Geometry Seminar
Fall 2008
Fridays at 4 pm in Milner 216

Format: Talks are 50-60 minutes, with the option to continue after a short break.
 * List below includes Department Colloquia and Frontiers Lectures with geometric content. * Approximately once a month the geometry group meets with the Physics Department for a joint Geometry & Physics seminar.

 August 29 Igor Zelenko (TAMU) Local geometry of vector distribution via geometry of curve of flags of isotropic/coisotropic subspaces. The talk is based on the joint work with Boris Doubrov. Equivalence of vector distributions is a classical problem which goes back to the end of 19th century and was studied by various mathematicians including Lie, Goursat, Darboux, Engel, Cartan and others. The basic notion here is a symbol of a distribution at a point, which is a graded nilpotent Lie algebra. The notion of the symbol is extensively used in works of N. Tanaka and his school who systematized and generalized the Cartan equivalence method. However, these tools become really effective only when the symbol algebras are isomorphic at different points, and all constructions strongly depend on the algebraic structure of the symbol. Note that the problem of classification of all symbols (graded nilpotent Lie algebras) is quite nontrivial already for small dimensions and it looks completely hopeless for arbitrary dimensions. The aim of my talk is to describe another approach, based on the ideas from geometric control theory, which allows to overcome the difficulties mentioned above. This approach allows to reduce the equivalence problem for vector distributions to the study of curves of flags of isotropic/coisotropic subspaces in a sympectic space. Our classification of distributions is done according to a so-called Young diagram of these curves of flags and is not directly related to Tanaka symbols of the distribution itself. The local geometry of distributions can be recovered from the properties of symmetry groups of so-called flat curves of flags associated with its Young diagram. For any given Young diagram one can describe the flat distribution and construct a canonical frame for any other distribution (with the same Young diagram). In the case of rank 3 distributions with non-rectangular Young diagram the infinitesimal symmetry algebra of the flat distribution can be described in terms of rational normal curves (their secants and tangential developable) in projective spaces.

 September 5 Igor Zelenko (TAMU) Local geometry of vector distribution via geometry of curve of flags of isotropic/coisotropic subspaces, Part 2. This is a continuation of last week's talk.

 September 12 Vadim Zharnitsky (UIUC) Integrability and periodic orbits in billiard systems.

 September 15 (Monday) in ENPH 501 (4 o'clock) Joint Geometry & Physics Seminar Dan Freed (U. Texas, Austin) Orientifolds and Topology

 September 26 Dennis The (TAMU) Contact geometry of hyperbolic equations of generic type. In the geometric theory of differential equations (founded by Lie and Darboux, and developed extensively by Goursat, Cartan, and many others), there is a natural notion of equivalence of differential equations associated with point transformations (mixing the independent and dependent variables) and, more generally, contact transformations. In this theory, one seeks to understand differential equations through their invariants under suitable types of coordinate transformations such as those mentioned above. The classification of (in general nonlinear) scalar 2nd order PDE in the plane into elliptic, parabolic, hyperbolic classes is a contact-invariant classification. Moreover, in the hyperbolic case, a finer contact-invariant classification reveals three subclasses: equations of Monge--Ampere, Goursat, and generic type. An intriguing property about hyperbolic equations of generic type is that any equation in this class admits at most a nine-dimensional (contact) symmetry group. This is in stark contrast to the Monge--Ampere class which contains the wave equation, admitting an infinite-dimensional symmetry group. In this talk, I will describe some of the basic contact invariants that arise in the theory and give an outline of how the nine-dimensional bound is established using further tools from exterior differential systems and Cartan's method of equivalence. The nine-dimensional bound is sharp: I'll also describe how normal forms for the contact-equivalence classes of maximally symmetric generic hyperbolic equations were found as well as the symmetry algebras which arise.

 October 3 at 1:45 pm in ENPH 501 Joint Geometry & Physics Seminar Andrew Neitzke (IAS, Princeton U.)

 October 10-12 Texas Geometry and Topology Conference at the University of Texas, Austin. (No seminar Oct. 10.)

 October 17 Leonid Gurvits (Los Alamos National Labs) Sharp bounds on the Waring rank and product rank of the permanent. Postponed to January.

 October 24 at 2 o'clock in Milner 317 Joint Geometry & Physics Seminar Charles Doran (U. Alberta) Normal Forms for K3 Surfaces and Modular Parametrization. Motivated in part by a string theoretic duality, we consider special classe of algebraic surfaces of Calabi-Yau type (i.e., K3 surfaces) of high Picard rank with a pair of canonically associated fibration structures. The related problems of describing normal forms, generalizing that of Weierstrass for elliptic curves, and modular parametrizations, generalizing the J-line for elliptic curve moduli, are considered. The solution to both these problems follows from a careful description of an algebraic correspondence between these K3 surfaces and certain abelian surfaces arising either as products of elliptic curves or as the Jacobian of a curve of genus two. We then apply these normal forms and modular parametrizations to characterize modular curves, Humbert surfaces, and Shimura curves via their uniformizing differential equations. This is joint work with Adrian Clingher and my students Jacob Lewis and Ursula Whitcher.

 October 24 Guoliang Yu (Vanderbilt U.) An equivariant index theorem and its applications to geometry. I will discuss an equivariant index theorem for the Dirac operator on noncompact manifolds and its applications to geometry and topology of three dimensional manifolds. This is joint work with Stanley Chang and Shmeul Weinberger.

 October 31 Andreas Cap (U. of Vienna) Curved Casimir operators. Curved Casimir operators provide a new, systematic approach to the construction of invariant differential operators for parabolic geometries. They tie in nicely with tractor calculus and the machinery of BGG sequences, and to apply them to specific problems, usually only requires verifications from finite dimensional representation theory. In the talk, I will briefly review the representation theory origins of curved Casimirs and then describe the general construction. Finally, I will outline how Curved Casimirs can be used both for the construction of specific examples of invariant differential operators and for general existence proofs. The talk is based on joint work with V. Soucek and A.R. Gover.

 November 7 Joint Geometry & Physics Seminar Paul Aspinwall (Duke U.) Probing geometry with D-branes.

 November 14 Jason Morton (Stanford U.) Tensor geometry and cumulants. We discuss a new statistical technique inspired by research in tensor geometry and making use of cumulants, which are the higher order tensor analogs of the covariance matrix. For non-Gaussian data not derived from independent factors, covariance matrix (PCA) and diagonal tensor (ICA) based tensor decomposition techniques for factor analysis are inadequate. Seeking a Zariski closed space of models which is computable and statistically meaningful leads to a proposed extension of PCA and ICA named Cumulant Component Analysis (CCA). Estimation is performed by maximization over a Grassmannian. Joint work with L.-H. Lim.

 November 21 Boris Kruglikov (U. of Tromso) Compatibility of overdetermined systems of PDEs, multi-brackets of nonlinear differential operators and applications. I will describe a compatibility criterion of overdetermined systems of differential equations. It involves a new algebraic object: multi-bracket of differential operators, which can be also treated as a differential syzygy. Among applications I will consider description via differential invariants of the local degree of mobility of a 2-dimensional metric (which depends only on its projective class).

 November 28 No seminar. (Thanksgiving.)

 December 4 Pawel Nurowski (SUNY, Stonybrook) Colloquium: 2 o'clock, Milner 317. Lorentzian approach to CR geometry. Many important Lorentzian manifolds of General Relativity theory, such as for example the Kerr manifold describing a rotating black hole, are foliated by null geodesics in such a way that the foliation has vanishing shear. The shear-free property of such a foliation means that its 3-dimensional leaf space is naturally equipped with a CR-structure. An example of a CR-structure is a real 3-dimensional manifold embedded in C2. However, the majority of abstractly defined 3-dimensional CR manifolds can not, even locally, be embedded in C2 in such a way that the CR-structure induced from the complex ambient space coincides with the abstract one. On the other hand every 3-dimensional CR manifold N naturally defines a class of Lorentzian metrics [g] on M=NxR, in which N parametrizes null geodesics without shear in M. This gives a local one-to-one correspondence between 3-dimensional CR-structures and foliations by null geodesics without shear in spacetimes. In this lecture we discuss in detail relations between metrics [g] and their corresponding 3-dimensional CR manifolds. In particular, we give a criterion for the local embeddability of 3-dimensional CR manifolds in terms of curvature conditions for metrics from the corresponding class [g]. An example of such a condition is the existence of an Einstein metric in the class [g].

 December 5 Pawel Nurowski (SUNY, Stonybrook) Conformal geometry of differential equations. Given two differential equations it is often useful to know invariants which guarantee that there exists a transformation of variables (independent, dependent or both) that transforms one of the equations into the other. Recently it has been observed, that various classes of ODEs and PDEs, when considered modulo some specific kinds of transformations of the variables, fall into nonequivalent classes of equations, whose local invariants are conformal invariants of apropriately defined pseudo-riemannian metrics on manifolds. In this talk we provide some examples of this phenomenon. The most striking of them associates a conformal 5-dimensional geometry of signature (2,3), with the equation z'=F(x,y,y',y'',z). This conformal geometry has Cartan normal conformal connection reduced from so(3,4) Lie algebra to the exceptional g2 Lie algebra. This implies in particular that Cartan's invariants of 2-dimensional nonintegrable distributions in dimension five are just conformal invariants of this (2,3)-signature conformal geometry.

Previous Semesters
Spring 2008
Fall 2007
Spring 2007
Fall 2006
Spring 2005