Although K3 surfaces and Enriques surfaces are closely related, their moduli spaces behave very differently. As the degree increases, one one obtains an infinite series of different moduli spaces of polarized K3 surfaces., In contrast, there are only finitely many different moduli spaces of (numerically) polarized Enriques surfaces. In fact there are exactly 87 orthogonal modular varieties which appear in context with moduli of polarized Enriques surfaces. Together with Dutour-Sikiric we classified the corresponding arithmetic groups. I will discuss this classification and related questions.

Moduli of polarized Enriques surfaces Sponsored by the Simons Foundation

Feb. 23, 2023 2:30pm (MATH 3…

Functional Analysis

Angel Roman (University of Washington at St. Louis)

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In the 1970's, George Mackey proposed an analogy between some unitary representations of a semisimple Lie group and unitary representations of its associated semidirect product group, known as the Cartan motion group. Recently, Nigel Higson and Alexandre Afgoustidis made this analogy precise between equivalence classes of irreducible tempered representations of a reductive groups and equivalence classes of irreducible unitary representations of the associated Cartan group. First, I will introduce tempered unitary representations of a reductive Lie group to develop the Mackey analogy. The goal of this talk is to show that the reduced group C*-algebra of the Cartan motion group can be embedded into the reduced group C*-algebra of the reductive group. To accomplish this, I will show the construction of a continuous fields of C*-algebras generated from a Lie groupoid known as deformation to the normal cone. I will also discuss the Fourier transform picture.