for this workshop

## Analytic combinatorics in several variables

at the

American Institute of Mathematics, San Jose, California

organized by

Yuliy Baryshnikov, Marni Mishna, Robin Pemantle, and Mark C. Wilson

This workshop, sponsored by AIM and the NSF, will be devoted to bringing researchers with expertise in asymptotic integrals, computer algebra and topology of complex varieties into contact with those working on analytic combinatorics in several variables (ACSV), in order to make progress on the technical infrastructure of ACSV results. Another goal is to connect ACSV community with potential users in combinatorics, representation theory and statistical physics. These fields generate many important and technically challenging application problems for ACSV methodology.

The main topics for the workshop are as follows.

Topic 1: Applications of ACSV.

- Several spectacular results at the integrable models of statistical physics were achieved using a collection of methods outside of ACSV toolbox. We aim to reinterpret, rederive, and hopefully to extend these results using ACSV methodology. These include integrals of symmetric functions and the tangent method.
- Establish universal Airy limits for a broad class of integrable systems, including lattice quantum walks and tiling ensembles.
- The theory of 3j, 6j, 9j symbols is another broad area where the tools of ACSV are not yet deployed broadly, but should be. In particular, understanding the behavior of matrix elements of representation of SU(2) (or more generally, SU(d)) near the boundary of the effective support would be very interesting.
- Generating functions arising as solutions of systems of algebraic equations (very common in formal languages, tree patterns, genomic analysis, etc.) is lacking a ACSV perspective. Often the results, such as the remarkable identities on frequencies of RNA patterns in the Knudsen–Hein model, appear as a miraculous coincidence. Computations involving the Weyl denominator formula also tend to be in a form relevant for study with these methods. This includes a wide variety of lattice walk enumeration problem Understanding those from more general principles is an important task.

- Effective topological decomposition of the chain of integration
in the multivariate Cauchy integral formula into cycles corresponding
to stratified critical points.
- Computation of singular integrals necessary to evaluate asymptotics near critical points where the tangent cone is a homogeneous hyperbolic polynomial of degree three or greater.

This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email *workshops@aimath.org*