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.

Friday, November 4th

Speaker:  Grace Work, Vanderbilt University

Title: Transversals to Horocycle Flow on the Moduli Space of Translation Surfaces

Abstract: Computing the distribution of the gaps between slopes of saddle connections is a question that was studied
first by Athreya and Cheung in the case of the torus, motivated by the connection with Farey fractions, and then in
the case of the golden L by Athreya, Chaika, and Lelievre. Their strategy involved translating the question of gaps
between slopes of saddle connections into return times under horocycle flow on the space of translation surfaces to
a specific transversal. We show how to use this strategy to explicitly compute the distribution in the case of the
octagon, the first case where the Veech group had multiple cusps, how to generalize the construction of the transversal
to the general Veech case (both joint work with Caglar Uyanik), and how to parametrize the transversal in the case of
a generic surface in $\mathcal{H}(2)$.

Friday, November 11th

Speaker:  Rares Rasdeaconu, Vanderbilt University

Title: Complex Manifolds and Special Hermitian Metrics

: Several classes of hermitian metrics on closed complex manifolds and the relations between them will be
discussed, and the equality between the balanced and the Gauduchon cones of metrics is addressed in several situations.
We will see that while the equality of such cones does not hold for arbitrary closed complex manifolds, but it holds on Moishezon
manifolds. Moreover, we prove that a SKT manifold of dimension three on which the balanced cone equals the Gauduchon
cone is in fact Kahler. (Joint work with I. Chiose and I. Suvaina)

Wednesday, November 16th,      4:10-5pm, SC1310
(joint with the Topology & Group Theory Seminar)

Speaker:  Vito Zenobi, Universite of Montpellier 2 (France), 

Title: The tangent groupoid and secondary invariants in K-theory

Abstract: I will explain how to define secondary invariants that detect exotic structures on smooth manifolds or metrics with positive
scalar curvatures on Spin Riemannian manifolds. These invariants are elements in the K-theory of the tangent groupoid C*-algebra,
introduced by Alain Connes to give a more conceptual viewpoint on index theory. These constructions easily generalize to more
involved geometrical situations (such as foliations), well encoded by Lie groupoids.

Friday, November 18th

Speaker:  Ronan J. Conlon, Florida International University

Title: New Examples of Gradient Expanding Kahler-Ricci Solitons

Abstract: A complete Kahler metric g on a Kahler manifold M is a gradient expanding Kahler-Ricci soliton if
there exists a smooth real-valued function f:M->R with ∇^{g}f holomorphic such that Ric(g)-Hess(f)+g=0.
I will present new examples of such metrics on the total space of certain holomorphic vector bundles. This is joint
work with Alix Deruelle (Universite Paris-Sud).


 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