Research

I am working on calculating the essential p-dimension of finite simple groups. The essential dimension of a group G is the minimum number of parameters needed to define a Galois G-algebra.  The essential p-dimension is the minimum number of parameters needed to define a Galois G-extension if we first allow any finite extension prime to p. This gives a lower bound for the essential dimension.

 For the symmetric group  S_n, we can relate this to solving polynomials of degree  n: Given a general polynomial z^n + a_1z^{n-1} + . . . + a_n, consider the set of roots,  {z_1, . . . , z_n}.  The essential dimension of S_n gives the minimum number d such that we can write z_1, . . . , z_n in terms of a_1, . . . ,a_n using arithmetic operations and a single algebraic function of d variables.

Below are the slides for my PhD defense. 

Publications/preprints:

•  "The essential p-dimension of the split finite quasi-simple groups of classical Lie type" [Journal of Algebra] 

• "The essential l-dimension at non-defining primes of finite groups of Lie type" [arxiv] 

• Translation of Tschebotarow's “The Resolvent Problem” (from Russian) [arxiv] 

• Translation of Tschebotarow's “The Problem of Resolvents and Critical Manifolds” (from Russian) [arxiv] 

The articles of Tschebotarow contain ideas which are precursors to the current ideas of essential dimension and resolvent degree.

Talks:

• UCLA Algebraic Topology Seminar: On the essential p-dimension of finite simple groups (3/29/2023)

• Séminaire Variétés Rationnelles:   On the essential p-dimension of finite simple groups (11/26/2021)