Spring 2024

** Organizers: Anna Marie
Bohmann, Rares Rasdeaconu, Ioana Suvaina**

__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.

__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).

__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).

__Speaker:__
Bar Roytman (University of Michigan)

__Title:__
Geometry of Operads in Equivariant Homotopical Algebra

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).