Fall 2016

** Organizers: Gennadi Kasparov, Ioana
Suvaina, ****Rares Rasdeaconu**

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

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**

Abstract**: **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