Geometry Seminar
Fall 2006
Fridays at 4 pm in Milner 216

Format: Talks are 50-60 minutes, with the option to continue after a short break.

Fall 2006

Friday, 9/1
Speaker: R. Douglas (A&M)
Title: Complex geometry and operator theory

Friday 9/8
Speaker: C. Robles (A&M)
Title: Rigidity and flexibility of homogeneous varieties

Note Wilkerson's general talk on Tues. 9/12

Wednesday 9/13 (NOTE SPECIAL DAY, TIME and ROOM)
Milner 317, 3-4pm
Speaker: C. Wilkerson (Purdue)

Title: Maps out of Classifying Spaces
(this will be an introduction to some of the tools needed in the program discussed in the departmental talk)
There will be two main topics:
1) If \pi is a finite p-group, for example, Z/pZ, how does one calculate
the homotopy classes of maps [B\pi, BG] where G is a compact connected Lie group..
More generally, how does one calculate the entire mapping space
Map(B\pi,BG) ( here we use the unpointed maps ). The case of \pi = Z/pZ
uses mod p cohomology, the Steenrod algebra, and the T-functor of Jean Lannes.
2) The classical method of calculating [X,Y] requires first knowledge
of H^*(X,\pi_*(Y)). This is rarely practical, since, for example, the
homotopy groups \pi_i(Y) are usually non-zero in infinitely many dimensions.
Thus to attack the problem of calculating [BG,BG], one needs a different
construction of BG, one that does not proceed cell by cell. I will briefly
talk about the centralizer decomposition of BG and homotopy limit constructions.

Friday  9/22
Speaker: J. Dilles (TAMU)
Title : Non-symplectic automorphisms of K3 surfaces
Abstract :  The study of non-symplectic automorphism of K3 surfaces is
partially motivated by the interests of phyisicists for CY 3-folds.
We will show how the study automorphism can in certain case reduce to
the underlying structure of lines. Othertimes, the canonical embedding
makes the study affordable.

Friday 9/29
no seminar as we are doubled, and even tripled up a few times later in the fall
don't miss the algebra/combinatorics seminar at 3pm 9/29 by
J. Morton (Berkeley) 

Friday 10/6

Speaker: Erxiao (Eric) Wang  (UT Austin)
Title: G_2 spetral curves and associative cones over tori
Abstract: I compute the genus, dimension of the moduli of G_2 spectral
curves, and also identify the Prym-Tjurin subtori of their Jacobians.
These computations will be used in a joint project with Emma Carberry
to describe the spectral data for superconformal almost complex curves
in S6, and to do a parameter count for solving periodicity equations to
obtain tori.

Friday 10/13
Matt Kerr (U. Chicago)
Title: Algebraic K-theory of Toric Hypersurfaces

Abstract: We describe how to use toric data to construct relative higher Chow cycles in CH^n(X,n) (n=2,3,4) for families of Calabi-Yau (n-1)-folds; and how to derive the inhomogeneous Picard-Fuchs equations satisfied by the regulator "periods" of such elements, which may be regarded as generalized normal functions.  This setting leads to motivic proofs of acceleration formulas for arithmetic constants and irrationality of \zeta(2) and \zeta(3), as well as relations to the Yukawa coupling, Meijer G-functions and local mirror symmetry.  This is (in part) joint work with C. Doran.

Friday 10/20
Speaker: S. Salur (U. Rochester)
Mirror Symmetry and Calibrated Geometries

Abstract: String theorists believe that every Calabi-Yau 3-fold X has a quantization, which is a Super Conformal Field Theory (SCFT) - a Hilbert space H with a collection of operators satisfying some relations - to be interpreted as the quantum theory of strings moving in X. Two different Calabi-Yau manifolds X and X' may have the same SCFT and in this case there are powerful relationships between the (topological) invariants of X and X'. This is the idea behind the Mirror Symmetry.

In this talk, I will first give brief introductions to Calabi-Yau
and G_2 manifolds, and then a short survey of my research (joint with Selman Akbulut) on relations between calibrated geometries and the Mirror Symmetry. 

Tues. 10/24 (Note special day!)
Speaker: N. Pali (Princeton)
Title: The Kahler-Ricci flow over Fano manifolds

October 27-29, Don't miss
Texas Geometry and topology conference at Rice Univ.

Monday 10/30 4pm  (Note Special day!)
Speaker: R. Herrera (CIMAT, Guanajuato)
Title: Parallel quaternionic spinors and Riemannian holonomy

Abstract: Spin manifolds are distinguished among oriented smooth manifolds
by admitting a principal bundle double-covering their
orthonormal-frame bundle, which gives rise to new vector bundles
whose sections are called spinors. The condition can be relaxed
to allow complex-spin structures (well-known due to Seiberg-Witten
theory) and, more generally, quaternionic-spin structures.
I will describe the geometric consequences of the existence of
a parallel spinor on quaternionic-spin manifolds from the holonomy
view-point, and how this generalizes the spin and complex-spin cases.

Tuesday 10/31 3-4pm (Note special day and time!)
Speaker: J. Buczynski (Poland)
Title: Smooth Fano Legendrian 8-fold.
I will explain a construction of new example of a smooth Legendrian 8-fold, which turns out to have extraordinary properties. It is a Fano variety of Picard number 1 and index 5 (so it gives rise to a Calabi-Yau 3-fold of Picard number 1). Also it is quasihomogeneous and it is a compactification of simple group SL(3).
  I will give a brief introduction to the subject. The talk will be  on an elementary level.

Thursday 11/2 (11am NOTE special day and time - joint meeting with algebra/combinatorics seminar)
Speaker: H. Salmasian (Queens)
Title: Rank, Small Principal series, and Representations of Rank Two
Abstract: We give an introduction to the construction of
singular unitary representations of non-compact
semisimple groups. For exceptional groups, we
describe a number of methods to construct
a class of such representations, and investigate
a possible connection with exceptional versions of
Howe duality.
Tuesday, 11/7 (NOTE SPECIAL DAY) 3pm Milner 216
Friday 11/10
Speaker: F. Zak (Moscow)
Title: Numerical invariants of projective varieties
Abstract: Let X be a nonsingular projective variety of dimension n,
codimension a, and degree d. A natural way to study X is
to compute its numerical invariants, such as Betti and Hodge
numbers, Chern numbers (such as self-itersection of canonical
class or Euler characteristic), classes (in particular, the
degree of dual variety) etc. The problem of what are the
possible values of numerical invariants of X and what is the
relationship between various invariants has been studied for
a century and a half, but little was known up to now. In
particular, Castelnuovo found a sharp bound for the genus in
the case when n=1, and the Riemann-Roch-Hirzebruch theorem
provides some relations between certain invariants. In my talk
I'll explain how to extend Castelnuovo's bound to varieties of
arbitrary dimension. I'll also explain why the numerical
invariants are all "asymptotically equivalent" to each other
and why, contrary to topologists' belief, the larger the
invariants (e.g. Betti numbers), the simpler is the variety.

Friday 11/17
Speaker: B. Doubrov (Minsk)
Title: Contact invariants of ordinary differential equations

Using the geometric interpretation of ordinary differential equation and the technique of
nilpotent differential geometry we constuct the canonical coframe associated with any
system of ordinary differential equations.  We describe the generators in the differential algebra of all contact invariants of such system and give the explicit formulas for these generators in the case of scalar ODE of any order. Further,
we discuss one class of invariats that appears as direct generalization of classical Wilczynski invariants of linear ODEs. These invariants can also be interpreted as differential invariants of non-parametrized curves  in $n$-dimensional projective space.
The equations with vanishing generalized Wilcynski invariants possess a remarkable property that
all other invariants of the canonical coframe become first integrals of the  original equation. Further, using the techniques from Koraira deformation theory, we show how to constuct ODEs with vanishing Wilczynski invariants and present several
non-trivial examples.

Friday  11/24 - no seminar (Thanksgiving weekend)

Monday 11/27 a double header on a very special day!
both in Milner 317

2pm Dr. A
nca. Mustata (UIUC) Mustata

The Kontsevich-Manin moduli spaces of stable maps have come to
play a central role in the enumerative geometry of curves in projective
varieties. In this talk I will discuss the structure of their intersection
rings, based on the analogy with Grassmannians and the flag varieties.

4pm Andrei Mustata

The space of quasi-maps (also known
as the linear sigma model) is a simple compactification for the space of
maps from the projective line to a variety.
I will present an extension of the relation between the homology of the
space of quasi-maps and the small quantum cohomology of toric Fano
varieties (as presented by Kapranov) to rationally connected targets.
As a consequence I will describe the behavior of small quantum cohomology
under flops for rational connected varieties with the cohomology generated
by divisors.

Friday 12/1 - Joint meeting with Several Complex Variables seminar!
A. Nicoara (Harvard)

Title: Equivalence of Types on Smooth Domains

Abstract: In 1979, Joseph J. Kohn defined the first multiplier ideal sheaf while investigating the subellipticity of the $\bar\partial$-Neumann problem. He designed an algorithm that generates an increasing chain of ideals, whose termination implies subellipticity. This termination condition is called Kohn finite ideal type. In that same paper, Kohn proved that for a domain in $C^n$ with real-analytic boundary, subellipticity of the $\bar\partial$-Neumann problem on the domain for (p,q) forms is equivalent to Kohn finite ideal type and also equivalent to the property that all holomorphic varieties of complex dimension q have finite order of contact with the boundary of the domain, known as finite D'Angelo type. The equivalence of these two notions of finite type for domains with smooth boundary is known as the Kohn Conjecture. I will present my very recent proof of the Kohn Conjecture and perhaps explain a little bit how this equivalence works on domains with Denjoy-Carleman quasianalytic boundary.