"Kahler manifolds, line bundles, and Toeplitz operators"
Time: 10:30 AM
Location: Hill 705
Abstract: Let L be a holomorphic hermitian line bundle on a compact Kahler manifold
M. (Berezin-)Toeplitz operators
act on spaces of holomorphic sections of powers of L. They have been used
for many years in geometric quantization,
and more recently in other areas (including work by J. Andersen and by L.
Polterovich). I will give an introduction
to this topic and will present some very recent results from our joint
paper with X. Ma, G. Marinescu and M. Pinsonnault.
Friday, March 8th
Igor Zelenko, Texas A&M University
"Geometry of vector distributions: from Cartan to Tanaka and beyond"
Time: 10:30 AM
Location: Hill 705
Abstract: My talk is devoted to local equivalence problem for vector distributions (subbundles of tangent bundles) on manifolds with respect to the action of the group of diffeomorphisms. Vector distributions appear naturally in Geometric Control Theory as the sets of admissible velocities for control systems linear with respect to control parameters, in Geometric Theory of Differential Equations as natural distributions on submanifolds of jet spaces, and CR geometry as complex subbundles of the tangent bundle of a real submanifold of C^N. The general way to solve such equivalence problems is to assign to a geometric structure the (co)frame (or the structure of absolute parallelism) on some (fiber) bundle over the ambient manifold in a canonical way. In my talk first I will review the classical approaches to this problem, making special emphasis to the algebraic version of Cartan's method of equivalence developed by N. Tanaka in 1970s. The central object in the Tanaka approach is the notion of a symbol of a distributions at a point, which is a graded nilpotent Lie algebra. The prolongation procedure (i.e.
the procedure of getting a canonical frame) can be described in terms of natural algebraic operation in the category of graded Lie algebras. Through this review of Tanaka theory I will motivate the recent approach of B. Doubrov and myself to this problem reducing the original problem to the problems of equivalence of curves of symplectic flags. The latter problem is simpler in many respects than the original one. Our approach is a combination of a certain symplectification of the problem (taking its origin in Pontryagin theory in Optimal Control) and various novel Tanaka type prolongations. This approach allowed us to make a unified construction of canonical frames for distribution of arbitrary rank independently of their Tanaka symbols, avoiding the problem of classification of graded nilpotent Lie algebras with given number of generators, which is important for the application of the Tanaka theory. Our approach significantly extends the set of distributions for which the canonical frame can be explicitly constructed.
Friday, February 22nd
Samuel Grushevsky, State University of New York at Stony Brook
"Commuting differential operators and spectral curves"
Time: 10:30 AM
Location: Hill 705
Abstract: We will explain how two commuting differential
operators give rise to an algebraic plane curve (in the complex projective
plane), called the spectral curve.
This lecture is a pre-colloquium lecture
addressed to introduce graduate students to concepts and ideas that will
be developed further in the departmental colloquium later at 4.00pm.
OF NOTE:
Light refreshments will be served for the audience and speaker after the talk between 11:30 and noon in 703.
Friday, February 15th
Guo-Zhen Lu, Wayne State University
"Recent development on sharp Moser-Trudinger and Adams inequalities in the Euclidean spaces and on the Heisenberg group"
Time: 10:30 AM
Location: Hill 705
Abstract: In this talk, we will report some recent development on sharp subcritical and critical Moser-Trudinger inequalities on first order Sobolev spaces and Adams inequalities on high order Sobolev spaces on the entire Euclidean spaces. Sharp sub-critical and critical Moser-Trudinger inequalities on the whole Heisenberg group will also be discussed.
Moreover, we will mention some recent works on sharp affine Moser-Trudinger inequalities.
Due to the absence of the symmetrization on the Heisenberg group and also on the high order Sobolev spaces, a key point is that we develop a rearrangement-free method to establish these sharp inequalities. These are joint works with Nguyen Lam and Hanli Tang.
Friday, February 8th
Bianca Santoro, City University of New York
"Complete Ricci-flat Kahler metrics on resolutions of singularities"
Time: 10:30 AM
Location: Hill 705
Abstract: We will discuss some existence results for complete Calabi-Yau metrics
on crepant resolutions of singularities, and use these results to give simple
examples of ALF Ricci-flat manifolds
Friday, January 18th
Yu Ding, California State University, Long Beach
"TBA"
Time: 10:30 AM
Location: Hill 525
Abstract: TBA
This page was last updated on April 15, 2013 at 12:22 pm and is maintained by webmaster@math.rutgers.edu.
For questions regarding courses and/or special permission, please contact
mclausen@math.rutgers.edu.
