Calendar

In this talk, I will introduce a new construction for the K-theoretic mirror symmetry of toric varieties/stacks, based on the 3d N=2 mirror symmetry introduced by Dorey-Tong. Given the toric datum, i.e. a  short exact sequence 0 -> Z^k -> Z^n -> Z^{n-k} -> 0, we consider the toric Artin stack of the form [C^n / (C^*)^k]. Its mirror is constructed by taking the Gale dual of the defining short exact sequence. As an analog of the 3d N=4 case, we consider the K-theoretic I-function, with a suitable level structure, defined by counting parameterized quasimaps from P^1. Under mirror symmetry, the I-functions of a mirror pair are related to each other under the mirror map, which exchanges the K\”ahler and equivariant parameters and maps q to q^{-1}. This is joint work with Yongbin Ruan and Yaoxiong Wen.

Zoom: https://harvard.zoom.us/j/91780604388?pwd=d3BqazFwbDZLQng0cEREclFqWkN4UT09

We present recent progress on studies of 2-dimensional topological order in terms of tensor networks and its connections to subfactor theory. We explain how Drinfel’d centers and higher relative commutants naturally appear in this context and use of picture language in this study.

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

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

Positroid varieties are subvarieties of the Grassmannian obtained by intersecting cyclic rotations of Schubert varieties.  We show that the “top open positroid variety” has mixed Hodge polynomial given by the q,t-rational Catalan numbers (up to a simple factor).  Unlike the Grassmannian, the cohomology of open positroid varieties is not pure.

The q,t-rational Catalan numbers satisfy remarkable symmetry and unimodality properties, and these arise from the Koszul duality phenomenon in the derived category of the flag variety, and from the curious Lefschetz phenomenon for cluster varieties.  Our work is also related to knot homology and to the cohomology of compactified Jacobians.

This talk is based on joint work with Pavel Galashin.

Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09

TITLE: Is relativity compatible with quantum theory?

ABSTRACT: We review the background, mathematical progress, and open questions in the effort to determine whether one can combine quantum mechanics, special relativity, and interaction together into one mathematical theory. This field of mathematics is known as “constructive quantum field theory.” Physicists believe that such a theory describes experimental measurements made over a 70 year period and now refined to 13-decimal-point precision—the most accurate experiments ever performed.

Talk chair: Zhengwei Liu

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.

I discuss the interplay between non-Fermi liquid behaviour and pairing near a quantum-critical point (QCP) in a metal. These tendencies are intertwined in the sense that both originate from the same interaction mediated by gapless fluctuations of a critical order parameter. The two tendencies compete because fermionic incoherence destroys the Cooper logarithm, while the pairing eliminates scattering at low energies and restores fermionic coherence. I discuss this physics for a class of models with an effective dynamical interaction V (Ω) ~1/|Ω|^γ (the γ-model). This model describes, in particular, the pairing at a 2D Ising-nematic critical point in (γ=1/3), a 2D antiferromagnetic critical point (γ=1/2) and the pairing by an Einstein phonon with vanishing dressed Debye frequency (γ=2). I argue the pairing wins, unless the pairing component of the interaction is artificially reduced, but because of fermionic incoherence in the normal state, the system develops a pseudogap, preformed pairs behaviour in the temperature range between the onset of the pairing at Tp and the onset of phase coherence at the actual superconducting Tc. The ratio Tc/Tp decreases with γ and vanishes at γ =2. I present two complementary arguments of why this happens. One is the softening of longitudinal gap fluctuations, which become gapless at γ =2. Another is the emergence of a 1D array of dynamical vortices, whose number diverges at γ =2. I argue that once the number of vortices becomes infinite, quasiparticle energies effectively get quantized and do not get re-arranged in the presence of a small phase variation. I show that a new non-superconducting ground state emerges at γ >2.

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

The talk discusses a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes of side-lengths 2^j, j\in \N_0. Cubes belong to an admissible set such that if two cubes overlap, then one cube is contained in the other, a picture reminiscent of Mandelbrot’s fractal percolation model. I will present exact formulas for the entropy and pressure, discuss phase transitions from a fluid phase with small cubes towards a condensed phase with a macroscopic cube, and briefly sketch some broader questions on renormalization and cluster expansions that motivate the model. Based on arXiv:1909.09546 (J. Stat. Phys. 179 (2020), 309-340).

Zoom: https://harvard.zoom.us/j/98520388668

The Cohen–Lenstra–Martinet conjectures have been verified in
only two cases. Davenport–Heilbronn compute the average size of the
3-torsion subgroups in the class group of quadratic fields and Bhargava
computes the average size of the 2-torsion subgroups in the class groups of cubic fields. The values computed in the above two results are remarkably stable. In particular, work of Bhargava–Varma shows that they do not change if one instead averages over the family of quadratic or cubic fields satisfying any finite set of splitting conditions.

However for certain “thin” families of cubic fields, namely, families of
monogenic and n-monogenic cubic fields, the story is very different. In
this talk, we will determine the average size of the 2-torsion subgroups of
the class groups of fields in these thin families. Surprisingly, these
values differ from the Cohen–Lenstra–Martinet predictions! We will also
provide an explanation for this difference in terms of the Tamagawa numbers of naturally arising reductive groups. This is joint work with Manjul Bhargava and Jon Hanke.

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

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

p-Adic numbers have always been primarily associated with pure Mathematics, and have become especially relevant in algebra and modern number theory. But why did Computer Scientists become interested in them? In this talk we will introduce p-adic numbers and survey their main properties. We will then introduce Dixon’s algorithm, which is the first algorithm that used p-adic numbers to compute the exact rational solution to an integer linear system of equations. We will also explore the latest runtime improvements in p-adic linear algebra algorithms, and discuss whether we can solve linear equation systems faster than matrix multiplication.

Zoom: https://harvard.zoom.us/j/96759150216?pwd=Tk1kZlZ3ZGJOVWdTU3JjN2g4MjdrZz09

Ideas from the early 1990s for regulating chiral fermions in lattice gauge theory led to a number of developments which paralleled roughly concurrent and independent discoveries in condensed matter physics.  I show how the Integer Quantum Hall Effect, Chern Insulators, Topological Insulators, and Majorana edge states all play a role in lattice gauge theories, and how one can also find relativistic versions of the Fractional Quantum Hall Effect, the Quantum Spin Hall Effect and related exotic forms of matter.  How to construct a nonperturbative regulator for chiral gauge theories (like the Standard Model!)  remains an open challenge, however, one that may require new insights from condensed matter physics into exotic states of matter.

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

TITLE: Michael Atiyah: Geometry and Physics

ABSTRACT: In mid career, as an internationally renowned mathematician, Michael Atiyah discovered that some problems in physics responded to current work in algebraic geometry and this set him on a path to develop an active interface between mathematics and physics which was formative in the links which are so active today. The talk will focus, in a fairly basic fashion, on some examples of this interaction, which involved both applying physical ideas to solve mathematical problems and introducing mathematical ideas to physicists.

Talk chair: Peter Kronheimer

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.