Homology of symmetric semi-algebraic sets

Abstract: Studying the homology groups of semi-algebraic subsets of $\mathbb{R}^n$ and obtaining upper bounds on the Betti numbers has been a classical topic in real algebraic geometry beginning with the work of Petrovskii and Oleinik, Thom, and Milnor. In this talk I will consider semi-algebraic subsets of $\mathbb{R}^n$ which are defined by symmetric polynomials and are thus stable under the standard action of the symmetric group $\mathfrak{S}_n$ on $\mathbb{R}^n$. The homology groups (with rational coefficients) of such sets thus acquire extra structure as $\mathfrak{S}_n$-modules leading to possible refinements on the classical bounds. I will also mention some connections with a homological stability conjecture. Joint work (separately) with Daniel Perrucci and Cordian Riener.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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