Fall 2023

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

__Speaker:__
Yu-Shen Lin (Boston University)

__Title:__
**The Torelli theorem for ALH* gravitational
instantons**

__Abstract__**:**** ** K3
surfaces are 2-dimensional Calabi-Yau manifolds and are
usually the testing stone before conquering

the general Calabi-Yau problems. The moduli space of K3
surfaces and its compactification on their own form
important

problems in various branches in geometry. Gravitational
instantons were introduced by Hawking as the building block

for his Euclidean quantum gravity theory back in the 1970s.
These are non-compact Calabi-Yau surfaces with L2-curvature

and thus can be viewed as the non-compact analogue of K3
surfaces. In this talk, we will discuss the Torelli theorem
of

certain type of gravitational instantons, labeled by ALH*.
As a consequence, this leads to a description of the moduli
space

of ALH*-gravitational instantons. The talk is based on joint
works with T. Collins, A. Jacob and T.-J. Lee. (contact
person:

Ioana Suvaina)

__Speaker:__
Anna Marie Bohmann (Vanderbilt University)

__Title:__ **Assembly
in the Algebraic K-theory of Lawvere Theories**

__Abstract__**:**** **Lawvere's algebraic
theories are an elegant and flexible way of encoding
algebraic structures, ranging from

group actions on sets to modules over rings and
beyond. We discuss a construction of the algebraic
K-theory of such

theories that generalizes the algebraic K-theory of a
ring and show that this construction allows us to build
Loday

assembly-style maps. This is joint work with Markus
Szymik.

__Speaker:__
Shih-Kai Chiu (Vanderbilt University)

__Title:__
**Calabi-Yau manifolds
**

and mathematical physics. In this talk, a Calabi-Yau manifold is defined as a smooth Kaehler variety with trivial

canonical bundle. Calabi conjectured in 1954 that a compact Calabi-Yau manifold must admit a Ricci flat Kaehler

metric. The existence of such metrics has many important consequences. For example, this implies the existence

of a constant spinor, which makes Calabi-Yau 3-folds candidates of the hidden 6 dimensions of our universe.

Since Yau's celebrated solution to the Calabi conjecture in 1978, much progress has been made. However, due to

the highly nonlinear nature of the PDE, the geometry of Calabi-Yau metrics is far from well-understood.

After a brief survey of the Calabi-Yau theorem, I will focus on the case of complete, noncompact Calabi-Yau

manifolds. Contrary to the compact case, there are explicit examples thanks to symmetry reduction techniques.

However, general existence theory and classification of complete Calabi-Yau manifolds remain unsolved. I will

try to motivate the study of the noncompact version of Yau's theorem. If time permits, I will talk about some of

my own results in this direction.

__Speaker:__
Hannah Housden (Vanderbilt University)

**
Title: Equivariant Stable
Homotopy Theory for Diagrams**

In this context, a G-space is just a functor from G to Top. This immediately generalizes: for any category D,

a D-space is a functor from D to Top. We will explore this generalization, which relies heavily on the theory of

"orbits," a generalization G-sets of the form G/H. Our main example is when D is the category with just an

initial object and a terminal object; in this case, the category of D-orbits is equivalent to Set. The latter half of

the talk will focus on the stable case and what it would mean to invert a sphere with D-action. One issue is that

D-representations often fail to give rise to representation spheres. To work around this, we model D-spectra via

spectral Mackey Functors, which admit many familiar constructions, such as Eilenberg-MacLane spectra and

geometric fixed points.

__Speaker:__
Ben Spitz (UCLA)

**
Title: Mackey and Tambara
Functors Beyond Equivariant Homotopy**

**
Abstract:
**"Classically",
Mackey and Tambara functors are equivariant
generalizations of abelian groups and

commutative rings, respectively. What this means is that, in equivariant homotopy theory, Mackey functors appear

wherever one would expect to find abelian groups, and Tambara functors appear wherever one would expect to find

commutative rings. More recently, work by Bachmann and Hoyois has garnered interest in related structures which

appear in motivic homotopy theory - these Motivic Mackey Functors and Motivic Tambara Functors do not have

anything to do with group-equivariance, but have the same axiomatics. In this talk, I'll introduce a general context for

interpreting the notions of Mackey and Tambara Functors, which subsumes both the equivariant and motivic notions.

The aim of this approach is to translate theorems between contexts, enriching the theory and providing cleaner proofs

of essential facts. To this end, I'll discuss recent progress in boosting a foundational result about norms from equivariant

algebra to this more general context. (contact person: Anna Marie Bohmann)