Analysis-Geometry Seminar
Analysis-Geometry
Seminar
Meets Fridays
2-3 pm in 509 Lake Hall
We generally
go to lunch beforehand.
Please join us at 457 Lake Hall at 12:30 pm.
This seminar
features talks in the fields of partial differential equations,
functional analysis, differential geometry and topology, and
mathematical physics. The seminar is supplemented by the
Graduate
Students Analysis and Geometry seminar.
The organizers:
Maxim
Braverman, Bob McOwen,
Mikhail
Shubin, Peter
Topalov
Upcoming Talks:
- January 23, 2009
Speaker: Stanislav Dubrovskiy
( Northeastern University
) Title: Moduli space of general connections
Abstract: Finiteness of functional moduli is a recurring theme in local differential geometry.
In this talk we investigate the moduli space of general connections (with torsion). We consider the action of the group of origin-preserving diffeomorphisms on the space of germs of generic connections at a point. The resulting moduli space gives rise to a Poincare series. By analyzing the corresponding moduli spaces of k-jets we calculate the series and establish that it is in fact a rational function, indicating a finite number of functional invariants.
This conforms once again the finiteness conjecture of Tresse, that algebras of invariants of "natural" differential-geometric structures are finitely generated.
- January 30, 2009
Speaker: Sergei Yakovenko
( Weizmann Institute, Israel
) Title: Oscillatory properties of Fuchsian ordinary differential equations in the real and complex domain
Abstract: It is a well-known fact that solutions of second order linear ODE with bounded coefficients admit explicit estimate of the distance between consecutive zeros of its solutions (Sturm theory). It is virtually unknown that this fact is valid for any linear ODE away from singular points, and a complex generalization exists which allows to treat also complex zeros of (multivalued) solutions.
Under certain assumptions on the nature of singular points it is possible to extend the above "oscillation-control" type results onto sectors with vertices at the singularities.
Application of this simple but powerful theory allows to construct an explicit bound for the number of zeros of Abelian integrals (the Infinitesimal Hilbert 16th problem).
The results were obtained in a joint work with Gal Binyamini and Dmitry Novikov.
- February 13, 2009
Speaker: Peter Topalov
( Northeastern University
) Title: On the Integral Geometry of Liouville Billiard Tables
Abstract: A notion of Radon transform for completely integrable billiard tables is introduced. It will be shown that in the case of Liouville billiard tables of dimension 3 the Radon transform is one-to-one on the space of continuous functions K on the boundary of the billiard. This allows us to obtain spectral rigidity of the corresponding Laplace-Beltrami operator with Robin boundary conditions.
- February 20, 2009
Speaker: Ivan Horozov
( Brandeis University
) Title: Gravity and Regulators of Number Fields
Abstract: We are going to describe two applications of a new tool - higher dimensional iterated integrals. One of the applications is in number theory and the other - in quantum physics.
The number theoretic application is about Borel regulator of a number field: We express the values of the Dedekind zeta function at the positive in terms of multiple polylogarithms. Zagier has conjectured that single-valued polylogarithms are enough.
The second application is a new approach to quantum gravity. A starting point for the gravity that we consider is general relativity in terms of a connection, using spinors. (Some people who have worked in this direction are sir Penrose, Ashtekar, Gambini, Lano, Fedosin, Agop, Buzea and Ciobanu, Mashhoon, Gronwald, and Lichtenegger, Clark and Tucker. There are many relations between the two application. For example: the loop expansion in this approach to quantum gravity is done in terms of Borel regulators of number fields.
- February 27, 2009
Speaker: Dave Finn
( Rose-Hulman Institute of Technology
) Title: TBA
Abstract:
- March 24, 2009 (Note special date)
Speaker: Alexander Turbiner
( UNAM, Mexico
) Title: A new continuous family of two-dimensional exactly-solvable and (super)integrable Schroedinger equations
Abstract: It is shown that the Smorodinsky-Winternitz potential, BC2 rational model, 3-body Calogero model, Wolves potential (G2-rational model in Hamiltonian Reduction nomenclature) are the members of a continuous family of two-dimensional exactly-solvable and (super)integrable Schroedinger equations marked by some continuous parameter. Their spectra is always linear in quantum numbers. Hidden algebra of the family for integer values k of the parameter is uncovered. It is non-semi-simple Lie algebra gl(2) x Rk+1 realized as vector fields on line bundles over k-Hirzebruch surface.
- April 3, 2009
Speaker: Justin Holmer
( Brown University
) Title: Motion of mKdV 2-solitons in an external field
Abstract: We consider the mKdV equation with a slowly varying potential term, and show that both single and double solitons remain intact but move with parameters of motion described by ODEs. These ODEs are formally predicted by symplectic projection, although the rigorous proof relies on substituting an ansatz into the equation and controlling errors using the Lyapunov functional employed in stability theory. The results are valid on a long enough time scale to observe interesting dynamics in the semiclassical limit. We confirm the results with numerical simulations. This is joint work with Maciej Zworski and Galina Perelman.
- April 17, 2009
Speaker: Jonathan Weitsman
( Northeastern University
) Title: Fermionization, convergent perturbation theory, and correlations in quantum gauge theories
Abstract: The problem of understanding path integrals associated to quantum gauge theories is a longstanding issue in mathematical physics, and now also in differential geometry. We show that quantum gauge theories in three and four dimensions are equivalent to purely fermionic theories, where, with appropriate cutoffs, the perturbation series is convergent. Classical techniques, developed in the 1980''s, have been used in the past to understand the path measures in similar cases, and we hope that they are useful in this situation also. Meantime as a byproduct we obtain some natural conjectures about the behavior of correlations in four-dimensional Yang Mills theory, and explore connections with the invariants of three-manifolds.
The technique of fermionization was developed in the 1970''s in the context of two dimensional quantum field theories, where the fermions are quantum solitons or vertex operators. The fact that these ideas can be used in gauge theory is unexpected. Mathematically, this is an indication that in many cases which arise naturally in geometry, measures on Banach spaces are related to much simpler algebraic objects arising from the calculation of certain determinants of operators on these spaces. This kind of relation cannot occur for finite-dimensional vector spaces, and may be indicative of the surprising ways in which integrals on infinite-dimensional spaces differ from---and are much simpler than---their finite-dimensional analogs.
In this talk I will survey the basic ideas of integration on function spaces and then discuss these new developments.
You
can also view past talks
that we have had: Fall 2008, Spring
2008,
Fall 2007, Spring
2007,
Fall 2006,
Spring 2006,
Spring 2005,
Fall 2004,
Spring 2004,
Fall 2003,
Winter 2003, Fall 2002,
Spring 2002,
Fall 2001,
Spring 2001,
Winter 2001,
Fall 2000,
1999-2000,
1998-9,
1997-8
or
1996-7.
Created:
August 20, 2000.
Comments to: 
