Geometry Seminar

                                                                                                                      Vanderbilt University
                                                                                                                             Fall  2016

   Organizers:  Gennadi Kasparov, Ioana Suvaina, Rares Rasdeaconu

   Fridays, 3:10-4:00pm in SC 1310 (unless otherwise noted)


                September 15, 2016 (Thursday), 4:10 pm, (Colloquium talk) - SPECIAL EVENT
                Location: Stevenson 5211

                Speaker: Tian-Jun Li, University of Minnesota

                Title:  Symplectic Calabi-Yau Surfaces

                Abstract: I will survey what is known about the geometry and topology of the symplectic analogue of
                Calabi-Yau manifolds. The focus will be on four dimensional symplectic CYs, which resemble the Kahler
                CY surfaces topologically. In particular, their Betti numbers are bounded. In contrast, in higher dimension,
                they are known to be much more flexible via both topological and differential geometric constructions. For instance,
                any finitely presented group can be realized as the fundamental group of a six dimensional symplectic CY.
                Tea at 3:30 pm in SC 1425. (Contact Person: Ioana Suvaina)

Friday, September 23rd

Speaker:  Jonathan Campbell, Vanderbilt University

Title: Waldhausen's Algebraic K-Theory

Abstract: In preparation for my talk next week, I'll introduce algebraic K-theory. In particular, I'll introduce
a formulation due to Waldhausen, which is the most general version of the algebraic K-theory machine.
I'll discuss some examples, and discuss some places of interest where this machine fails to work (which I
will discuss in the next talk). The talk will require no previous knowledge of algebraic K-theory.

Friday, September 30th

Speaker:  Ioana Suvaina, Vanderbilt University

Title: Seiberg-Witten Theory and Geometry of 4-Manifolds

Abstract: The Seiberg-Witten theory provides a smooth invariant, which can be used to distinguish homeomorphic,
non-diffeomorphic, smooth structures. It also has a deep impact on the Riemannian properties of 4-manifolds.
We will discuss how obstructions to the existence of Einstein metrics arise, and how one can compute the Yamabe
invariant for Kahler surfaces and some symplectic 4-manifolds.

Friday, October 7th

Speaker:  Ioana Suvaina, Vanderbilt University

Title: ALE Kahler manifolds

Abstract: The study of asymptotically locally Euclidean Kahler manifolds had a tremendous development in the last
few years. This talk presents a survey of the main results and the open problems in this area. When the manifolds
support an ALE Ricci flat Kahler metric the complex surfaces and their metric structures are classified. The remaining
case to be studied is that of ALE scalar flat Kahler manifolds. In this direction, we have a description of the underlying
complex manifold. It is exhibited as a resolution of a deformation of an isolated quotient singularity. As a consequence,
there exists only finitely many diffeomorphism types of minimal ALE Kahler surfaces.

Friday, October 21st

Speaker:  Jonathan Campbell, Vanderbilt University

Title: The K-Theory of Varieties

Abstract: The Grothendieck ring of varieties is a fundamental object of study for algebraic geometers. As with all
Grothendieck rings, one may hope that it arises as π_0 of a K-theory spectrum, K(Var_k). Using her formalism
of assemblers, Zahkarevich showed that this is in fact that case. I'll present an alternate construction of the spectrum
that allows us to quickly see various structures on K(Var_k) and produce character maps out of K(Var_k). I'll end
with a conjecture about K(Var_k) and iterated K-theory.

Friday, October 28th

Speaker:  Grace Work, Vanderbilt University

Title: Translation Surfaces

Abstract: Translation surfaces arise naturally out of the study of the classical dynamical system of idealized
billiards in a rational polygon. We will provide several definitions and examples and explore dynamical properties
of flows on the moduli spaces of these surfaces.


 Old Seminar Web-Pages: Fall 2009, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Fall 2015, Spring 2016