Current Seminars - Fall 2009

Monday, November 2

GMA Visions Seminar

Professor Patrick Eberlein, UNC-CH, "Closed Orbits for Actions of Self Adjoint Groups,"Phillips 367, 4:00pm

                                                          ABSTRACT

Let G be a group acting by linear transformations on a finite dimensional real vector space V.  Let < , > be a positive definite inner product on V relative to which G is self adjoint ; that, is G is closed under the metric adjoint operation.  A nonzero vector v in V is minimal if it has the smallest norm of all elements in the orbit G(v).  It is known that an orbit G(v) is closed in V if and only if it contains a minimal vector.

Minimal vectors v typically have a special geometric significance.  For example, if G = GL(n,R) acts on V = M(n,R) by conjugation, then the minimal vectors are those matrices A in V that commute with their transposes.  I will describe a general method for determining minimal vectors and a linear algebra criterion for deciding when an orbit G(v) is closed.  Using this criterion one can show that if G(v) is closed with compact stabilizer G_{v} for one nonzero vector v, then G(w) is closed for w in a dense open subset O of V.  (However, the stabilizer G_{w} may not be compact for all w in O). I will present an application in geometry to finding an optimal Ricci tensor on a generic 2-step nilpotent Lie group.  This involves the group G = SL(q,R) x SL(p,R) acting appropriately on V = so(q,R) x ... x so(q,R) (p times), where p,q are positive integes with p at most (1/2)q(q-1) and so(q,R) denotes the vector space of skew symmetric q x q real matrices.

Tuesday, November 3

Ergodic Theory and Dynamical Systems Seminar

Professor Jane Hawkins, UNC Chapel Hill, "Iteration of Elliptic Functions and Parameter Space," Phillips 383, 3:30pm, Refreshments in Phillips 330 at 3:00pm

                                                          ABSTRACT

We study parameter space for eierstrass elliptic P functions with square period lattices since this restriction  gives a family of elliptic functions which can be parametrized by a single nonzero complex parameter. On the other hand this class already exhibits most of the richness of behavior typical of elliptic functions.

We discuss a specific  question posed by Bob Devaney about the parameter space. In his Ph.D. work Josh Clemons has shown that Mandelbrot-like bifurcations occur and that the maps in the family are quadratic-like. There appear to be large uncharted areas in parameter space and the natural question is: what occurs in parameter space "between Mandelbrot sets"?  We discuss an answer to this; it is joint work with Mark McClure of UNC Asheville.

Wednesday, November 4

Analysis/PDE Seminar

Professor Joseph Cima, UNC-CH, "Introduction to H1 ," Phillips 381, 4:00-5:00 pm

                                                          ABSTRACT

There will be a brief introduction to the analytic theory on the disc.Then we will define the "real" Fefferman and Stein Hardy space(s) on R^n. Much of this material is taken from Stein's book, Harmonic Analysis (Princeton Press- which I have checked out of the library!), Chapters 3,4.

I will give a few of the shorter proofs and just quote other pertinent results. Mainly how can one tell if an L1 function can be in the Hardy 1 space, definitions of atoms, definition of BMO, the topological dual etc.

The level of this talk will be suitable for graduate students. There will be a second lecture two weeks later.

Friday, November 6

Physically Inspired Mathematics Seminar

Professor Ivan Cherednik, UNC-CH, "Q-Whittaker Function in Rank One,"

Phillips 385, 3:00 pm

                                                          ABSTRACT

Q-Whittaker function are  special eigenfunctions of the q-Toda difference operators, which play important role in the modern theory of affine flag varieties. Analytically, they are limits of the global hypergeometric functions introduced by the speaker and Stokman (in the C-check-C case). There is significant recent progress in the algebraic and analytic theory of the latter functions, generalizing the celebrated Macdonald's polynomials; it helps a lot in the theory of the Q-Whittaker functions and opens a road to the "next" levels of the quantum geometric Langlands correspondence. The Q-Whittaker functions miss the key symmetries of the global hypergeometric functions. However their coefficients have important integrality-positivity properties (which are, generally, absent in the Macdonald theory). These coefficients are directly related to the level one Demazure characters of Kac-Moody algebras, Q-Hermite polynomials,the quantum K-theory of flag varieties Givental-Lee) and, hopefully, to the affine IC-theory. There will be two seminars devoted mainly to the rank one case (almost from scratch).

Applied Mathematics Colloquium

Sergey Panyukov, Lebedev Physics Institute, "Why Brownian yet Anomalous?"

Phillips 332, 4:00 pm

Geometric Methods in Representation Theory Seminar

Professor Anthony Henderson, "Enhancing the Nilpotent Cone,"
Phillips 367, 4:15-5:15 pm

                                                         ABSTRACT

Many features of an algebraic group are controlled by the geometry of its nilpotent cone, which in the case of GL_n(C) is merely the variety N of n×n nilpotent matrices. The study of the orbits of the group in its nilpotent cone leads to combinatorial data relating to the representations of the Weyl group, via the famous Springer correspondence. In the case of GL_n(C), the basic manifestation of this correspondence is the fact that conjugacy classes of nilpotent matrices and irreducible representations of the symmetric group are both parametrized by partitions of n.

Pramod Achar and I have shown that studying the orbits of GL_n(C) in the enhanced nilpotent cone C^n × N leads to exotic combinatorial data of type B/C (previously studied by Spaltenstein and Shoji). As I will explain, this is closely related to Syu Kato's exotic Springer correspondence for the symplectic group, and also to nilpotent orbits in characteristic 2.