Valley Geometry Seminar
Schedule for Fall 2006
-
September 15th: Jenia Tevelev (University of Massachusetts, Amherst)
Geometry of Chow Quotients of Grassmannians
Abstract: We consider Kapranov's and Lafforgue's compactification
of the moduli space of ordered n-tuples of hyperplanes in
the r-dimensional projective space in linearly general position.
For r=1 this is canonically identified with the Grothendieck-Knudsen
moduli space of stable rational curves. It turns out that for r=2
(and higher) this space also has a functorial meaning, the corresponding
stable surfaces can be described using membranes in the affine
building, but geometry of this moduli space turns out
to be (universally) bad. This is joint work with Sean Keel
(Duke Math. J. 134, no. 2 (2006), 259-311)
- September 22nd: Alan Durfee (Mount Holyoke College)
Polynomial knots
Abstract: A polynomial knot is a smooth embedding R -> R^n whose
components are
polynomials. The case $n = 3$ is of particular interest. It is both an
object of real algebraic geometry as well as being an open ended
topological knot. I will discuss basic properties of these knots and
give many examples. (Joint work with Don O'Shea)
-
September 29th: Ana-Maria Castravet (University of Massachusetts,
Amherst)
Hilbert's 14'th Problem and Cox Rings
Abstract: We give a description of the generators of the total
coordinate ring of
the
blow-up of a projective space in any number of points that lie on a
rational normal curve. As a corollary we show that the algebra of
invariants
of the action of a two-dimensional vector group introduced
by Nagata is finitely generated by certain explicit determinants.
We also prove the finite generation of the algebras of invariants of
actions of vector groups related to T-shaped Dynkin diagrams introduced
by Mukai. This is joint work with J. Tevelev.
-
October 13th: Elizabeth Denne (Harvard University)
Introduction to Geometric Knot Theory
Abstract: This area of knot theory is interested in the shape of knots.
For example finding the total curvature of a knotted curve or solving
the ropelength problem, the mathematical model of tying a knot tight in
rope of fixed thickness. This talk starts with some of the known
results in the area. Then we discuss the existence of an alternating
quadrisecant line for every nontrivial tame knot in $R^3$. This result
has applications to the total curvature, ropelength and distortion of
knots.
- October 20th: Theron Hitchman (Williams College)
Rigidity of discrete subgroups of Semisimple Lie groups: dynamics
Abstract: We shall discuss the overall program of dynamical rigidity for
lattices in
semisimple Lie groups. We'll spend most of our time on pre-historical
motivation (skipping the actual history) and basic examples.
Then we'll discuss some of my recent work on the program with David Fisher of
Indiana University. At the end we shall talk about some unfinished business.
- October 27th: Alina Marian (Yale University)
The level-rank duality for nonabelian theta functions
Abstract:
Spaces of sections of tensor powers of the theta line bundle on moduli spaces of
semistable arbitrary rank bundles on a compact Riemann surface are subject to a
level-rank duality: each space of sections is geometrically isomorphic to the dual of
the space of sections obtained by interchanging the tensor power (level) of the theta
bundle on the moduli space and the rank of the bundles that make up the moduli space.
This corresponds in representation theory to an isomorphism of conformal blocks of
representations of affine Lie algebras, when the rank of the algebra and the level of
the representation are interchanged.
I will describe a proof of the geometric statement, which is the result of joint work
with Dragos Oprea, and draws inspiration from work by Prakash Belkale who established
the isomorphism for a generic Riemann surface.
- November 3rd: Rob Kusner (University of Massachusetts, Amherst)
Recent Progress on Nondegeneracy of CMC Surfaces [joint
work
with Nick Korevaar (University of Utah) & Jesse Ratzkin (University
of
Connecticut)]
- November 10th: Teresa Krick (University of Buenos Aires)
Factorization of Sparse Polynomials
Abstract: In this talk I will discuss algorithms for factoring polynomials over number fields and
reach
recent results on sparse polynomials, where the complexity of the algorithm takes into account the
fact
that the polynomial may have many zero coefficients. For simplicity I'll focus on rational polynomials
in one or two variables.
- November 17th: Mauricio Velasco (Cornell University)
Gr\"obner bases, monomial group actions and the Cox rings of Del Pezzo surfaces.
We will describe the total coordinate rings (Cox rings) of the surfaces obtained by blowing up P^2 at 4, 5 or 6 general points.
We prove a conjecture of V. Batyrev and O. Popov which yields a presentation of these rings as a quotient of a polynomial ring by an ideal generated by quadrics. (Joint work with
Mike Stillman and Damiano Testa.)
- December 1st: Izzet Coskun (Massachusetts Institute of
Technology)
The Geometry of Grassmannians
A Littlewood-Richardson rule is a positive rule for computing
the structure constants of the cohomology ring of flag varieties with
respect to their Schubert basis. In recent years new geometric
Littlewood-Richardson rules have led to the solution of many important
problems, including Klyachko, Knutson and Tao's solution of Horn's
conjecture and Vakil's solution of the reality of Schubert calculus. In
this talk I will survey some of the basic geometric ideas that underlie
these Littlewood-Richardson rules. I will also describe new rules for
Grassmannians, two-step flag varieties and isotropic Grassmannians.
- December 8th: Ilia Zharkov (Harvard University)
Integral affine structures
I will discuss integral affine structures with singularities on
spheres, which conjecturally arise as the metric limit of maximally
degenerate Calabi-Yau families. Large part of the talk will be in
dimension
two.
- December 15th: Diane Maclagan (Rutgers University)
Note special time: 2pm
Equations and degenerations of \overline{M}_{0,n}.
Curves are one of the basic objects of algebraic geometry, and
so much attention has been paid to the moduli space of all curves of a
given genus. This talk will focus on the moduli space of genus zero
stable curves with n marked points, which is a compactification of the
space M_{0,n} of isomorphism classes of n points on the projective line.
After introducing this space, I will describe joint work with Angela
Gibney on explicit equations for it (as a subscheme of a special toric
variety), which lets us see degenerations to toric varieties.
Past seminars
General information
The Valley Geometry Seminar meets Friday afternoons in 1634 LGRT at the
University of Massachusetts. It is a five-college seminar (Amherst
College, Hampshire College, Mount Holyoke College, Smith College and the
University of Massachusetts). The seminar generally is 4:00-5:00 PM
with refreshments at 3:45. Occasionally the long format is used:
first part of talk 3:30-4:15, tea 4:15-4:30, second part of talk
4:30-5:15.
The topics include geometry in all its various forms (algebraic and
differential geometry, geometric representation theory, topology,
combinatorics, symbolic computation, commutative algebra, applications
to and from physics, etc.) as well as other topics. The talks are
intended for a general audience with geometric interests of some form
or other.
The webpage is maintained by Jessica Sidman (email:
jsidman at mtholyoke dot edu).