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Analysis-Geometry
Seminar
Meets
Fridays 2:00-3:00 pm in 509 Lake Hall
We
generally go to lunch beforehand.
Please join us at 457 Lake Hall at 12:30
This seminar features talks in the fields of
partial differential equations, functional analysis, differential geometry and
topology, and mathematical physics.
The organizers are: Maxim Braverman,
Bob McOwen, Mikhail Shubin, Peter
Topalov
Talks in 2010-2011:
·
September 10,
2010
Speaker: Frederick
E. Daum (Raytheon Company)
Title: Ten dubious methods to solve a first order linear underdetermined
single PDE for exact particle flow in nonlinear filters
Abstract: We analyze ten methods to solve a first order linear (highly) underdetermined scalar PDE that arises in nonlinear filtering, including: generalized inverse of the differential operator, irrotational flow, incompressible flow, variational principle, optimal control, method of characteristics, separation of variables, direct integration, the hprinciple, as well as the most general solution. The key issues are uniqueness of the solution of the PDE and stability of the particle flow. This new theory improves the computational complexity of particle filters by many orders of magnitude. Moreover, it is a new method to multiply two functions in a high dimensional space.
·
October 1, 2010
Speaker:
Chris Beasley (Northeastern
University)
Title: Non-Abelian
Duality on a Three-Manifold
Abstract: Non-abelian duality is a quantum version of Poincare duality in dimensions two, three, and four. In dimension two, non-abelian duality is manifested as mirror symmetry, and in dimension four, it is manifested as electric-magnetic duality. Today I will explain aspects of non-abelian duality in dimension three.
·
October 15, 2010
Speaker:
Gerard Misiolek
(
Title: Euler-Arnold equations of right-invariant metrics on diffeomorphism groups
Abstract: In 1966
·
October 29, 2010
Speaker: Vasilisa Shramchenko (
Title: Riemann-Hilbert problems associated with Hurwitz spaces
Abstract: There are two (dual to each other) Riemann–Hilbert problems naturally associated with every semisimple Frobenius manifold. In the case of Frobenius structures on Hurwitz spaces (moduli spaces of functions over Riemann surfaces), the corresponding Riemann–Hilbert problems turn out to be solvable in terms of meromorphic bidifferentials defined on the underlying Riemann surface. In this talk, I will define the Hurwitz spaces, present solutions to one of the Riemann–Hilbert problems (the one with Fuchsian singularities) arising in the Frobenius manifolds theory and discuss their properties and monodromy.
·
November 5, 2010
Speaker: Roman Shterenberg (
Title: Complete asymptotic expansion of the integrated density of states of
multidimensional almost periodic Schrödinger operators
Abstract: We prove
the complete asymptotic expansion of the integrated density of states of a
Schrodinger operator acting in R^d when the potential
b is either smooth periodic, or generic quasi-periodic (finite linear
combination of exponentials), or belongs to a wide class of almost-periodic
functions.
·
November 19, 2010
Speaker: Vladimir Chernov (
Title: Causality, Low
conjecture and the partial order on the space of Legendrian
spheres
Abstract: Let X be
a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset in the Euclidean spce. The Legendrian Low
conjecture formulated by Natario and Tod says that two events x, y in X are causally related if
and only if the Legendrian link of spheres whose
points are light geodesics passing through x and y is non-trivial in the
contact manifold of all light geodesics in X. The Low conjecture says that for
dimension two the events x, y are causally related if and only if the Legendrian link
is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar
statements hold for any globally hyperbolic X such that the universal cover of
its Cauchy surface is an open manifold. The conjecture follows from the
existence of the natural partial order on the space of Legendrian
spheres in the spherical cotangent bundle of such Cauchy surfaces. We also
discuss related joint works with Yuli Rudyak.
·
January 21, 2011
Speaker: Robert McOwen (Northeastern University)
Title: The Calderon
Conductivity Problem
Abstract: In 1980, A.P. Calderon proposed the inverse problem of determining the conductivity of a region from boundary measurements, namely the so-called Dirichlet-to-Neumann map. This problem is "ill-posed" but has important applications in medical imaging, mineral/oil prospecting, etc. Results on the problem have relied upon sophisticated mathematics (PDE theory, harmonic analysis, scattering theory, quasi-conformal maps, etc) and there are still many interesting open questions.
·
February 11, 2011
(2:00-3:00pm)
Speaker: Feride Tiglay (The Fields Institute)
Title: Integrable evolution
equations on spaces of tensor densities
Abstract: In a pioneering paper V. Arnold presented a general framework within which it is possible to employ geometric and Lie theoretic techniques to study a variety of equations of interest in mathematical physics. I will describe how to extend his formalism using tensor densities and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. These two PDE possess all the hallmarks of integrability: the existence of a Lax pair formulation, a bihamiltonian structure, the presence of an infinite family of conserved quantities and the ability to write down explicitly some of its solutions. I will also talk about local well-posedness of the corresponding Cauchy problem and some results on blow-up as well as global existence of solutions. Time permitting, I will describe the “peakon” solutions for these equations.
·
February 11, 2011
(3:15-4:15pm)
Speaker: Roman Belavkin (
Title: Optimal measures and transition kernels
Abstract: We study positive measures that are solutions to an abstract optimisation problem, which is a generalisation of a classical variational problem with a constraint on information of a Kullback Leibler type. The latter leads to solutions that belong to a one parameter exponential family, and such measures have the property of mutual absolutely continuity. Here we show that this property is related to strict convexity of a functional that is dual to the functional representing information, and therefore mutual absolute continuity characterises other families of optimal measures. This result plays an important role in problems of optimal transitions between two sets: Mutual absolute continuity implies that optimal transition kernels cannot be deterministic, unless information is unbounded. For illustration, we construct an example where, unlike non-deterministic, any deterministic kernel either has negatively infinite expected utility (unbounded expected error) or communicates infinite information.
·
February 18, 2011
(3:15-4:15pm)
Speaker: Paul Bressler (
Title: Formality for algebroids
Abstract: The celebrated formality theorem of Kontsevich establishes a bijection beween the collection of equivalence classes of (non-commutative) deformations of the algebra of functions on a manifold on the one hand, and the collection of equivalence classes of deformations of the trivial Poisson bracket. I will describe a generalization of this result to algebroids which are twisted forms of the sheaf of functions on the manifold (a.k.a. gerbes).
·
March 18, 2011
(2:00-3:00pm)
Speaker: Grigory
Litvinov (
Title: Integral geometry, Hypergroups and
I.M. Gelfand's Question
Abstract: It is well known that the Radon Transform is closely related to the classical Fourier transform and harmonic analysis on the additive groups of finite dimensional real linear spaces. In this talk we discuss ``similar" interrelations between standard problems of Integral Geometry (in the sense of Gelfand and Graev) and harmonic analysis on certain commutative hypergroups (in the sense of J. Delsarte). These interrelations may be interpreted as an answer to an old question of I.M. Gelfand concerning algebraic foundations of Integral Geometry.
·
April 1, 2011
(10:35-11:40pm)
Speaker: Stefan Rosemann (Friedrich-Schiller-Universität
Title: Yano-Obata conjecture for holomorph-projective transformations
Abstract: In this
talk, i want to present a joint work with V. Matveev. Given a Kaehler
manifold, one associates a class of distinguished
curves to the Kaehler metric called h(olomorphically)-planar curves. Such curves can be seen as
some kind of generalisation of geodesics on Kähler manifolds. One
problem of interest is to understand whether the group of
holomorph-projective transformations (i.e.,
the group of all bi-holomorphic mappings which
preserve the set of all h-planar curves) is really bigger then the group
of holomorphic isometries
of the Kaehler manifold. We have proven a classical
conjecture attributed to Yano and Obata stating that on a compact, connected
Riemannian Kaehler manifold, the connected components
of both groups coincide unless the metric has constant positive holomorphic
sectional curvature.
·
April 1, 2011
(2:00-3:00pm)
Speaker: Chris Kottke (
Title: Fiber products for manifolds with
corners and generalized blow-up
Abstract:. I will give a brief introduction
to the theory of manifolds with corners and `b-maps'. Then I will discuss a new
result with Richard Melrose which characterizes those transversal maps for
which smooth fiber products exist in this category, along with a theory for
resolving the fiber products when they fail to be smooth. This result is an
application of a new construction called ``generalized boundary blow-up,''
meant to extend the iterative process of successive radial blow-up of boundary
faces. This construction uses ideas similar to those in toric
geometry to approach complicated resolution problems through combinatorial
methods
.
·
April 8, 2011
(10:35-11:40am), room LA 575 !!! (Note the unusual time and place.)
Speaker: Alberto de Sole
(
Title: Bi-Hamiltonian integrable systems
Abstract: We will describe the relation between the theory of Poisson vertex algebras and the theory of Hamiltonian equations. In particular, we will see how to use the conformal structure to prove integrability of a bi-Hamiltonian system. Time permitting, we will describe some recent results on the classification of rank 1 Hamiltonian operators, and on the cohomology of the generalized variational complex.