Ben Williams
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The A1- and classical homotopy theory of algebraic groups, and their applications to algebra.
A strong background in algebra and (algebraic) topology, and a fondness for matrices.
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Supervision Enquiry
Great Supervisor Week Mentions
I am very grateful for all the support that you have extended to me during my time here at UBC. I deeply appreciate how you have been continuously encouraging and guiding me in the last two years, and also how you have always been so friendly and supportive of all of my efforts and struggles. Working under your supervision has been very enjoyable and I have learned and grown a lot.
Thanks Ben!
I am very grateful for all the support that you have extended to me during my time here at UBC. I deeply appreciate how you have been continuously encouraging and guiding me in the last two years, and also how you have always been so friendly and supportive of all of my efforts and struggles. Working under your supervision has been very enjoyable and I have learned and grown a lot. Thanks, Ben!
Graduate Student Supervision
Doctoral Student Supervision (Jan 2008 - April 2022)
In this thesis, we establish decomposition theorems for topological Azumaya algebras, and topological Azumaya algebras with involutions of the first kind. Decomposition of topological Azumaya algebrasLet A be a topological Azumaya algebra of degree mn over a CW complex X. We prove that if m and n are relatively prime, m
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Master's Student Supervision (2010 - 2021)
In this thesis, we study a family of smooth varieties, whose members are denoted B_n^r(C), that bears a similar relationship to topological Azumaya algebras as the Grassmannians G_{n,r}(C) do to complex vector bundles. Specifically, we will show that the varieties B_n^r(C) form homotopical approximations to the classifying space BPGL_n(C). The varieties B_n^r(C) are obtained by first considering the variety of r-tuples of nxn complex matrices that generate the matrix algebra Mat_n(C), and then taking the quotient by an evidently free PGL_n(C)-action. The focus of this thesis is a computation of the singular cohomology groups of B_n^r(C) when n=2. We will show how these cohomological computations have applications in bounding the minimal number of generating sections of a topological Azumaya algebra over a paracompact space.
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