Geometry Seminar Schedule, Spring 2007

January 16

Organizational meeting.

January 23

Michael Otto, “New results regarding the moment map” at  4:00 PM in Math 402.

The moment map is a central object of study in symplectic geometry. I will give an overview over some of its most interesting properties and applications. In addition, I will present a new result which can be thought of as a generalization of Duistermaat's convexity theorem.

January 30

Arlo Caine, “Abelian Localization and the Duistermaat-Heckman Formula” at  4:00 PM in Math 402.

Suppose that M is a compact symplectic manifold equipped with a Hamiltonian action by the circle.  The Duistermaat-Heckman forumla shows that the integral over M of the exponential of the moment map against the Liouville measure "localizes" to a computation about the fixed point sets of the circle action.  The formula can be deduced as a consequence of what is known as the abelian localization formula of Berline and Vergne for equivariant differential forms.  The main point of this talk will be to give a proof, due to Bismut, of this localization formula and derive the Duistermaat-Heckman formula as a corollary.  No familiarity with equivariant differential forms will be assumed.

February 6

Milen Yakimov,  The University of California, Santa Barbara, “Poisson structures on flag varieties” at  4:00 PM in Math 402.

The geometry of Poisson structures originating from Lie theory found numerous applications in representation theory, ring theory, and dynamical systems. The linear Poisson structure on the dual of a Lie algebra provides the setting for the orbit method of Kirillov, Kostant, and Dixmier for the study of the unitary duals of Lie groups and the spectra of universal enveloping algebras.  In this talk we will describe in detail the geometry of a class of Poisson structures on complex flag varieties and some of their relations to combinatorics (Schubert cells and their Deodhar partitions, cluster algebras, total positivity, the Springer and the Lusztig partitions of wonderful compactifications), ring theory (spectra of algebras of quantum matrices and other quantized algebras), integrable systems (Kogan-Zelevinsky systems). In the special case of hermitian symmetric spaces of compact type, these Poisson structures further elucidate the works of Wolf, Richardson, Roehrhle, and Steinberg on the structure of the orbits of certain Levi factors.

February 13

Luis Garcia-Naranjo, "(Almost) Poisson Geometry of Mechanical Systems with  Nonholonomic Constraints” at  4:00 PM in Math 402.

Mechanical systems with restrictions on the velocities that do no arise from position constraints are termed nonholonomic. A simple example is a sphere that rolls without slipping on a table. The equations of motion for this type of systems do not arise from a variational principle and are not Hamiltonian. In this talk I will explain how the equations can be formulated with respect to a bracket of functions that fails to satisfy the Jacobi identity (hence the name "Almost Poisson"). I will then present some techniques to reduce these systems in the case where the configuration space is a Lie group and the kinetic energy metric and the constraints have invariance properties.

February 20

Ana-Maria Castravet, U Mass Amherst,  “Hilbert's 14th Problem and Cox Rings” at  4:00 PM in Math 402

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.

February 27

Matthew Salomone, "Negative curvature problems in Lagrangian dynamics" at  4:00 PM in Math 402

The characteristics of the geodesic flow on a Riemannian manifold are strongly affected by the geometry and topology of the manifold. In this talk we explore how intrinsic geometry is also reflected in more general Lagrangian flows. We will concentrate in particular on how a metric of negative curvature recently allowed for the classification of the bounded dynamics of a certain three-body problem.

March 6

Arlo Caine,  “Abelian Localization and Duistermaat-Heckman Theorem Part II” at  4:00 PM in Math 402.

The title advertises this talk as part II of my first lecture on this topic from January 30 of this year.  Given the length of time between the two talks, I will treat it rather as an independent lecture, so attendance at the first one will not be a prerequisite.  I will start with a Riemannian example of an equivariantly closed differential form, then state the abelian localization theorem, and sketch Bismut's proof.  If time permits, I'll make some remarks on some applications of the the localization formula.

March 13

Spring Break

March 20

David Glickenstein, “Polygon shortening flows” at  4:00 PM in Math 402.

In this talk I will present some easy results on a linear ODE on polygons which is, in some sense, analogous to a curve shortening flow (a PDE on curves). We will be able to completely analyze the flow, showing that (possibly nonplanar, nonconvex, self-intersecting) polygons become planar and affinely regular as they shrink to points. I will try to draw some analogies to curve shortening flow and, if there is time, I will talk a little about how one might construct a nonlinear flow which finds regular polygons (not only affinely regular). This is based on work to appear in April in the American Mathematical Monthly, and so it should be accessible not only to graduate students, but also to advanced undergrads with a basic knowledge of linear ODE.

March 27

Ben Polletta,  “A Scalar Curvature Measure for Simplicial Complexes” at  4:00 PM in Math 402.

In this talk I will describe a scalar curvature measure which lives on simplicial complexes. This measure has the nice property that, if we approximate a manifold by finer and finer simplicial complexes, the measure approaches the smooth scalar curvature measure. In two dimensions, if we approximate a surface with finer and finer triangulations, this is the result of local calculuations. In higher dimensions, some new tricks must be used.

April 3

Lennie Friedlander, "Regularized determinants of elliptic operators” at  4:00 PM in Math 402.

This will be mostly an expository talk. The determinant of an elliptic operator was defined by Ray and Singer in 1974. Their definition involves the zeta-function of the operator. I will start with the definition, then I will discuss the properties that the determinant of an operator should satisfy. There are certain anomalies that arise from the regularization procedure, e.g. the determinant of a product does not necessarily equal the product of determinants. All the anomalies turn out to be local: they can be computed explicitly in terms of the symbols of the operators involved.

April 10

Philip Foth, "Linear geometry in the Minkowski space" at 4:00 PM in MATH 402.

The first part of the talk will be fairly elementary: I will explain the main features of the Minkowski 3-space in terms of linear algebra and basic geometry. The second part will be spent on the symplectic geometry of the moduli space of polygons in this space, illuminating similarities and differences with the Eucledian case and constructing a completely integrable system on it.

April 17

Ben Pittman-Polletta, “A Scalar Curvature Measure for Piecewise Linear Spaces” at  4:00 PM in Math 402.

For a variety of reasons, we might like to approximate a smooth manifold by a piecewise linear space, for example by triangulating a surface. In these cases it is also useful to approximate the geometric structures on the manifold, like the metric, connection, and curvature, with analogous structures on the approximating space. In this talk, I will describe a scalar curvature measure on piecewise linear spaces. I will then show that on an approximating sequence of piecewise linear spaces, this measure converges to the smooth scalar curvature.

April 24

Kirti Joshi  “Theta divisors of vector bundles; A survey” at  4:00 PM in Math 402.

This is a survey of results on theta divisors of vector bundles on curves. I will explain some recent results and open questions in the area.

May 1