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Topological field theories and holomorphic field theories have each had a substantial impact in both physics and mathematics, so it is natural to consider theories that are hybrids of the two, which we call topological-holomorphic and denote as THFTs. Examples include Kapustin’s twist of N=2, D=4 supersymmetric Yang-Mills theory and Costello’s 4-dimensional Chern-Simons theory. In this talk about joint work with Rabinovich and Williams, I will define THFTs, describe several examples, and then explain how to quantize them rigorously and explicitly, by building on techniques of Si Li. Time permitting, I will indicate how these results offer a novel perspective on the Gaudin model via 3-dimensional field theories.

Zoom: https://harvard.zoom.us/j/91780604388?pwd=d3BqazFwbDZLQng0cEREclFqWkN4UT09

In this talk, I will first briefly review the Deligne-Mostow theory of moduli spaces of weighted points on the projective line, and a construction of ball quotients from periods of (possibly singular) K3 surfaces with non-symplectic group action. Then I will discuss how these two constructions can be unified for some examples. I will focus on a new case about a 6-dimensional family of K3 surfaces with D4-singularity. This is a joint work with Yiming Zhong.

Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

TITLE: Moment maps and the Yang-Mills functional

ABSTRACT: In the early 1980s Michael Atiyah and Raoul Bott wrote two influential papers, ‘The Yang-Mills equations over Riemann surfaces’ and ‘The moment map and equivariant cohomology’, bringing together ideas ranging from algebraic and symplectic geometry through algebraic topology to mathematical physics and number theory. The aim of this talk is to explain their key insights and some of the new directions towards which these papers led.

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

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We consider spin-S quantum spin chains with a family of O(2S+1)-invariant nearest-neighbor interactions and discuss the ground state phase diagram of this family of models. Using a graphical representation for the partition function, we give a proof of dimerization for an open region in the phase diagram, for all sufficiently large values of S. (Joint work with Jakob Bjoernberg, Peter Muehlbacher, and Daniel Ueltschi).

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

K-stability is an algebraic condition that characterizes the existence of K\”ahler-Einstein metrics on Fano varieties. Recently there has been a lot of work on the construction of the K-moduli space, i.e. a good moduli space parametrizing K-polystable Fano varieties.  Motivated by results in differential geometry, it is conjectured that this K-moduli space is proper and projective. In this talk, I’ll discuss some recent progress in birational geometry that leads to a full solution of this conjecture. Based on joint work with Yuchen Liu and Chenyang Xu.

Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09

I will use Walker-Wang models to demonstrate the connection between 1-form symmetry-protected topological phases and topological measurement-based quantum computation. I will describe the classification of these phases in terms of symmetry domain walls and how these lead to “anomalous” 1-form symmetry actions on the boundary. I will also demonstrate that when the symmetries are strictly enforced these phases persist to finite temperatures and use this to explain symmetry-protected self-correction properties of the boundary topological phases.

Zoom: https://harvard.zoom.us/j/977347126

The Rogers-Ramanujan identity is an equality between a certain “q-series” (given as an infinite sum) and a certain modular form (given as an infinite product). Motivated by ideas from physics, Nahm formulated a necessary condition for when such q-hypergeometric series were modular. Perhaps surprisingly, this turns out to be related to algebraic K-theory. We discuss a proof of this conjecture. This is joint work with Stavros Garoufalidis and Don Zagier.

Zoom: https://harvard.zoom.us/j/99334398740

Password: The order of the permutation group on 9 elements

We are interested in the minimization problem of a functional in which the perimeter is competing with a nonlocal singular term comparable to a fractional perimeter, with volume constraint. We prove that minimizers exist and are radially symmetric for small mass, while minimizers cannot be radially symmetric for large mass. For large mass, we prove that the minimizing sequences either split into smaller sets that drift to infinity or develop fingers of a prescribed width. We connect these two alternatives to a related minimization problem for the optimal constant in a classical interpolation inequality.

Zoom: https://harvard.zoom.us/j/98248914765?pwd=Q01tRTVWTVBGT0lXek40VzdxdVVPQT09

(Password: 419419)

We know that different systems with the same unbroken global symmetry can nevertheless be in distinct phases of matter.  These different “symmetry-protected topological” phases are characterized by protected (gapless) surface states.  After reviewing this physics in interacting systems with global symmetries, I will describe how a different class of symmetries known as subsystem symmetries, which are neither local nor global, can also lead to protected gapless boundaries.  I will discuss how some of these subsystem-symmetry protected phases are related (though not equivalent) to interacting higher-order topological insulators, with protected gapless modes along corners or hinges in higher dimensional systems.

Zoom: https://harvard.zoom.us/j/977347126

Speaker: Kenz Kallal (Harvard)
Title: The Arthur–Selberg trace formula and some applications to arithmetic statistics

Speaker: Lux Zhao (Harvard)
Title: The history and mathematics of the axiom of choice

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