Geometry Seminar
Fridays, 4-5pm, Milner 216

Organized by J.M. Landsberg


Friday, 1/21
Peter Kuchment
On an integral geometry problem involving the circular Radon transform
The talk will address the following problem that arises in different reformulations in many areas of pure and applied mathematics from approximation theory, to complexanalysis, to mathematical physics, to computerized tomography: Consider a "suitable" class of functions on $R^n$ (or on a moregeneral Riemannian manifold) and a set $S$. Does the knowledge ofintegrals of this function over all spheres centered at the pointsof $S$ allow one to recover the function uniquely? The answer is certainly negative in general (look at a hyperplane set $S$), so one wantsto describe all "bad" non-uniqueness sets $S$. These happen to be algebraic varieties. The conjecture is that any bad set $S$ must belong to the set of zeros of a homogeneous harmonic polynomial plus an algebraic variety of co-dimension two. This has been proven in 2D, but very littleis known in 3D. Even in 2D, there are open questions when one deals with functions with non-compact support.

Friday, 1/28
Jenia Tevelev (UT Austin)
Tropical Compactifications
Many important varieties of algebraic geometry
are not compact.  For many practical purposes (e.g. to understand
their intersection theory) it is necessary to find
reasonably good compactifications.
We study a new class of compactifications of very affine varieties
(closed subvarieties of an algebraic torus) defined
by imposing the polyhedral structure on their tropicalization
(a non-archimedean analogue of a complex amoeba of complex analysis).
Find out more


Friday, 2/4
Speaker:  Niranjan Ramachandran (U. Maryland)
Values of zeta functions at s=1/2
Abstract: Many conjectures relate special values of zeta functions of
varieties to deep arithmetical invariants.  These conjectures address only
special values at integers. We present a result about values at 1/2 for
varieties over finite fields. This involves supersingular
elliptic curves.

Friday, 2/11
G. Farkas (UT Austin)
Syzygies and effective divisors on moduli spaces of curves
One of the most important invariants of the moduli space
of curves is its cone of effective divisors which loosely speaking,
determines all the ways in which M_g maps to other projective
varieties. The shape of this cone used to be governed by the
Harris-Morrison Slope Conjecture which singled out the classical
Brill-Noether divisors of curves with special linear systems as those
having minimal slope. We construct a new stratification of M_g
defined in terms of syzygies of curves with the top stratum being
a divisor that violates the Slope Conjecture for infinitely many genera.
As a consequence we can prove that various moduli spaces of pointed
curves are of general type.

Friday, 2/18
Speaker:  E. Allaud (Univ. of Utah)
Geometric infinitesimal variations of Hodge structures
The concept of infinitesimal variations of Hodge structures  invented by Griffiths in the 60s has been the cornerstone for many results showing  the interplay between Hodge theory (complex analysis) and algebraic geometry. But the  object itself remains a bit mysterious. Griffiths proved that  the infinitesimal variations of Hodge structures satisfy a  "transversality condition": they are integral elements of an exterior  differential system. But it is known that in general the geometric  infinitesimal variation of Hodge structures are "lost" among the crowd  of those integral elements. I will discuss new conditions  satisfied by the geometric infinitesimal variations of Hodge structures  for certain class of varieties.

Friday, 2/25
Speaker: Brendan Hassett (Rice U.)
Sections of rationally connected fibrations through prescribed points

Let X-->B be a projective variety fibered over a smooth complex curve.
 `Weak approximation holds' if, given anarbitrary collection of horizontal jet data,
there exists a section s:B-->X with these jets. We prove this provided the fibers
are smooth and rationally connected. (This is joint with Yuri Tschinkel.)


Friday, 3/4
Speaker: Alex Yong (U.C. Berkeley)
: On Smoothness and Gorensteinness of Schubert varieties 


The study of singularities of Schubert varieties in the flag manifold 
involves interesting interplay between algebraic geometry, representation
theory and combinatorics.

Although all Schubert varieties are Cohen-Macaulay, few are smooth.
An explicit combinatorial characterization of the smooth ones was given
by Lakshmibai and Sandhya (1990). The singular locus of an arbitrary
Schubert variety was determined around 2001 by several authors.

Gorensteinness is a measurement of the ``pathology'' of the
singularities of an algebraic variety; it logically sits between
smoothness and Cohen-Macaulayness. We explicitly characterize which
Schubert varieties are Gorenstein, analogous to Lakshmibai and
Sandhya's theorem. Here is the geometric interpretation: a Schubert
variety is Gorenstein if and only if it is Gorenstein at the generic
points of the singular locus. We also compute the canonical sheaf of a
Gorenstein Schubert variety as a line bundle in terms of the Borel-Weil

I will discuss the geometric corollaries and questions that arise in this
work. This is a joint project with Alexander Woo.

Friday, 3/11
B. McKay (U. Southern Florida and Cork)
Title: Complete Cartan connections

Friday, 3/18
No seminar- spring break

Friday, 3/25
Speaker: F Sottile (TAMU)
The Horn recursion in the minuscule Schubert calculus


A consequence of Knutson and Tao's proof of the saturation 
conjecture is a conjecture of Horn, which implies that the
non-vanishing of Littlewood-Richardson numbers is recursive:
A Littlewood-Richardson number is non-zero if and only if
its partition indices satisfy the Horn inequalities imposed
by all `smaller' non-zero Littlewood-Richardson numbers.
A way to express this Horn Recursion is that non-vanishing
in the Schubert calculus of a Grassmannian is controlled by
non-vanishing in the Schubert calculus of all smaller Grassmannians.

This talk will discuss joint work with Kevin Purbhoo
extending this Horn recursion to the Schubert calculus for
all cominuscule flag varieties, which are analogs of
Grassmannians for other reductive groups. Our goal will be
to describe the geometric version of the Horn problem and
outline the source of our inequalities and the scheme
of our proof.

Friday, 4/1
Speaker: P. Lima-Filho (TAMU)
On the equivariant cohomology of geometrically cellular real varieties

We study the Bredon equivariant cohomology of geometrically cellular real varieties. These are real varieties X which have a nice cellular decomposition after base extension to the complex numbers. We present a family of very simple spectral sequences that converge to Bredon cohomology in special cases, and we use them to compute several examples. These spectral sequences have the same E_2-terms of some spectral sequences constructed by Bruno Kahn using motivic homotopy theory and converging, within a range, to the Lichtenbaum etale-motivic cohomology. This talk complements a previous talk given at this same seminar.

Friday,  4/8
Speaker: F. Eastabrook (NASA  JPL)
Exterior differential systems for embeddings in frame bundles

Cartan’s method of the moving frame, describing curved Riemannian manifolds of dim n, is best set on the othonormal frame bundle over a flat space of dim n(n+1)/2. An exterior differential system (EDS) generated by n 2-forms for vanishing induced torsion and n n-1-forms for vanishing Ricci tensor is shown to be well posed by calculation of its Cartan characters (for n=4, and the right signature, this is vacuum relativity.) Calculation of these uses a Monte Carlo character program due to H. D. Wahlquist. I also have come upon another, possibly new, family of well posed EDS generated solely by n(n+1)/2 2-forms. Its solutions are ruled by flat n-1 spaces. With hyperbolic signature, all these EDS are field theories. Cartan n-forms for variational principles can be written for these well-posed embedding systems.

Monday,  4/11 (NOTE: special day)
Speaker: S. Basu (Georgia Tech)
Title: Efficient Algorithms for Computing the Betti Numbers of Semi-algebraic Sets.

Computing homological information of semi-algebraic sets (or more generally
constructible sets) is an important problem
for several reasons. From the point of view of computational complexity theory,
it is the next logical step after the problem of deciding
emptiness of such sets, which is the signature NP-complete problem
in appropriate models of computation.

In this talk I will describe some recent progress in designing
efficient algorithms for computing the Betti
numbers of semi-algebraic sets in several different settings.
I will describe a single exponential time algorithm for computing
the first few Betti numbers in the general case and polynomial
time algorithms in case the set is defined in terms of quadratic
inequalities. One common theme underlying these algorithms is the use
of certain spectral sequences -- namely, the Mayer-Vietoris spectral
sequence and the ``cohomological descent'' spectral sequence first
introduced by Deligne.
Certain parts of this work  is joint with R. Pollack, M-F. Roy
and (separately) with T. Zell.

Friday, 4/15: Weyman seminar will
meet because of Monday's talk
Speaker: F. Sottile
Combinatorics III

Friday, 4/22 : Chris Hillar (UC Berkeley)
Title: TBA

Friday, 4/29
Speaker: Y.P. Lee (Utah)
Quantum K-theory

I will explain the construction of quantum K-theory, which is the K-theoretic counterpart of quantum cohomology. Some open problems will be discussed in this talk.

Friday,  5/6
Speaker: P. Magyar (MSU)
Geometry of Affine Schubert varieties