Professor Shing-Tung Yau (Harvard University / Tsinghua University)
The Gravity of Math
Special Colloquium - Wednesday, January 10, 17:00-18:30
A Talk by Shing-Tung Yau -- based on the book, The Gravity of Math: How Geometry Rules the Universe by Steve Nadis and Shing-Tung Yau (Basic Books, April 16, 2024)
On November 25, 1915, Albert Einstein published the field equations of general relativity, which afforded a totally new picture of gravity: Rather than being a mysterious force that operates between each and every massive object, he told the world, gravity is instead a manifestation of the curvature of space-time caused by the presence of massive objects. Gravity, in other words, springs from geometry itself—geometry being the cause and gravity merely the effect.
Robbert Dijkgraaf, director of the Institute for Advanced Study, called the formulation of this gravitational theory “perhaps the greatest achievement of a single human mind.” While Einstein’s accomplishment surely ranks as one of the greatest theoretical breakthroughs in the history of science, his creation was not entirely the product of “a single human mind.” The theory Einstein unveiled in 1915 rested on complicated mathematics that he needed substantial help in understanding. He received critical assistance in this effort from Marcel Grossmann, David Hilbert, and Tullio Levi-Civita, among others. He derived the first draft of his celebrated equations with Grossmann, who introduced Einstein to Riemannian geometry, upon which his theory was built.
Mathematicians played a pivotal role in exploring the implications of general relativity, gaining insights on phenomena like black holes, gravitational waves, and the Big Bang—in some cases many decades before any experimental evidence became available. Mathematics has also been essential in the quest to understand concepts such as the positivity of mass and even how such basic notions as mass and angular momentum ought to be defined in general relativity.
This talk, in broad terms, will explore the ways in which our views of the universe have been shaped and informed by mathematics, particularly when it comes to gravity.
Professor Richard Schoen (University of California, Irvine)
How Minimal Hypersurface and MOTS Singularities Affect Relativity Theorems
Plenary Talk 1 - Wednesday, January 10, 15:30-16:30
Some theorems in relativity work directly only in low dimensions because of the possibility
of singularities in area minimizing hypersurfaces and MOTS. These include both Riemannian
and spacetime positive mass theorems as well as the Riemannian Penrose inequality. In some
cases the singularities are relatively easy to work around and in others they present significant problems to overcome. This will be a general lecture which will attempt to give perspective and more clarity to this issue.
Professor Mu-Tao Wang (Columbia University)
Angular Momentum in General Relativity
Plenary Talk 2 - Thursday, January 11, 17:00-18:00
Two black holes rotate about each other and eventually merge into a single black hole. How does one measure the angular momentum carried away by gravitational radiation during this merger? This has been a subtle issue since the 1960’s due to the existence of "supertranslation ambiguity": the angular momentums recorded by two observers of the same system may not be the same. In this talk, I shall describe how the mathematical theory of quasilocal mass and optimal isometric embedding identifies a new definition of angular momentum that is free of any supertranslation ambiguity. In addition, some recent development of the cross-section continuity of the angular momentum definition will also be discussed. This is based on joint work with Po-Ning Chen, Jordan Keller, Daniel Paraizo, Robert Wald, Ye-Kai Wang, and Shing-Tung Yau.
Professor Sergei Gukov (California Institute of Technology)
Topological Strings, the Smooth Poincare Conjecture in d=4, and my Work with Professor Yau
Plenary Talk 3 - Friday, January 12, 10:50-11:50
Over twenty years ago, I had the honor of having Professor Yau as my postdoctoral mentor. Since then, he has imparted numerous valuable lessons to me and shared profound insights across different fields of mathematics. His guidance has been instrumental in inspiring me to forge novel connections, especially between geometry and topology, as in his renowned proof of the Calabi conjecture. In this talk, which partially draws upon our recent collaborative work with Professor Yau and Artan Sheshmani, I will concentrate on exploring a new intersection between enumerative geometry and low-dimensional topology.
Short Talks
Professor Rak-Kyeong Seong (UNIST)
From Calabi-Yau Manifolds to Machine Learning
Short Talk 1 - Wednesday, January 10, 14:10-14:40
This talk will give a brief overview of how work with Professor Yau on minimum volumes corresponding to toric Calabi-Yau manifolds led to the introduction of machine learning techniques at the interface of mathematics and string theory in 2017. The talk will then discuss how our current applications of machine learning techniques, combined with Calabi-Yau mirror symmetry and tropical geometry, enhance our understanding of the phase structure of supersymmetric gauge theories realized in string theory.
Professor Tristan Collins (Massachusetts Institute of Technology)
Complete Calabi-Yau Metrics
Short Talk 2 - Wednesday, January 10, 14:40-15:10
Professor Chin-Lung Wang (National Taiwan University)
A Complete Solution to Lamé Equations with Finite Monodromy
Short Talk 3 - Thursday, January 11, 10:00-10:30
It is a well known result of Beuker and Waall that the classical Lamé equation w'' = (n(n + 1) P(z) + B) on a torus has finite monodromy only if the parameter n belong to a finite set of rational numbers ni mod 1 . By combining the geometry of spherical tori and the combinatorics of dessins d'enfants, we give an exact counting formula for every given n = ni mod 1 and describe the construction of each solution. This is a joint work with You-Cheng Chou and Po-Sheng Wu.
Professor Zhouping Xin (Chinese University of Hong Kong)
On Some Free Boundary Value Problems Arising from Subsonic-Sonic Jet Flows and Rigidity
Short Talk 4 - Thursday, January 11, 11:00-11:30
In this talk, I will discuss some results on steady compressible potential jet flows from a finite converging nozzle, which are free boundary problems for a nonlinear degenerate elliptic
equation. An important feature is that such problems do not have a variational structure. Formulation of the problems and the existence (and non-existence) of solutions will be discussed. Both finite jets and infinite jets can be obtained by a PDE approach and regularity and properties of the solutions. In particular,a general result on the rigidity of the location of sonic degeneracy will be established. This talk is based on joint works with Chunpeng Wang.
Professor Young-Hoon Kiem (KIAS)
Shifted Lagrange Multipliers Method
Short Talk 5 - Thursday, January 11, 11:30-12:00
Modern enumerative invariants (GW, DT, PT and FJRW etc) are defined as integrals of cohomology classes against virtual fundamental classes constructed by Li-Tian and Behrend-Fantechi. When the obstruction sheaf admits a cosection, the virtual fundamental class is localized to its zero locus. When the cosection is further enhanced to a (-1)-shifted closed 1-form, the zero locus admits a (-2)-shifted symplectic structure and thus we have another virtual fundamental class by a recent work of Oh-Thomas. In this talk, we will see that the two virtual fundamental classes coincide. Moreover, we will see that (-1)-shifted closed 1-forms arise naturally from a shifted version of the Lagrange multipliers method. A generalized quantum Lefschetz principle will follow as an immediate consequence. Based on a joint work with Hyeonjun Park.
Professor Chenglong Yu (Tsinghua University)
Complex Hyperbolic Structures on Moduli Spaces of Calabi-Yau Varieties
Short Talk 6 - Thursday, January 11, 14:00-14:30
There are many works realizing moduli spaces as quotients of complex hyperbolic balls, including Deligne-Mostow on moduli of points on projective line, Allcock-Carlson-Toledo on moduli of cubic surfaces and cubic threefolds, Kondo on moduli of curves of genus three and four, etc. A common feature of these construction is cyclic covers. In this talk, I will discuss constructions of Calabi-Yau varieties by cyclic covers and the complex hyperbolic structures on their moduli. Some of the examples are related to Deligne-Mostow varieties and induces commensurability relations among those ball quotients. The talk is based on joint work with Zhiwei Zheng.
Professor Li-Sheng Tseng (University of California, Irvine)
From Mirror Symmetry to Mapping Cones
Short Talk 7 - Thursday, January 11, 14:30-15:00
For manifolds with a geometric structure described by a closed differential form, we will describe a natural mapping cone cohomology that intrinsically depends on the geometric structure and originates from the de Rham cohomology. The inspiration for this cohomology is motivated by mirror symmetry, from previous joint works with S.-T. Yau. We will further discuss implications of this mapping cone perspective for Morse theory and Yang-Mills theory, based on joint works with David Clausen, Xiang Tang, and Jiawei Zhou.
Professor Bong Lian (Brandeis University)
Fractional Complete Intersections
Short Talk 8 - Thursday, January 11, 15:30-16:00
We will consider a class of (typically) singular Calabi-Yau varieties given by cyclic branched covers of a fixed semi Fano manifold. The first prototype example goes back to Euler, Gauss and Legendre, who considered 2-fold covers of P1 branched over 4 points. Two-fold covers of P2 branched over 6 lines have been studied more recently by many authors, including Matsumoto, Sasaki, Yoshida and others, mainly from the viewpoint of their moduli spaces and their comparisons. I will outline a higher dimensional generalization from the viewpoint of mirror symmetry, and discuss the Riemann-Hilbert problem for periods of these singular varieties. The new insight here is the idea of ‘abelian gauge fixing’ and ‘fractional complete intersections’ that leads to a new interpretation of those classical results. The idea further points to a construction of large class of Calabi-Yau mirror pairs. The lecture is based on joint work with S. Hosono, T.-J. Lee, M. Romo, L. Santilli, H. Takagi, S.-T. Yau.
Professor Yong-Geun Oh (IBS, Center for Geometry and Physics)
Contact Instantons and Entanglement of Legendrian Links
Short Talk 9 - Thursday, January 11, 16:00-16:30
In this lecture, we first introduce the new nonlinear elliptic system of bordered contact instantons with Legendrian boundary condition on contact manifolds in the quantitative study of contact topology. Then we illustrate how the study of compactified moduli spaces of contact instantons can be combined with the contact Hamiltonian geometry and dynamics to prove the Shelukhin's conjecture by studying the dis-entanglement energy of Legendrian links.
Professor Yong Lin (Tsinghua University)
Normalized Discrete Ricci Flow and Community Detection
Short Talk 10 - Friday, January 12, 10:00-10:30
We prove the existence and uniqueness of solution of normalized discrete Ricci flow on graphs. We also use the discrete Ricci flow on the graph cut problems. These are based on the joint works with Bai, Lai, Lu, Wang and Yau.
