are ubiquitous in quantum field theory and provide new constraints on renormalization group flows and the IR phase diagram. In this talk I’ll review some recent progress in identifying and understanding fusion category symmetries in 1+1d conformal field theories. Time permitting, I’ll also comment on higher dimensional generalizations.
—–
Subscribe to Harvard CMSA seminar videos (more to be uploaded):
https://www.youtube.com/channel/UCBmPO-OK1sa8T1oX_9aVhAg/playlists
https://www.youtube.com/channel/UCM06KiUOw1vRrmvD8U274Ww
—–
Subscribe to Harvard CMSA seminar videos (more to be uploaded):
https://www.youtube.com/channel/UCBmPO-OK1sa8T1oX_9aVhAg/playlists
https://www.youtube.com/channel/UCM06KiUOw1vRrmvD8U274Ww

I will speak about recent joint work with Bell, Diller, and Jonsson in which we refute a conjecture of Bellon-Viallet by constructing (mostly) explicit examples of birational maps of projective 3-space with transcendental dynamical degree, also known as algebraic entropy.  The set of possible dynamical degrees for birational maps of projective space is known to be a countable set, with nearly all examples given by eigenvalues of integer matrices (and thus algebraic), yet we demonstrate the existence of infinitely many transcendental values in this set.  The proof builds on previous work of Bell-Diller-Jonsson, combining the study of monomial maps of toric varieties with classical techniques from Diophantine approximation.