Mathematical Physics

Semisimple categorical symmetries, and their physical implications have been studied in great detail in the continuum and on the lattice. In particular, the nice properties of semisimple categories allows one to construct hamiltonians which enjoy this symmetry, and study its symmetric gapped phases…

I will say some words about nonunital categories, and tell you an awesome new theorem of McNamara and Wang: unitary QFTs are determined by their partition functions.

The amplituhedron was introduced by Arkani-Hamed and Trnka as a geometric object which "encodes" scattering amplitudes in N=4 SYM theory. One of their motivations was the Britto-Cachazo-Feng-Witten (BCFW) recursion for amplitudes, which in particular produced many different formulas for the same…

The theory of quiver varieties provides a fundamental bridge between representation theory, enumerative geometry, and physics. From 3d mirror symmetry, any quiver variety comes with a dual variety known as the Coulomb branch. A conjecture proposed by Bullimore-Dimofte-Gaiotto-Hilburn-Kim and…

Cosmological correlation functions probe the origins of structure in the universe. We view them as a prototype for time-dependent correlators and finite-temperature correlators in QFT. I will share recent work on the simple discrete geometries and spacetime pictures that control these correlators at…

Identifying which boundary conditions preserve integrability in two-dimensional sigma-models is a basic structural question, relevant from open string dynamics to impurity problems. I will discuss a simple analytic approach that determines integrable boundary conditions through the divisor structure…

I will explain a recent joint work with A. Shapiro in which we prove that the quantized K-theoretic BFN Coulomb branch ring associated to a quiver gauge theory is isomorphic to a quantum upper cluster algebra, when the gauge theory quiver is without 1-cycles. The new ingredient needed to upgrade our…

Given a vertex operator algebra V, one can define sheaves of conformal blocks on moduli spaces of curves following constructions of Ben-Zvi--Frenkel and Damiolini--Gibney--Tarasca. When V is strongly rational, these sheaves are vector bundles equipped with a projectively flat connection. In this…

In recent years, non-onsite symmetry representations on the lattice have enabled new insights in both anomalous and non-invertible symmetries. In (1+1)D, these can be represented as matrix product operators (MPO), a type of tensor network that captures the correlated action on neighbouring sites. In…

Quantum group and its application to knot polynomial was the theme of Witten-Reshtikhin-Turaev theory. The categorfication of quantum group and application to knot homology has been discovered and developed by Khovanov-Lauda, Rouquier, and Webster, in the form of KLRW algebra. With Mina Aganagic and…

Discs are among the simplest manifolds, but their groups of diffeomorphisms can be very complicated. I will describe the techniques from geometry, topology, and dynamics that were used to understand these groups in low dimensions, the relationship of these groups to stable homotopy theory and number…

The construction of symplectic singularities using notions of dressed monopole operators, otherwise known as 3d N=4 Coulomb branch, led to many interesting new developments, adding the known construction of Hamiltonian reduction. In this talk we will discuss a certain type of symplectic…