Abstract : Recent years have shed new light on the theory of hypergeometric functions.
They appear as marginal likelihood integrals in statistics and as Feynman integrals in quantum field theory.
Evaluating these integrals is a central, but challenging problem in these areas.
Advances in algebraic methods provids new insights into this problem.
Such methods rest on graph theory, combinatorics of convex bodies, Gröbner bases, D-modules, and toric geometry, to name a few.
These new perspectives also raise fascinating new questions, both theoretical and computational.
We gather active researchers in this area to identify such questions, and to accelerate the progress.
Abstract : A hypergeometric function is an analytic function defined in terms of an elementary integral. A hypergeometric system is a system of differential (or difference) equations that it satisfies, which ties analysis to various fields of algebra, such as ring/module theory (D-modules), homological algebra, or combinatorics of polytopes. This talk aims to provide the speaker's view on the current state of the art. A unifying thread is a study of algebraic equations.
[03843] D-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals
Author(s) :
Anna-Laura Sattelberger (Department of Mathematics, KTH Royal Institute of Technology)
Abstract : I explain how to compute series solutions of regular holonomic D-ideals with Gröbner basis methods via an algorithm due to Saito, Sturmfels, and Takayama. As a point in case, I consider a D-ideal originating from a triangle Feynman diagram. This talk is based on joint work (arXiv:2303.11105) with Johannes Henn, Elizabeth Pratt, and Simone Zoia. Therein, we compare D-module techniques to dedicated methods developed for solving differential equations in the context of Feynman integrals.
[04120] Distribution of eigenvalues of a singular elliptical Wishart matrix
Author(s) :
Koki Shimizu (Tokyo University of Science)
Hiroki Hashiguchi (Tokyo University of Science)
Abstract : We derive the exact distributions of eigenvalues of a singular Wishart matrix under the elliptical model. These distributions cover the results under the Gaussian model as a special case. The joint density of eigenvalues and distribution function of the largest eigenvalue for a singular elliptical Wishart matrix are represented in terms of generalized hypergeometric functions. Numerical computations for the distribution of the largest eigenvalue are conducted under Gaussian and Kotz-type models.
[04121] Restriction algorithms for holonomic systems and their applications
Author(s) :
Nobuki Takayama (Kobe University)
Abstract : Definite integrals with parameters (in statistics and physics) satisfy holonomic systems (maximally overdetermined systems of linear PDEs). When parameters are restricted to an algebraic set, these integrals satisfy smaller or simpler holonomic systems. We will survey algorithms of finding these smaller or simpler systems with applications and numerical examples.
Abstract : A likelihood function on a smooth very affine variety gives rise to a twisted de Rham complex. We show how its top cohomology vector space degenerates to the coordinate ring of the critical points defined by the likelihood equations. We obtain a basis for cohomology from a basis of this coordinate ring. We investigate the dual picture, where twisted cycles correspond to critical points. We show how to expand a twisted cocycle in terms of a basis, and apply our methods to Feynman integrals from physics.
[04152] Algebraic A-hypergeometric Laurent series and residues
Author(s) :
Alicia Dickenstein (University of Buenos Aires)
Abstract : A-hypergeometric systems of partial differential equations (introduced by Gelfand, Kapranov and Zelevinsky) have natural geometric solutions, with singularities on the associated discriminant. We describe A-hypergeometric algebraic Laurent series associated with Cayley configurations of n lattice configurations in n space. These algebraic series are generated by certain combinatorially defined sums of point residues, whose computation can be interpreted in terms of a toric degeneration. Joint work with E. Cattani and F. Martinez.
[04334] Sampling from toric models and hypergeometric functions
Author(s) :
Shuhei Mano (The Institute of Statistical Mathematics)
Abstract : The toric model is an important class of stochastic models, and sampling from toric models has various applications including statistics. The sampling problem is related with hypergeometric functions, because the normalizing constant of the probability function is a multi-variable polynomial and satisfies a GKZ-hypergeometric system. In this talk, I will review several problems in which the relationship works effectively.
