# Berkeley Nonlinear Algebra Seminar

# Schedule

**Nov 15 (Tuesday)**: CANCELLED DUE TO THE UNION STRIKE.

**Room: Evans 1015**

11am-12pm: Bernd Sturmfels.

*Geometry of dependency equilibria*. (rescheduled)

1-2pm: Yelena Mandelshtam.

*From translation surfaces to algebraic curves*. (rescheduled)

**Nov 21 (Monday)**: SCHEDULED.

**Room: Evans 748**

11am-12pm: Andrei Okounkov.

*Take a residue, leave a residue*.

1-2pm: Serkan Hoşten.

*Gram spectrahedra of binary vs. symmetric binary forms*.

**Nov 29 (Tuesday)**: SCHEDULED.

**Room: Evans 1015**

11am-12pm: Bernd Sturmfels.

*Geometry of dependency equilibria*.

1-2pm: Vera Serganova.

*Lie superalgebras and finite groups*.

**Dec 6 (Tuesday)**: SCHEDULED.

**Room: Evans 1015**

11am-12pm: Persi Diaconis.

*Statistical enumeration in groups by double cosets.*

1-2pm: Benson Au.

*Spectral asymptotics for contracted tensor ensembles.*

2-3pm: Yelena Mandelshtam.

*From translation surfaces to algebraic curves*.

# Abstracts

**Bernd Sturmfels:**An n-person game is specified by n tensors of the same format. Its equilibria are points in that tensor space. Dependency equilibria satisfy linear constraints on conditional probabilities. These cut out the Spohn variety, named after the philosopher who introduced the concept. Nash equilibria are tensors of rank one. We discuss the real algebraic geometry of the Spohn variety and its payoff map, with emphasis on connections to oriented matroids and algebraic statistics. This is joint work with Irem Portakal.

**Yelena Mandelshtam:**A translation surface is obtained by identifying edges of polygons in the plane to create a compact Riemann surface equipped with a nonzero holomorphic one-form. Every Riemann surface can be given as an algebraic curve via its Jacobian variety. We aim to construct explicitly the underlying algebraic curves from their translation surfaces, given as polygons in the plane. The key tools in our approach are discrete Riemann surfaces, which allow us to approximate the Riemann matrices, and then, via theta functions, the equations of the curves. In this talk, I will present our algorithm and numerical experiments. From the newly found Riemann matrices and equations of curves, we can then make several conjectures about the curves underlying the Jenkins-Strebel representatives, a family of examples that until now, lived squarely on the analytic side of the transcendental divide between analysis and algebra.

**Andrei Okounkov:**Residue calculus is, of course, a very powerful way to evaluate many different kinds of integrals. For multivariate integrals, however, it may get really cumbersome, to the point that one sometimes gets stuck in purely combinatorial complexity. I will discuss some ideas that have recently been used with some success to circumvent such problems. Of course, they are much more specialized than residue calculus and apply in only very special situations. Still, these situations are not without interest in concrete applications, including solutions of q-difference equations and some questions in analysis of automorphic forms.

**Serkan Hoşten:**I will report on the geometric and combinatorial structure of symmetry adapted Gram spectrahedra of symmetric binary forms. In particular, I will present a characterization of extreme points of these spectrahedra for symmetric binary forms that are of rank two. This is complementary to a classical result in the non-symmetric case. I will also report what we know about the facial structure and combinatorics of the same spectrahedra.

**Vera Serganova:**Lie superalgebras generalize Lie algebras and provide mathematical foundations of supersymmetry. I will explain the similarity between representation theory of simple Lie superalgebras over complex numbers and representation theory of finite groups in positive characteristic. This analogy is due to the fact that in both cases we deal with Frobenius tensor categories.

**Svala Sverrisdottir:**The projective variety of Lie algebra structures on a 4-dimensional vector space has four irreducible components of dimension 11. We compute their prime ideals in the polynomial ring in 24 variables. By listing their degrees and Hilbert polynomials, we correct an earlier publication and we answer a 1987 question by Kirillov and Neretin.

**Persi Diaconis:**Let H and K be subgroups of the finite group G. This divides G up into H-K double cosets. We ask: pick g in G at random, what double coset is it likely to be in? This (perhaps strange sounding) question connects to the 'real world' in surprising ways. If G is Gl(n,q) and H=K is the subgroup of upper-triangular matrices the double cosets are indexed by permutations, the induced measure is 'Mallows measure' and one has dozens of papers to call on--The group theory situation has q a power of a prime (of course) and calls for new tools. This illuminates the basic task of 'row reduction' (sorry, almost sounds related to linear algebra). If H and K are parabolic subgroups of the symmetric group S_n,(corresponding to partitions lambda and mu) the double cosets are indexed by statisticians 'contingency tables' and the induced measure is the 'Fisher-Yates distribution'. Which double coset is largest (and how large is it?) lets us use 100 years of tools from mathematical statistics. Enumerating double cosets is NP hard, even for this example but much can be said. If H=K is the Sylow-p subgroup of G the problem appears in modular representation theory and is largely open). All of this is joint work with Mackenzie Simper. I will try to explain it all in 'mathematical English'.

**Benson Au:**We study the family of random matrices obtained by contracting a random real symmetric Wigner-type tensor with unit vectors. We show that the joint spectral distribution of this ensemble is well-approximated by a semicircular family whose covariance is given by the rescaled overlaps of the corresponding symmetrized contractions. In the single-matrix model, this implies that the empirical spectral distribution of the contracted tensor ensemble is close to the semicircle distribution in Kolmogorov-Smirnov distance with high probability. We further characterize the extreme cases of the variance. Our analysis relies on a tensorial extension of the usual graphical calculus for moment method calculations in random matrix theory. This is a joint work with Jorge Garza-Vargas (UC Berkeley).