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Events happening on November 5, 2019

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Nov. 05, 2019 10am (MATH 350)	Geometry/Analysis Mitsuru Wilson (CU Boulder) X I will give an introductory talk on pseudo-differential operators and their symbols. I will also present my recent results for the case of associated vector bundles of a homogeneous space wherein I generalized Ruzhansky's construction of symbolic calculus on compact Lie groups to symbolic calculus of operators acting on the smooth sections of homogeneous vector bundles. Pseudo-differential operators are a generalization of differential operators on the space of smooth functions on a manifold. One of the most fundamental constructions for a pseudo-differential operator is its symbol. I will draw analogies from $ℝ^{n}$ to the compact Lie group case and to vector bundles. Global symbolic calculus on homogeneous vector bundles.
Nov. 05, 2019 1pm (MATH 220)	Peter Mayr (CU Boulder) X It is easy to see that the direct product of two finitely generated groups or finitely generated lattices is finitely generated again. We investigate under which conditions generating sets for subdirect products are finite. Finiteness conditions for subdirect products 1
Nov. 05, 2019 2pm (MATH 350)	Lie Theory Mandi Schaeffer-Fry (MSU) X A theorem of Thompson studies the structure of finite groups with exactly one irreducible character degree not divisible by a given prime. I will discuss two recent variations of Thompson’s theorem. Namely, in joint work with E. Giannelli and C. Vallejo, we prove a variation for pairs of primes, and with E. Giannelli and N. Rizo, we prove a variation in the case that exactly two degrees are not divisible by a given prime. On Some Extensions of a Theorem of Thompson
Nov. 05, 2019 3pm (MATH 350)	Algebraic Geometry Abhiram Natarajan (Purdue) X Joint seminar with the Theoretical CS seminar, and the Topology seminar. One object that is pervasive throughout this talk is random polynomials, i.e. polynomials sampled from a specific distribution over the space of polynomials of fixed degree and dimension. We are going to be interested in bounds on the average Betti numbers of the zero sets of random polynomials in various situations. The first part of the talk is about the topological complexity of real algebraic sets restricted to sets that arise in o-minimal geometry. Given any sequence of smooth and compact hypersurfaces in R^{n-1}, we prove that (up to extracting subsequences) there exists a regular definable (in a particular o-minimal structure over R) hypersurface $Γ$ in $R P^{n}$ such that each hypersurface in the sequence appears as a component of the zero set of some polynomial on $Γ$ . On the other hand, we show that given $Γ$ , for "most" polynomials, a Bezout-type bound holds for the restriction of $Γ$ on the zero set of the polynomial. In the second part of the talk, we study the arrangement of hypersurfaces defined in projective space by the zero sets of s randomly and independently chosen (Kostlan) homogeneous polynomials. Our first theorem here uses a random spectral sequence which gives information about the top Betti number of the union of the zero sets of the polynomials. The next result here deals instead with the asymptotic structure of the union of the zero sets when we have s quadrics. It turns out that in this case, the problem of understanding the number of connected components of the union of zero sets is related to the connectivity of a certain random graph model, and can be studied in a precise way. Specifically, our second theorem gives an upper bound on the average number of connected components in a random arrangement of quadrics' zero sets. This is joint work with Saugata Basu and Antonio Lerario. Quantitative bounds on the topology of "random" sets in o-minimal geometry and real algebraic hypersurface arrangements
Nov. 05, 2019 4pm (MATH 350)	Topology Caroline Matson (CU Boulder) Heights of Formal Groups

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