Abstract: The quantum Knizhnik-Zamolodchikov (qKZ) equation is a difference generalization of the famous Knizhnik-Zamolodchikov (KZ) equation. The problem to explicitly capture the monodromy of the qKZ equation has been open for over 25 years. I will describe the solution to this problem, discovered jointly with Andrei Okounkov. The solution comes from the geometry of Nakajima quiver varieties and has a string theory origin.
Part of the interest in the qKZ monodromy problem is that its solution leads to integrable lattice models, in parallel to how monodromy matrices of the KZ equation lead to knot invariants. Thus, our solution of the problem leads to a new, geometric approach, to integrable lattice models. There are two other approaches to integrable lattice models, due to Nekrasov and Shatashvili and to Costello, Witten and Yamazaki. I’ll describe joint work with Nikita Nekrasov which explains how string theory unifies the three approaches to integrable lattice models.
Abstract: An old problem is to find a unified approach to the knot categorification problem. The new string theory perspective on the qKZ equation I described in the first talk can be used to derive two geometric approaches to the problem.The first approach is based on a category of B-type branes on resolutions of slices in affine Grassmannians. The second is based on a category of A-branes in a Landau-Ginzburg theory. The relation between them is two dimensional (equivariant) mirror symmetry. String theory also predicts that a third approach to categorification, based on counting solutions to five dimensional Haydys-Witten equations, is equivalent to the first two.This talk is mostly based on joint work with Andrei Okounkov.
New research by Alex Teytelboym et al. presented at the 2018 Big Data Conference was recently published in Science Advances. The article, “Interpreting Economic Complexity,” explores patterns related the economic complexity index (ECI) and product complexity index (PCI). The paper can be found here.
Abstract: Two network measures known as the economic complexity index (ECI) and product complexity index (PCI) have provided important insights into patterns of economic development. We show that the ECI and PCI are equivalent to a spectral clustering algorithm that partitions a similarity graph into two parts. The measures are also closely related to various dimensionality reduction methods, such as diffusion maps and correspondence analysis. Our results shed new light on the ECI’s empirical success in explaining cross-country differences in gross domestic product per capita and economic growth, which is often linked to the diversity of country export baskets. In fact, countries with high (low) ECI tend to specialize in high-PCI (low-PCI) products. We also find that the ECI and PCI uncover specialization patterns across U.S. states and U.K. regions.
Ravi Jagadeesan, an economic design fellow at CMSA and frequent collaborator with Center Affiliate Scott Kominers, is the recipient of the 2019 AMS-MAA-SIAM Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student. Ravi’s “fundamental contributions across several topics in pure and applied mathematics, including algebraic geometry, statistical theory, mathematical economics, number theory, and combinatorics” have qualified him for this prize. Jagadeesan is currently a PhD student in Business Economics at Harvard. He graduated from Harvard with an A.B. in Mathematics and an A.M. in Statistics in Spring 2018. In his response he thanked the Center of Mathematical Sciences and Applications for support.
Stephen Hawking passed away yesterday. He was 76. He visited the Black Hole Initiative in 2016 (pictured above). In 2006, Prof. Shing-Tung Yau helped arrange Prof. Hawking’s visit to China, where he has remained a popular cultural figure.
In the words of Prof. Yau, “He was very friendly and was willing to explain physics to laymen. His smile attracted the attention of everybody… the Chinese are grateful for his generosity in spending time in China.”