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We show that specific exponential integrals serve as generating functions of labeled edge-colored graphs. Based on this, we derive asymptotics for the number of edge-colored graphs with arbitrary weights assigned to different vertex structures. The asymptotic behavior is governed by the critical points of a polynomial. As an application, we discuss the Ising model on a random graph and show how its phase transitions arise from our formula.

K-theoretical Coulomb branches are expected to have cluster structure. Cautis and Williams categorified this expectation. In particular, they conjecture (and prove in type A) that the category of perverse coherent sheaves on the affine Grassmannian is a cluster monoidal categorification. We discuss recent progress on this conjecture. In particular, we construct cluster short exact sequences of certain perverse coherent sheaves. We do that by constructing a bridge, relating this (geometric) category to the (algebraic) category of finite dimensional modules over the quantum affine group. This is done by relating both categories to the notion of Feigin--Loktev fusion product.

Recent work by Davide Gaiotto and collaborators introduced a new type of parametric Feynman integrals to compute BRST anomalies in topological and holomorphic quantum field theories. The integrand of these integrals is a certain differential form in Schwinger parameters. In a new article together with Simone Hu, we showed that this "topological" differential form coincides with a "Pfaffian" differential form that had been used by Brown, Panzer, and Hu, to compute cohomology of the odd graph complex and of the linear group. In my talk, I will review some aspects of the graph complex and the role played by the Pfaffian form there, sketch the proof of equivalence, and comment on various observations on either side of the equivalence and their natural counterparts on the other side.

In this talk, I will discuss a birational realization of King's conjecture which is indeed true, and its connections with noncommutative algebraic geometry and mirror symmetry. In particular, I will also establish the A-side analog of this result using constructible sheaves and promote the celebrated Coherent-Constructible Correspondence to a global family. The talk is based on recent joint work https://arxiv.org/abs/2501.00130 and work in preparation with David Favero.