Geometry and Topology Seminar

                                                                                                                                  Vanderbilt University
                                                                                                                                                    Spring 2024

Fridays, 1:25-2:15pm in SC 1312

    Organizers: Anna Marie Bohmann, Rares Rasdeaconu, Ioana Suvaina

                Friday, March 8th

Speaker: Rares Rasdeaconu (Vanderbilt University)

Title: The loss of maximality in Hilbert squares

Abstract:  In an ongoing joint work with V. Kharlamov, we investigate the maximality of the Hilbert square of maximal
real varieties. We found that starting from dimension two many of deformation classes of algebraic varieties do not contain
any real variety whose Hilbert square is maximal. For example, the K3-surfaces have never a maximal Hilbert square.
The talk will be an introduction to the maximality of real algebraic manifolds and Hilbert squares, outlining the current
state of affairs and open problems.

                Friday, March 22nd

Speaker: Qi Yao (Stony Brook University)

Title: Asymptotic behaviors of solutions to homogeneous complex Monge-Ampere equations on ALE K\"ahler manifolds

Abstract: Initiated by Mabuchi, Semmes, Donaldson, homogeneous complex Monge-Ampere (HCMA) equations become
a central topic in understanding the uniqueness and existence of canonical metrics in K\"ahler classes. Under the setting of
asymptotically locally Euclidean (ALE) K\"ahler manifolds, one of the main difficulties is the asymptotic behaviors of solutions
to HCMA equations. In this talk, I will give an introduction to canonical metric problems under the setting of ALE K\"ahler
manifolds and discuss the recent progress in studying asymptotic behaviors of solutions to HCMA equations. It is still an
ongoing project.
(Contact person: Ioana Suvaina). 

Friday, March 29th

Speaker: Chloe Lewis (University of Wisconsin-Eau Claire)

Title: Tools for computing Real topological Hochschild homology

Abstract: In the trace methods approach to studying algebraic K-theory, we work with more computationally accessible invariants
of rings and their topological analogues. One such invariant is topological Hochschild homology (THH), which has proven quite
computationally tractable, in part due to the existence of the Bokstedt spectral sequence and the Hopf algebra structure of THH.
In this talk, we'll investigate an equivariant generalization of THH called Real topological Hochschild homology (THR) which encodes
the C_2-action of involution. We'll develop equivariant analogues of these computational tools by constructing a Real Bokstedt spectral
sequence and will give a description of the algebraic structures that are present in THR.
(Contact person: Anna Marie Bohmann and
Hannah Housden).


                Friday, April 19th

Speaker: Bar Roytman (University of Michigan)

Title: Geometry of Operads in Equivariant Homotopical Algebra

                  Abstract: A space with an action by a finite group G-admits corresponding actions on its homotopy, homology, and cohomology
                groups. In equivariant homotopy theory, by keeping track of fixed point loci of subgroups, one obtains far richer algebraic structures
                than sets, groups, and rings with G-action. We will review this on the levels of equivariant analogues of sets, groups, and rings.
                Graduating to the topological case, we examine the special roles of little disk and linear isometry operads in equivariant homotopy
                theory and some of their convenient geometric properties, including some recent discoveries. We will discuss how these properties
                are particularly helpful for study of Thom spectra and the Fujii-Landweber Real bordism spectrum in particular. (Contact person:
                Anna Marie Bohmann and Hannah Housden).



 Old Seminar Web-Pages