Jan de Gier

We derive a determinantal expression for the inhomogeneous asymmetric five vertex model in a rectangular geometry with arbitrary boundary conditions at the bottom and top. Standard non-intersecting lattice path, or free fermion, approaches are not applicable and the determinantal form, thus, is not immediate.

This workshop explored how modern machine learning can both accelerate mathematical discovery and preserve rigorous standards. It focused on three angles: using AI techniques to help mathematicians make advances on challenging problems; using mathematics to understand AI predictions; and using deep-learning models for automated theorem proving. Key discussions included using machine learning as a tool for constructing interesting mathematical constructions and navigating in mathematical search spaces, to uncover conjectures and high-quality examples (e.g., sphere packings via DiffuseBoost, combinatorial objects via AlphaEvolve); Integrating Large Language Models (LLMs) with formal systems (e.g., Lean/mathlib) to create scalable, certifiable AI-based automated theorem prover; Collaborative formalization (e.g., the Carleson theorem project), autoformalization for high-quality supervised data, and reinforcement learning/search methods for proof generation and algorithmic reasoning.

We present the transition probability for the asymmetric simple exclusion process on the half-space for general initial conditions and particle insertion at the boundary. In the limit of total asymmetry, where particles only jump to the right, we show that the transition probability reduces to a single Pfaffian for a large class of initial conditions. We further derive a Fredholm Pfaffian formula for the multipoint distribution of the totally asymmetric model conditioned on the number of particles in the system using a connection with a Pfaffian point process supported on Gelfand-Tsetlin patterns.

We present explicit formulas for total crossing events in the multi-species asymmetric exclusion process (r-ASEP) with underlying U q ( s l ˆ r + 1 ) symmetry. In the case of the two-species TASEP these can be derived using an explicit expression for the general transition probability on Z in terms of a multiple contour integral derived from a nested Bethe ansatz approach. For the general r-ASEP we employ a vertex model approach within which the probability of total crossing can be derived from partial symmetrisation of an explicit high rank rainbow partition function. In the case of r-TASEP, the total crossing probability can be show to reduce to a multiple integral over the product of r determinants. For two-TASEP we additionally derive convenient formulas for cumulative total crossing probabilities using Bernoulli-step initial conditions for particles of type 2 and type 1 respectively.