Geometry Seminar
Fridays, 4-5pm, Milner 216
Organized by J.M. Landsberg
jml@math.tamu.edu
January
Friday, 1/21
Speaker: Peter
Kuchment
Title: On an integral geometry problem involving
the circular Radon transform
Abstract: 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
Speaker:
Jenia Tevelev (UT Austin)
Title: Tropical
Compactifications
Abstract: 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
February
Friday, 2/4
Speaker: Niranjan Ramachandran (U. Maryland)
Title: 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
Speaker: G. Farkas (UT Austin)
Title: Syzygies and effective divisors on moduli
spaces of curves
Abstract: 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)
Title: Geometric
infinitesimal variations of Hodge structures
Abstract: 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.)
Title: Sections
of rationally connected fibrations through prescribed points
Abstract: 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.)
March
Friday, 3/4
Speaker: Alex Yong (U.C. Berkeley)
Title: On Smoothness and Gorensteinness of
Schubert varieties
Abstract:
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
construction.
I will discuss the geometric corollaries and questions that arise in this
work. This is a joint project with Alexander Woo.
Friday, 3/11
Speaker: 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)
Title: The
Horn recursion in the minuscule Schubert calculus
Abstract:
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)
Title: On the equivariant cohomology of
geometrically cellular real varieties
Abstract: 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)
Title: Exterior differential systems for
embeddings in frame bundles
Abstract:
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.
Abstract: 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
Title: Combinatorics III
Friday,
4/22 : Chris Hillar (UC Berkeley)
Title: TBA
Friday,
4/29
Speaker: Y.P. Lee (Utah)
Title: Quantum
K-theory
Abstract: 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)
Title: Geometry
of Affine Schubert varieties
Abstract: