Kapustin and Witten have studied a one-parameter family of topological twists of 4d N=4 super Yang–Mills. They have shown that the categories of boundary conditions on a surface are exactly the categories participating in the geometric Langlands program of Beilinson and Drinfeld. Moreover, S-duality is manifested as a quantum geometric Langlands duality after the topological twist. In this talk I will describe some mathematical formalizations of Hilbert spaces of states on a 3-manifold. I will outline an equivalence between two such possible formalizations: complexified Floer homology of Abouzaid–Manolescu and skein modules. This is a report on work in progress joint with Sam Gunningham.

Zoom: https://harvard.zoom.us/j/94717938264

I will discuss the conditions under which Higgs and confining regimes in gauge theories with fundamental representation matter fields can be sharply distinguished. It is widely believed that these regimes are smoothly connected unless they are distinguished by the realization of global symmetries. However, I will show that when a U(1) global symmetry is spontaneously broken in both the confining and Higgs regimes, the two phases can be separated by a phase boundary. The phase transition between the two regimes may be detected by a novel topological vortex order parameter. I’ll illustrate these ideas by explicit calculations in gauge theories in three spacetime dimensions, and then explain the generalization to four dimensions. One important implication of our results is that nuclear matter and quark matter are sharply distinct phases of QCD with an approximate SU(3) flavor symmetry.

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

Kontsevich’s recursion, proved in the early 90s, is a recursion formula for the counts of rational holomorphic curves in complex manifolds. For complex fourfolds and sixfolds with a real structure (i.e. a conjugation), signed invariant counts of real rational holomorphic curves were defined by Welschinger in 2003. Solomon interpreted Welschinger’s invariants as holomorphic disk counts in 2006 and proposed Kontsevich-type recursions for them in 2007, along with an outline of a potential approach of proving them. For many symplectic fourfolds and sixfolds, these recursions determine all invariants from basic inputs. We establish Solomon’s recursions by re-interpreting his disk counts as degrees of relatively oriented pseudocycles from moduli spaces of stable real maps and lifting cobordisms from Deligne-Mumford moduli spaces of stable real curves (which is different from Solomon’s approach).

Zoom: https://harvard.zoom.us/j/94717938264

The Eisenstein-Kronecker function is a useful object in number theory and physics. It is a tool for proving rationality for period integrals of cusp forms. It generates integration kernels of elliptic polylogarithms which express Feynman diagrams on the torus. It is the central object in the construction of elliptic R-matrices. We propose a new description in terms of iterated convolutions of the Eisenstein zeta function and discuss its features in the various settings.

Zoom: https://harvard.zoom.us/j/779283357

I will talk about a novel way of constructing (2+1)d topological phases using M-theory.
They emerge as macroscopic world-volume theories of M5-branes wrapped on non-hyperbolic 3-manifolds. After explaining the algorithm of extracting modular structures of the topological phase from topological data of the 3-manifold, I will discuss the possibility of full classification of topological orders via the geometrical construction.

Zoom: https://harvard.zoom.us/j/94717938264

A given compact Lie group, G, admits many left-invariant Riemannian metrics. Typically, they form a finite dimension cone L(G). Up to a multiplicative constant, the Riemannian measure for such metrics is the Haar measure of the group. Because the group is compact, each metric g in L(G) has the property that there exists a constant C(G,g)—called the doubling constant—such that, for any radius r, the volume of the ball of radius 2r is at most C(G,g) times the volume of the ball of radius r. The title of this presentation asks the question: does there exist a constant C(G) such that, for all g in L(G), C(G,g) is bounded above by C(G). Is any compact Lie group uniformly doubling? We conjecture that this is the case. The only cases for which the conjecture is known are Riemannian tori and the group SU(2). The result for U(2) is work in progress. This reports on joint work with Maria Gordina (University of Connecticut) and Nathaniel Eldredge (University of Northern Colorado).

Zoom: https://harvard.zoom.us/j/779283357

We present recent advances in constructions of globally consistent F-theory compactifications, which result in the exact chiral spectrum of the three-family Standard Models. We highlight geometric tools to determine the gauge degrees, including continuous and discrete Abelian symmetries. We also outline techniques that determine the chiral matter spectrum. We present the first examples of such constructions and then turn to a subsequent systematic exploration of the landscape of F-theory three-family Standard Models with a gauge coupling unification. We also outline recent progress toward calculations of Yukawa couplings and calculations of the number of charged-vector matter pairs.

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

Many salient features of elliptic fibrations leave intriguing imprints on the physics of F-theory compactifications. In this talk, we will focus on the Mordell—Weil group of sections. Beyond the well-known relationship to abelian gauge symmetries, we will discuss how the global structure of the gauge group, e.g., the Standard Model [SU(3) x SU(2) x U(1)]/Z6, is encoded in the geometry. By re-interpreting a non-trivial global structure as having gauged 1-form center symmetries, we will further explain how geometric restrictions of having Mordell—Weil torsion can be understood field theoretically asanomalies of 1-form symmetries. A comparison between these anomalies and elliptic K3 surfaces points towards String Universality for 1-form center symmetries in 8d.

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

On August 24-25, 2020 the CMSA will be hosting our sixth annual Conference on Big Data. The Conference will feature many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.

Registration for this event is required, details on how to join the webinar will be sent to registered participants before the event. Register here using this Google form.

For the schedule details and additional information, please visit https://cmsa.fas.harvard.edu/2020-big-data-conference/.

Organizers: 

Shing-Tung Yau, William Caspar Graustein Professor of Mathematics, Harvard University

Scott Duke Kominers, MBA Class of 1960 Associate Professor, Harvard Business

Horng-Tzer Yau, Professor of Mathematics, Harvard University

Sergiy Verstyuk, CMSA, Harvard University

Speakers:

Sanjeev Arora, Princeton University

Juan Camilo Castillo, University of Pennsylvania

Joseph Dexter, Dartmouth College

Nicole Immorlica, Microsoft

Amin Saberi, Stanford University

Vira Semenova, University of California, Berkeley

Varda Shalev, Tel Aviv University

Information about last year’s conference can be found here: cmsa.fas.harvard.edu/2019-big-data/

On August 24-25, 2020 the CMSA will be hosting our sixth annual Conference on Big Data. The Conference will feature many speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.

Registration for this event is required, details on how to join the webinar will be sent to registered participants before the event. Register here using this Google form.

For the schedule details and additional information, please visit https://cmsa.fas.harvard.edu/2020-big-data-conference/.

Organizers: 

Shing-Tung Yau, William Caspar Graustein Professor of Mathematics, Harvard University

Scott Duke Kominers, MBA Class of 1960 Associate Professor, Harvard Business

Horng-Tzer Yau, Professor of Mathematics, Harvard University

Sergiy Verstyuk, CMSA, Harvard University

Speakers:

Sanjeev Arora, Princeton University

Juan Camilo Castillo, University of Pennsylvania

Joseph Dexter, Dartmouth College

Nicole Immorlica, Microsoft

Amin Saberi, Stanford University

Vira Semenova, University of California, Berkeley

Varda Shalev, Tel Aviv University

Information about last year’s conference can be found here: cmsa.fas.harvard.edu/2019-big-data/

I will describe how the structure of supersymmetric boundary correlators in 4d N=4 SYM can be encoded in a class of associative algebras equipped with twisted traces. In the case of interfaces, this yields a new connection to integrability.

Zoom: https://harvard.zoom.us/j/94717938264

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

Gauge theories often admit discrete theta angles. Theories with different discrete theta angles can be constructed by gauging a global symmetry in a parent theory with different additional symmetry protected topological (SPT) phases. I will discuss how the global symmetries of these theories and their ‘t Hooft anomalies are controlled by the SPT phases. I will then present examples in various dimensions that exhibit two-group symmetries and non-invertible symmetries. The talk is based on https://arxiv.org/abs/2007.05915 in collaboration with Po-Shen Hsin.

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