Mathematical Physics

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.

The search for algebraic foundations of colour-kinematics duality and the double-copy construction has brought into focus a generalization of Batalin--Vilkovisky algebras, referred to here as coexact BV-algebras and as $\textrm{BV}^\square$-algebras in other sources. While these structures capture the cubic sector, they fail to encode higher-valence phenomena, for which a homotopy-theoretic extension becomes necessary. This work introduces a conceptual notion of homotopy coexact BV-algebra, defined through the homotopical interplay of commutative and BV structures, and provides a concrete model in terms of generators and relations, obtained through an extension the theory of Koszul duality for operads.

Topologically ordered phases of matter have long been regarded as lying beyond the Landau–Ginzburg paradigm of symmetry breaking. Our recent studies of anyon condensation, however, suggest that such phases may, in fact, be brought back within the Landau–Ginzburg framework. In this talk, I will present a framework that brings topological phases back into the Landau–Ginzburg picture by reformulating the string-net model to be a genuine lattice gauge theory coupled to anyonic matter fields. Within this modified model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated as Landau's effective theory of the Higgs mechanism.

Recent work of Bonifacio-Hinterbichler, Bonifacio, and Kravchuk-Mazáč-Pal introduced an analogy at the level of representation theory between conformal field theories and hyperbolic surfaces. In the spectral theory of hyperbolic surfaces, there are analogs of scaling dimensions of local operators and OPE coefficients, and these numbers obey an analog of crossing symmetry. The conformal bootstrap can thus be adapted to constrain these numbers. How strong are the resulting constraints? The main result of this talk is that they are complete constraints: every solution to the crossing equations must come from a hyperbolic surface (this is a rigorous theorem).