#
Analytic combinatorics in several variables

April 4 to April 8, 2022
at the

American Institute of Mathematics,
San Jose, California

organized by

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

## Original Announcement

This workshop 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 Knudsenâ€“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.

Topic 2: ACSV infrastructure.
- 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.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.

A list of open problems.

Workshop Videos