# Berkeley Nonlinear Algebra Seminar

# Schedule

**Nov 15 (Tuesday)**:

**Room: Evans 1015**

11am-12pm: Bernd Sturmfels.

*Geometry of dependency equilibria*.

1-2pm: Yelena Mandelshtam [title].

**Nov 21 (Monday)**:

**Room: Evans 748**

11am-12pm: Andrei Okounkov [title].

1-2pm: Serkan Hoşten [title].

**Nov 29 (Tuesday)**:

**Room: Evans 1015**

11am-12pm: Svala Sverrisdottir.

*Four-dimensional Lie algebras revisited*.

1-2pm: Vera Serganova [title].

**Dec 6 (Tuesday)**:

**Room: Evans 1015**

11am-12pm: Persi Diaconis.

*Statistical enumeration in groups by double cosets.*

1-2pm: Benson Au [title].

# 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.

**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'.