Calendar

In the 90s’, Witten gave a physical derivation of an isomorphism between the Verlinde algebra of $GL(n)$ of level $l$ and the quantum cohomology ring of the Grassmannian $\text{Gr}(n,n+l)$. In the joint work arXiv:1811.01377 with Yongbin Ruan, we proposed a K-theoretic generalization of Witten’s work by relating the $\text{GL}_{n}$ Verlinde numbers to the level $l$ quantum K-invariants of the Grassmannian $\text{Gr}(n,n+l)$, and refer to it as the Verlinde/Grassmannian correspondence.

The correspondence was formulated precisely in the aforementioned paper, and we proved the rank 2 case (n=2) there. In this talk, I will discuss the proof for arbitrary rank. A new technical ingredient is the virtual nonabelian localization formula developed by Daniel Halpern-Leistner.  At the end of the talk, I will describe some applications of this correspondence.

Zoom: https://harvard.zoom.us/j/93926041523

TITLE: Kunihiko Kodaira and complex manifolds.

ABSTRACT: Kodaira’s motivation was to generalize the theory of Riemann surfaces in Weyl’s book to higher dimensions.  After quickly recalling the chronology of Kodaira, I will review some of Kodaira’s works in three sections on topics of harmonic analysis, deformation theory and compact complex surfaces.  Each topic corresponds to a volume of Kodaira’s collected works in three volumes, of which I will cover only tiny parts.

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

For more information, please visit the event page.

Register here to attend.

In this talk, I will introduce the Gopakumar-Vafa(GV) invariant and show one calculation on the nonreduced cycle. The GV invariant is an integral invariant predicted by physicists that counts the number of curves inside a given Calabi-Yau threefold. The definition has been conjectured by Maulik-Toda in 2016 in terms of perverse sheaf. I’ll use this definition on the total space of the canonical bundle of P2 and compute the associated invariants. This verifies a physical formula based on the work of Katz-Klemm-Vafa in 1997.

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

Inspired by fractional quantum Hall physics and Tannaka-Krein duality, it is conjectured that every modular tensor category (MTC) or (2+1)-topological quantum field theory (TQFT) can be realized as the representation category of a vertex operator algebra (VOA) or chiral conformal field theory (CFT).  It is obviously true for quantum group/WZW MTCs, but it is not known for MTCs appeared in subfactors such as the famous double Haagerup.  After some general discussion, I will focus on pointed MTCs or so-called abelian anyon models.  While all abelian anyon models can be realized by lattice VOAs, it is not clear whether or not they can be realized by non-lattice VOAs.  The trivial MTC is realized by the Monster moonshine module, which is a non-lattice realization.  I will provide evidence that this might be true for all abelian anyon models.  The talk is partially based on a joint work with Liang Wang: https://arxiv.org/abs/2004.12048 

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

Deep learning has revolutionized our ability to generate novel images and 3D shapes. Typically neural networks are trained to map a high-dimensional latent code to full realistic samples. In this talk, I will present two recent works focusing on generation of handwritten text and 3D shapes. In these works, we take a different approach and generate image and shape samples using a more granular part-based decomposition, demonstrating that the whole is not necessarily “greater than the sum of its parts”. I will also discuss how our generation by decomposition approach allows for a semantic manipulation of 3D shapes and improved handwritten text recognition performance.

Zoom: https://harvard.zoom.us/j/98231541450

In this talk, I will describe how symmetry can enrich strong-randomness quantum critical points and phases, and lead to robust topological edge modes coexisting with critical bulk fluctuations. Our approach provides a systematic construction of strongly disordered gapless topological phases. Using real space renormalization group techniques, I will discuss the boundary and bulk critical behavior of symmetry-enriched random quantum spin chains, and argue that nonlocal observables and boundary critical behavior are controlled by new renormalization group fixed points. I will also discuss the interplay between disorder, quantum criticality and topology in higher dimensions using disordered gauge theories.

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

Mazur conjectured, after Faltings’s proof of the Mordell conjecture, that the number of rational points on a curve of genus g at least 2 defined over a number field of degree d is bounded in terms of g, d and the Mordell-Weil rank. In particular the height of the curve is not involved. In this talk I will explain how to prove this conjecture and some generalizations. I will focus on how functional transcendence and unlikely intersections are applied in the proof. If time permits, I will talk about how the dependence on d can be furthermore removed if we moreover assume the relative Bogomolov conjecture. This is joint work with Vesselin Dimitrov and Philipp Habegger.

Zoom: https://harvard.zoom.us/j/96767001802

Password: The order of the permutation group on 9 elements.

Graph representation learning has emerged as a dominant paradigm for networked data. Still, prevailing methods require abundant label information and focus on representations of nodes, edges, or entire graphs. While graph-level representations provide overarching views of graphs, they do so at the loss of finer local structure. In contrast, node-level representations preserve local topological structures, potentially to the detriment of the big picture. In this talk, I will discuss how subgraph representations are critical to advance today’s methods. First, I will outline Sub-GNN, the first subgraph neural network to learn disentangled subgraph representations. Second, I will describe G-Meta, a novel meta-learning approach for graphs. G-Meta uses subgraphs to adapt to a new task using only a handful of nodes or edges. G-Meta is theoretically justified, and remarkably, can learn in most challenging, few-shot settings that require generalization to completely new graphs and never-before-seen labels. Finally, I will discuss applications in biology and medicine. The new methods have enabled the repurposing of drugs for new diseases, including COVID-19, where predictions were experimentally verified in the wet laboratory. Further, the methods identified drug combinations safer for patients than previous treatments and provided accurate predictions that can be interpreted meaningfully.

Zoom: https://harvard.zoom.us/j/91458092166

Orbifolds are ubiquitous in physics, not just explicitly in CFT, but going undercover with names like Kramers-Wannier duality, Jordan-Wigner transformation, or GSO projection. All of these names describe ways to “topologically manipulate” a theory, transforming it to a new one, but leaving the local dynamics unchanged. In my talk, I will answer the question: given some (1+1)d QFT, how many new theories can we produce by topological manipulations? To do so, I will outline the relationship between these manipulations and (2+1)d Dijkgraaf-Witten TFTs, and illustrate both the conceptual and computational power of the relationship. Ideas from high-energy, condensed-matter, and pure math will show up in one form or another. Based on work with Davide Gaiotto [arxiv:2008.05960].

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