2006-07 Geometry seminar
Fridays, Milner 216 at 4pm
Generally the speaker will give a 50-60 minute
talk, followed by a break, followed by more for
those who are interested.
Speaker: R. Douglas (A&M)
Title: Complex geometry and operator theory
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.
Speaker: J. Dilles (TAMU)
Title : Non-symplectic automorphisms of K3
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.
no seminar as we are doubled, and even tripled up a few times later in
don't miss the algebra/combinatorics
seminar at 3pm 9/29 by
J. Morton (Berkeley)
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
Matt Kerr (U. Chicago)
Title: Algebraic K-theory of Toric
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.
Speaker: S. Salur (U. Rochester)
Title: 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
Tues. 10/24 (Note special day!)
Speaker: N. Pali (Princeton)
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
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.
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
Speaker: H. Salmasian (Queens)
Title: Rank, Small Principal series, and
Representations of Rank Two
Abstract: We give an introduction to the construction of
Tuesday, 11/7 (NOTE SPECIAL DAY)
3pm Milner 216
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
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.
B. Doubrov (Minsk)
Title: Contact invariants of ordinary
Abstract: 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
Friday 11/24 - no seminar (Thanksgiving weekend)
Monday 11/27 a double header on a very special day!
both in Milner 317
2pm Dr. Anca. Mustata (UIUC) Mustata
THE TAUTOLOGICAL RINGS OF STABLE MAP SPACES
Abstract: 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
Title THE HOMOLOGY OF QUASI-MAP SPACES AND FLOPS OF RATIONALLY CONNECTED
Abstract: 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
Friday 12/1 - Joint meeting with Several Complex Variables seminar!
Speaker: 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.
Friday 2/9 a double header:
G. Farkas (UT Austin)
A. Ortega (Morelia)
A. Zelevinsky (Northeastern) week of March
19-23 (Frontiers lectures)
Texas Geometry and Topology Conference (TGTC) at TCU,
Y. Siu (Harvard) week of March 26-30 (Frontiers lectures)
Texas Algebraic Geometry Seminar (TAGS) at Austin, March 30
Speaker: D. Knopf (UT Austin)