2023 Spring Talks

Title: Symmetry-Resolved Density of States 

• • Date: January 18, 2023

  • Speaker:  Hirosi Ooguri  (Caltech, Kavli IPMU)

  • Abstract: It is well-known that the density of states of a unitary quantum field theory has a universal behavior at high energy. In two dimensions, it is known as the Cardy formula. When the theory has a global symmetry, it is interesting to find out how the Hilbert space is decomposed into irreducible representation of the symmetry. In this talk, I will derive universal formulas regarding the decomposition of states at high energy. The formulae are applicable to any unitary quantum field theory in any spacetime dimensions. When the AdS/CFT correspondence applies, our formulae agree with the entropy formula for the Kerr and Reissner-Nordstrom black holes. We also settle some question on the Reissner-Nordstrom black hole. 

Title:  Castelnuovo Bound and Higher Genus Gromov-Witten Invariants of Quintic 3-fold 

• • Date: February 1, 2023

  • Speaker:  Yongbin Ruan (IASM at Zhejiang University)

  • Abstract: One of most difficult problems in geometry and physics is to compute higher genus Gromov-Witten (GW) invariants of compact Calabi-Yau 3-folds such as quintic 3-folds. The effort to solve the problem leads to the inventions of several subjects such as mirror symmetry and FJRW theory. Almost twenty years ago, physicist Albrecht Klemm and his group shocked the community to produce explicit predications of higher genus GW invariants up to $g=51$! Their calculation is based on five mathematical conjectures, four BCOV conjectures from B-model and one A-model conjecture called Castelnuovo bound. Several years ago, a spectacular progress has been made to solve four BCOV conjectures. In this talk, I will report the solution of Castelnuovo bound conjecture. This is a joint work with Zhiyu Liu.

Title:  Infra-red phases of class R theories 

• • Date: February 15, 2023

  • Speaker:  Dongmin Gang (Seoul National University)

  • Abstract: Motivated by the physics of  M5-branes in M-theory, there is a class of 3D gauge theories (called class R theories) labeled by 3-manifolds. 

After briefly reviewing the field theoretic construction, I will explain how to understand various infra-red (macroscopic) behavior of the 3D gauge theories from basic topological properties of 3-manifolds. 

Title:  What can the working (pure) mathematician expect from deep learning? 

• • Date: March 8, 2023

  • Speaker:  Geordie Williamson (Sydney Mathematical Research Institute and University of Sydney)

  • Abstract:  Deep learning (the training of deep neural nets) is a simple idea, which has had many extraordinary applications throughout industry and science over the last decade. In mathematics the impact has so-far been modest at best. I will discuss a few instances where it has proved useful, and led to interesting (pure) mathematics. I will also discuss what can be learned from these examples, and try to guess an answer to the question in the title. I will also reflect on my experience as a pure mathematician interacting with deep learning. 

Title:  Holography with End-of-the-World Branes and Quantum Entanglement 

• • Date: March 22, 2023

  • Speaker:  Tadashi Takayanagi (YITP, Kyoto University)

  • Abstract:  Holography relates quantum many-body systems to gravitational theories. Quantum entanglement plays a key role to explain how the spacetime geometries in gravity emerge from quantum systems. A new class of holography can be found by introducing so called end-of-the-world branes and has been actively studied recently. Such holographic models describe quantum systems with boundaries, such as boundary conformal field theories (BCFTs), where the boundary dynamics is described by gravitational degrees of freedom localized on the branes. Considerations of quantum entanglement in these setups help us to have deep insights on the black hole information problem. At the same time, they can also be used to analyze quantum information aspects of non-equilibrium processes. In this talk, I will explain these developments. 

Title:  What is a holomorphic quantum modular form? 

• • Date: April 5, 2023

  • Speaker: Stavros Garoufalidis (Southern University of Science and Technology / Max Planck Institute for Mathematics )

  • Abstract: Holomorphic quantum modular forms arose naturally from asymptotic questions of quantum invariants in dimension three. HQMFs, along with resurgence and p-adic analytic continuation are three known realizations of functions that appear in mathematical physics. I will give a historical introduction on the subject, drawn from many years of joint work with Don Zagier, and illustrated with examples of the simplest hyperbolic 3-manifolds. 

Title:  Orthosymplectic bow varieties 

• • Date: April 19, 2023

  • Speaker: Hiraku Nakajima (Kavli IPMU, the University of Tokyo)

  • Abstract: Bow varieties are introduced by Cherkis as the analog of ADHMN description of instantons on multi-Taub NUT spaces for unitary groups. They are closely related to quiver varieties and Coulomb branches of quiver gauge theories of affine type A. They are also useful to understand brane configurations that appeared in Hanany-Witten's work. In an on-going joint project with Finkelberg and Hanany, we introduce their variants, called orthosymplectic bow varieties. In this talk, I will give introduction to orthosymplectic bow varieties. 

Title:  Twisted Real quasi-elliptic cohomology 

• • Date: May 3, 2023

  • Speaker: Zhen Huan (Center for Mathematical Sciences, Huazhong University of Science and Technology)

  • Abstract: Quasi-elliptic cohomology is closely related to Tate K-theory. It is constructed as an object both reflecting the geometric nature of elliptic curves and more practicable to study than most elliptic cohomology theories.  It can be interpreted by orbifold loop spaces and expressed in terms of equivariant K-theories. We formulate the complete power operation of this theory. Applying that we prove the finite subgroups of Tate curve can be classified by the Tate K-theory of symmetric groups modulo a certain transfer ideal.  In this talk we construct twisted Real quasi-elliptic cohomology as the twisted KR-theory of loop groupoids. The theory systematically incorporates loop rotation and reflection. After establishing basic properties of the theory, we construct Real analogues of the string power operation of quasi-elliptic cohomology. We also explore the relation of the theory to the Tate curve. This is joint work with Matthew Spong and Matthew Young.

Title:  On the statistics of indecomposable components in large tensor products of representations Lie algebras and quantum groups.

• • Date: May 17, 2023

  • Speaker: Nicolai Reshetikhin (YMSC, Tsinghua University & University of California, Berkeley)

  • Abstract:  Let $V$ is a finite dimensional representation of a simple Lie group. Characters, being evaluated on positive elements, define statistics on irreducible components of the tensor power $V^{\otimes N}$ for each $N=1,2,\dots$. In the talk I will explain how this distribution behave in the limit $N\to \infty$. If time permit, I will explain a similar problem for representations of quantum groups at roots of unity where certain characters define statistics on indecomposable components of large tensor products.