Mathematical Physics

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 of the Lax connection, giving a generalisation of previous constructions.  As an application, I will treat open strings on AdS3 x S3 with mixed NSNS-RR flux, where the standard approaches cease to work at generic flux, and show how the background supports integrable D-branes that wrap twisted conjugacy classes in SL(2,R)xSU(2).  If time permits I will discuss this example and construction from the point of view of 4d Chern-Simons theory.

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 earlier partial results in this direction is a precise description of the image of the Coulomb ring for gauge group G inside the localized equivariant K-theory of the affine Grassmannian Gr_T, where T \subset G is the maximal torus. This description combines conditions on residues at possible simple poles at shifted root hyperplanes with divisibility conditions controlled by the representation N determining the matter content of the theory.

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 talk, I will explain how these bundles satisfy the compatibility conditions required to form a modular functor. A key consequence of this result is that the category C_V of admissible V-modules is a modular fusion category. This provides a purely algebro-geometric construction of the tensor product on C_V, which is expected to agree with the tensor product defined by Huang and Lepowsky using analytic methods. Time permitting, I will discuss open questions concerning extensions beyond the rational setting. This is joint work with Lukas Woike.

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 this talk, I will present the
underlying mathematical structures that govern these symmetries, and
show that these MPOs provide the lattice representation theory of
generalised symmetries in (1+1)D.  Time permitting, I will discuss some
applications of these ideas, in particular how they can be leveraged to
improve computational performance in state of the art in tensor network
algorithms.

Based on arXiv:2509.03600, arXiv:2408.06334

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 collaborators, we developed a new approach using Fukaya category and relate to KLRW algebra. The geometric origin allows us to give a geometric construction for the categorified coproduct. The talk is based on past work and work in progress with Mina Aganagic, Vivek Shende, Elise LePage, Ivan Danilenko and Yixuan Li.

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 theory in high dimensions, and recent breakthroughs in understanding their rational homotopy type. This talk will be aimed at a broader audience.