Geometry and Topology Seminar

                                                                                                                                  Vanderbilt University
                                                                                                                                                    Fall 2023

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

    Organizers: Anna Marie Bohmann, Rares Rasdeaconu, Ioana Suvaina

              Thursday, Aug 31st (12:20pm-1:10pm SC1310)

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)


              Friday, Sep 15th (1:25pm-2:15pm SC1312)

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.


              Friday, Sep 29th (1:25pm-2:15pm SC1312)

Speaker:  Shih-Kai Chiu (Vanderbilt University)

Title: Calabi-Yau manifolds

                  Abstract: The study of Calabi-Yau manifolds lies at the intersection of algebraic geometry, geometric analysis,
                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.


              Friday, Nov 3rd (1:25pm-2:15pm SC1312)

Speaker:  Hannah Housden (Vanderbilt University)

                  Title: Equivariant Stable Homotopy Theory for Diagrams

                  Abstract: A group can be viewed as a category with one object where every morphism is an isomorphism.
                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.

Friday, Nov 17th (1:25pm-2:15pm SC1312)

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
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)


 Old Seminar Web-Pages