Analysis-Geometry Seminar

• October 20, 2000
  Speaker: Michael Farber  (Tel Aviv University)
  Title: On the zero in the spectrum conjecture
  Abstract:  In the talk I will describe a negative solution to the zero-in-the-spectrum conjecture obtained in a recent joint work with S. Weinberger. We constructed for any n>5 a closed smooth n-dimensional manifold M such that the Laplace-Beltrami operator acting on L² forms of all degrees on the universal covering M^~ of M is invertible. Our work was inspired by the well-known paper of M.A. Kervaire who constructed homology spheres with prescribed fundamental groups. We exploit the method of the extended L²-cohomology which treats the Novikov-Shubin phenomenon ( "zero in the continuous spectrum").
   
• November 3, 2000
  Speaker: Burkhard Wilking (UPenn)
  Title: New Examples of Manifolds with Positive Sectional Curvature Almost Everywhere
  Abstract: There are only very few examples of Riemannian manifolds with positive sectional curvature known.  In fact ,in dimensions larger than 24, the known examples are diffeomorphic to locally rank 1 symmetric spaces.
      We will construct metrics with positive sectional curvature on a open and dense set of points on the projective tangent bundles of RPⁿ, CPⁿ and HPⁿ.
      The so called deformation conjecture says that these kind of metrics can be deformed into metrics with positive sectional curvature everywhere.
      However, the simplest new example within our class,  the projective tangent bundle of RP³, is diffeomorphic to the product RP³xRP².  This non-oriented manifold is known not to admit a metric with positive sectional curvature. Thus the construction provides a counterexample to the deformation conjecture.
   
• November 10, 2000
  Speaker: Alexander G. Abanov (Department of Physics, SUNY at Stony Brook) 
  Title: Topological terms in effective action induced by Dirac fermions
  Abstract: We derive an effective action for Dirac fermions on three-dimensional sphere coupled to O(3) non-linear sigma model through the Yukawa-type interaction.  The non-perturbative (global) quantum anomaly of this model results in a Hopf term for the effective non-linear sigma model.  We obtain this term using the "embedding'' of the CP¹ model into the CP^M generalization of the model which makes the quantum anomaly perturbative.  This perturbative anomaly is calculated by means of a gradient expansion of a fermionic determinant and is given by the Chern-Simons term for an auxiliary gauge field.
   
   
• November 17, 2000
  Speaker: Yulij S. Ilyashenko  (Cornell University)
  Title: Some bounds of the number of limit cycles for Abel and Lienard equations
         (partly based on a joint work with A.Panov).
Abstract: We estimate the number of limit cycles of planar vector field through the size of the domain of the Poincaré map,  the increment of this map and the width of the complex domain to which the  Poincaré map  may be analytically extended. The estimate is based on the relation between growth and zeros of holomorphic functions. This estimate is then applied to getting the upper bound of the number of limit cycles of Lienard equation dx/dt  = y - F(x),  d y/dt = -y through the (odd) power of the monic polynomial F and magnitudes of its coefficients. In the same way, an upper bound of the number of limit cycles of the Abel equation is obtained.
 
 
November 24, 2000
Speaker: Mathai Varghese (University of Adelaide and MIT)  Cancelled
   Professor Mathai will give a talk sometime during the winter quarter. 
 
December 1, 2000
Speaker: Sergei  Yakovenko (Weizmann Institute) 
Title: Rolle theorem for complex- and vector-valued functions (joint work with A. Khovansjkii) 
Abstract: The classical Rolle theorem implies an inequality between the number of zeros of a smooth function of one real variable, and that of its derivative. This real-Rolle inequality is very important for applications, and it would be highly desirable to have it also for complex-valued functions.
    Alas, the simplest examples show that no such  inequality may exist. Instead we establish a geometric inequality
between curvature and spherical length of spatial curves, which translates into an inequality for analytic functions very
similar to the Rolle inequality - except that it does not concern zeros...
   Suprisingly, despite this drawback the "complex-Rolle" inequality allows for very accurate counting of complex zeros  for exponential sums.
 
December 1, 2000,   3:30PM (Note the special time)
Speaker: Stanislav Molchanov (University of North Carolina, Charlotte)
Title:  Schroedinger operators with the mixed spectrum
Abstract: The central part of the famous (and still open) Anderson conjecture tells about possible coexistence of dense point and
absolutely continuous spectra for multidimensional Schroedinger operators with "weak disorder".
   The talk will describe several recent results on the models with mixed (dense point plus absolutely continuous) spectra. These models include Schroedinger operators with random sparse potentials and the surface Anderson model.
 
 
December 8, 2000
Speaker:   Steven Rosenberg (Boston University)
Title:  Gauge theory techniques for flat connections in quantum cohomology
Abstract: Quantum cohomology gives a finite dimensional integrable system via the Dubrovin connection.  We use gauge theory techniques to help find flat sections for the Dubrovin connection, which are key ingredients in Givental's approach to mirror symmetry.

January 26, 2001
Speaker: Vladimir Kondratiev  (Moscow State University)
Title: Qualitative properties of solutions for elliptic semilinear equations
Abstract: For the equations of the form
        Du + |u|^q-1u=0,         q = const 1,
we will discuss properties of solutions in unbounded domains (such as cone, cylinder etc.), in particular, existence or non-existence of positive solutions, and asymptotic behavior of solutions.
 

February 2, 2001
Speaker:  Andrzej Borisovich (Gdansk University)
Title:  Bifurcations in Plateau problem
Abstract: The talk is devoted  to a general method of the study of  bifurcations in the Plateau problem with parameters  in the boundary  conditions. It is based on the Crandall-Rabinowizt bifurcation theorem, the finite-dimensional  Lyapunov-Schmidt type reduction for Fredholm  maps of index 0, the key function by Sapronov and some others  constructions. Many new bifurcations of the minimal surfaces were found and  studed.
  All the results found by the author's method fit the results of physical experiments. In particular,  the experiments by J.Plateau with minimal films was described mathematically.
 

February 9, 2001
Speaker:  Bruno De Oliveira (Harvard University and University of Pennsylvania)
Title:  Complex cobordisms and the non-embeddability of CR-manifolds
Abstract: We give results on complex cobordisms whose ends are strictly pseudoconvex Cauchy-Riemann-manifolds. Suppose the complex cobordism is given by a complex 2-manifold X with one pseudoconvex and one pseudoconcave end. We introduce two methods to construct pseudoconcave surfaces that show that the complex 2-manifold X giving a complex cobordism is not determined by the pseudoconvex end. These two constructions give new methods to construct non-embeddable Cauchy-Riemann 3-manifolds and prove that embeddability of a strictly pseudoconvex Cauchy-Riemann 3-manifold is not a complex-cobordism invariant. We show that a new phenomenon occurs: there are CR-functions on the pseudoconvex end that do not extend to holomorphic functions on X. We also show that the extendability of the CR-functions from the pseudoconvex end is necessary but not sufficient for embeddability to be preserved under complex cobordisms.
   In the talk we will use differential/complex/algebraic geometric arguments so any geometer is welcome.
 

February 16, 2001
Speaker:  Alexander Turbiner (Institute for Nuclear Sciences - National University of Mexico)
Title:  Hydrogenic chains in magnetic field (state-of-the-art variational calculations)
Abstract: It was anticipated by Kadomtsev-Kudriavtsev and Ruderman at 70es, that unusual chemical compounds can appear in a strong magnetic field, which do not exist without magnetic field. One-electron exotic molecular systems are of main content of the talk. State-of-art choosing variational trial functions is used. In particular, it admits to carry out the most accurate study of H₂⁺ molecular ion. It is shown that in a strong magnetic field the systems (pppe) and (ppppe) form bound states giving rise to exotic molecular ions H₃⁺⁺ and H₄⁺⁺⁺. In the contrary, H₂⁺ ion becomes unstable for some orientations of molecular axis towards a magnetic field line. It leads to a conceptual question about a content of neutron star atmosphere.
 

March 2, 2001
Speaker: Mikhail Shubin (Northeastern University)
Title:  Semiclassical asymptotics and gaps in the spectra of magnetic Schrödinger operators
Abstract:  I will discuss an L²version of the semiclassical approximation for  a magnetic Schrödinger operator with periodic electric
and magnetic fields and a Morse type electric potential. In particular, the existence of arbitrarily large number of gaps in the spectrum can be established for a  small coupling constant.
     This is a joint work with V. Mathai.
 

March 9, 2001
Speaker: Mathai Varghese (University of Adelaide and MIT)
Title:   On some aspects of noncommutative Bloch theory
Abstract:  I  will discuss the noncommutative Bloch theory of Hamiltonians appearing in the quantum Hall effect and also the mathematics of a model of the quantum Hall effect.

March 30, 2001
Speaker: Yuri A. Kordyukov (Ufa State Aviation Technical University)
Title:   Adiabatic limits and spectral theory for Riemannian foliations.
 

April 6, 2001
Speaker: Mikhail Shubin (Northeastern University)
Title:  A discreteness of spectrum criterion for the magnetic Schrödinger operators
Abstract:  A necessary and sufficient condition for the discreteness of spectrum of a magnetic Schr\"odinger operator will be explained. In case when the magnetic field vanishes it becomes the Molchanov criterion (1953).

(This is joint work with V.Kondratiev.)

April 13, 2001
Speaker: Weiping Zhang  (Nankai Institute of Mathematics, visiting  MIT)
Title:   Toeplitz operators and index theorems on odd dimensional manifolds.
Abstract:  We discuss various index theorems for Topelitz operators on odd dimensional manifolds. On closed manifolds, classically, the corresponding index theorem can be derived from the Atiyah-Singer index theorem. It can also be proved by computing the variations of eta invariants, in using a result of Booss-Wojciechowski which expresses the Toeplitz index via spectral flows. We also descibe a recent result with Xianzhe Dai on an extension of the above index theorem to the case of manifolds with boundary.
 

April 27, 2001
Speaker: Gang Liu  (UCLA)
Title:   Moduli space of J-holomorphic curves in contact geometry.
Abstract:  We will describe some basic properties of J-holomorphic curves in the symplectization of a contact manifold, which will lay down an analytic foundation for the applications of J-holomorphic curves in  contact geometry.

• September 28, 2001
  Speaker:   Andrzej Borisovich (Gdansk University)
  Title:   Nielsen fixed  points theory, symplectic maps and Poincare-Birkhoff theorem
  Abstract:  The talk is devoted to the study of the fixed points set of the area preserving  (simplectic) selfmaps of the plane annulus.
  The study of such maps, which describe the motions of  a non-compressible  fluid, began with  works of  Poincare and Birkhoff. The fist result says that, if the map is homotopic to  identity, satisfies the twist condition and the rotation number on the boundary of annulus is nonzero, then it has at least two distinct fixed points. More general results in this direction were obtained by Franks,
  who poved that, under more general assumptions, number  of fixed points is more then  2n, where n is the rotation
  number.
      The author's approach is based on the Nielsen fixed points theory and the notion  of special degree for symplectic maps. It allows to prove analogous results in more general situation  and without the twist condition.
   
• October 12, 2001
  Speaker:   Patrick Iglesias (CNRS)
  Title:   Extension and examples of the moment map for singular and infinite differentiable spaces
   
• October 19, 2001
  Speaker:   Tatiana Toro (University of Washington)
  Title: What happens when the Poisson kernel equals 1 a.e.?
  Abstract:  In this talk we would like to convey the idea that Poisson kernel of a domain W being 1 almost everywhere is a very rigid condition.  If  W is bounded Lewis and Vogel showed it must be a ball.  We will discuss what happens in the unbounded case.
   
• October 26, 2001
  Speaker:   Patrick McDonald (New College of USF)
  Title:   Exit time moments for Brownian motion and spectral geometry
  Abstract:  Given a compact Riemannian manifold with boundary, we use exit time moments for Brownian motion to construct a sequence of geometric invariants for the underlying manifold.  While these invariants are not spectral, they determine much of the spectral and Riemannian geometry of the manifold.  In particular, we discuss a number of comparison theorems for the invariants, we prove that the invariants always determine the heat content asymptotics associated to the manifold, and, for generic domains in Euclidean space, we prove that the invariants determine the Dirichlet spectrum.
   
• November 2, 2001
  Speaker: Robert Brooks (Technion-Israel Institute of Technology)
  Title:   A Statistical Model for Riemann Surfaces
  Abstract:  (joint work with Eran Makover) We study the question: What does a typical Riemann surface look like geometrically? We model the problem of picking a Riemann surface at random on the problem of picking a 3-regular graph at random, and show that this gives an interesting picture of a typical Riemann surface.
   
   
• November 16, 2001
  Speaker: András Vasy (MIT)
  Title:   Semiclassical estimates in asymptotically Euclidean scattering
  Abstract:  In this joint work with Maciej Zworski, we study long-range perturbations of the Laplacian on an asymptotically Euclidean space. We show how positive commutator estimates obtained via the symbol calculus can be used to show the limiting absorption principle, and give estimates for the resolvent, under non-trapping assumptions, as Planck's constant h goes to 0.
   
   
• November 30, 2001
  Speaker: Paul Kirk  (Indiana University)
  Title:   A splitting theorem for the spectral flow of a path of Dirac operators
  Abstract:  We explain how to calculate the spectral flow of  a path D_t  of Dirac operators on a closed manifold M decomposed along a hypersurface in terms of the spectral flow of D_t on the pieces with respect to elliptic boundary conditions, Maslov indices of the Boundary conditions, and terms coming from an adiabatic stretching procedure. The method is elementary and we show how other similar theorems in the literature follow easily.
   
   
• December 7, 2001
  Speaker: Victor Nistor (Penn State University)
  Title:   Geometric operators on manifolds with cylindrical ends and generalizations
  Abstract:  I will begin by recalling a few classical results on the analysis on manifolds with cylindrical ends, including Fredholm conditions and a determination of the spectrum of the Laplace operator. Then I will describe a class of manifolds for which one can obtain similar results. This class of manifolds is described in terms of Lie algebras of vector fields, as in geometric scattering theory. These results are based, in part, on joint works with B. Ammann, R. Lauter, R. Melrose, and B. Monthubert.

January 4, 2002
Speaker:   Mikhail Katz  (Bar Ilan University)
Title: Cup length, systolic geometry, and surjectivity of period map
Abstract:  The result concerns a 2-dimensional conformal invariant of Riemannian metrics on 4-manifolds, similar to conformal length for Riemann surfaces.  The approach makes use of two ingredients.  The first is the Conway-Thompson unimodular lattices of high density, known to exist through probabilistic or averaging procedures, but difficult to pin down explicitly.  The second ingredient is the current work in gauge theory, which targets the surjectivity of the period map for the class of 4-manifolds with B⁺ = 1.  The outcome is a polynomial asymptotic lower bound for the conformal invariant as the Betti number increases.  On the other hand, a polynomial upper bound is the result of joint work with V. Bangert.  The polynomial upper and lower bounds can be viewed as a higher-dimensional analog of the logarithmic upper and lower bounds for conformal length on Riemann surfaces, due to P. Buser and P. Sarnak.

January 18, 2002  Cancelled
Speaker:   Leonid Friedlander  (The University of Arizona)
Title: On the density of states of periodic media in the large coupling limit
Abstract:  Let W₀ be a domain in the cube (0,p )ⁿ, and let c_t(c) be a function that equals 1 inside W₀, equals t in
(0,p )ⁿ \ W₀, and that is extended periodically to Rⁿ. It is known that, in the limit  t®¥, the spectrum of the operator -c_t
exhibits the band-gap structure. We establish the asymptotic behavior of the density of states function in the bands.
 

January 25, 2002
Speaker:    Vladimir Kondratiev  (Moscow State University)
Title: Existence and non-existence of positive solutions for second order non-linear ellitpic equations in unbounded domains
Abstract:  We consider a non-linear elliptic second-order equation, which has a linear divergence type ellpitic operator as its  principal part. It is considered in domains like cone, cylinder, paraboloid or strip. Existence and non-existence conditions are obtained. They depend on the type of domain and  the non-linear terms in the equation.
 

February 1, 2002
Speaker:   Ognjen Milatovich (Northeastern University)
Title: Essential self-adjointness of  Schrödinger type operators.
Abstract:  Several essential self-adjointness conditions for the Schrödinger  type operators on manifodls and in sections of vector bundles will be explained in the talk. These conditions are expressed in terms of completeness of certain metrics on the manifold. These metrics are  naturally associated with the operator.
                (Joint work with M.Braverman and M.Shubin.)
 

February 8, 2002
Speaker:   Alexander Kozhevnikov (University of Haifa)
Title: On a Complete scale of isomorphisms for elliptic and parabolic pseudodifferential boundary-value problems.
Abstract:  In monographs by J.-L. Lions and E. Magenes (1968) and  Ya. Roitberg (1996) a theorem on complete scale of isomorphisms has been established which, roughly speaking, means that  operators generated by elliptic  differential boundary-value problems are  isomorphisms (Fredholm operators) between Sobolev-type spaces of functions ''with  s  and '' s-d - derivatives'', where  d  is the order of the elliptic operator. The completeness of the scale means that  s  can be an arbitrary  real number.
      In a monograph by S. Eidelman and N. Zhitarashu (1990) a theorem on a complete scale of isomorphisms has been obtained for parabolic differential initial boundary-value problems.
      Due to the fact, that for  elliptic and parabolic pseudodifferential initial boundary-value problems there exist  parametrices belonging to the Boutet de Monvel algebra, a much shorter proof has been found as well as some applications of the results.
 

February 22, 2002
Speaker:   Mikhail Shubin (Northeastern University)
Title: New criteria of discreteness of spectrum for Schrödinger  operators
Abstract:  New necessary and sufficient conditions for the discreteness of spectrum  for (magnetic)  Schrödinger operators will be explained. The fact that  all these conditions are equivalent to the discreteness of spectrum leads to the equivalence of these conditions between each other. This leads to new properties  of the Wiener capacity, which at the moment have no direct proofs.
 

April 5, 2002
Speaker:   Vadim Tkachenko (Ben-Gurion University)
Title: 1d Periodic Differential Operator of Order 4
Abstract:  We consider a differential operator
                L = d⁴/dx⁴+ d/dx p(x) d/dx + q(x),           x Î R¹
with 1-periodic functions  p(x)  and q(x). We prove that characteristic equation for its Floquet multipliers is inverse and hence its spectrum in  L²(R¹)   may be described using some hyper-elliptic Riemann surface.
     We prove the following uniqueness theorem:   Let  U(l) be the monodromy matrix of operator  L  and let its characteristic determinant   det(U(l)-r I)  be the same as of operator L₀=d⁴/dx⁴.  Then  p(x)ºq(x)º0.
 

April 26, 2002
Speaker:   Maxim Braverman (Northeastern University)
Title: New proof of the Cheeger-Muller Theorem
Abstract:  We present a short analytic proof of the equality between the analytic and combinatorial torsion. We use the same approach as in the proof given by Burghelea, Friedlander and Kappeler, but avoid using the difficult Mayer-Vietoris type formula for the determinants of elliptic operators. Instead, we provide a direct way of analyzing the behaviour of the determinant of the Witten deformation of the Laplacian. In particular, we show that this determinant can be written as a sum of two terms, one of which has an asymptotic expansion with computable coefficients and the other is very simple (no zeta-function regularization is involved in its definition).
 

May 10, 2002
Speaker: Benji Fisher  (Boston College)
Title: Quasicrystals:  Algebra, Geometry, Number Theory, and Physics
Abstract:  TIn the 1980's, almost simultaneously, the first mathematical and physical quasicrystals were discovered; the first mathematical one was the Penrose tiling of the plane.  These structures are not periodic, but still have long-range order.  One of the most striking features of quasicrystals is their symmetry groups:  these may include five-fold rotations and other symmetries that are "crystallographically forbidden'' for ordinary  (periodic) crystals in two and three dimensions.
       This talk will describe some recent work on classifying the symmetry types of quasicrystals in two and three dimensions.  Another, related question,  is how to detect the symmetry type of a physical quasicrystal.  For example, certain symmetry types reveal themselves in the X-ray diffraction spectrum.  The talk will conclude with a description of current work on this question, using spaces of almost-periodic functions to study the quantum mechanics of quasicrystals.
 
 

May 17, 2002
Speaker: Jacek Szmigielski    (University of Saskatchewan and Yale)
Title: Non smooth water waves and continued fractions
Abstract:  One model of nonlinear, strongly dispersive water waves, called the CH equation (Camassa, Holm) admits non smooth travelling waves which have corners in their profiles, and yet they behave in many respects like the smooth solitons of other integrable models of nolinear waves.  Those special non smooth solitons are dubbed peakons and the intention of this talk is to give a gentle overview of their collision properties.  The mathematical locale within which all their properties become transparent is an inverse spectral problem for the  discrete Dirichlet string which in turn is an implicit part of the famous treatise by Stieltjes on continued fractions.  An appropriate adaptation of Stieltjes's work provides a decisive insight into some mechanical questions regarding collisions of peakons.  This is a part of joint work with R.Beals, D. Sattinger.

September 20, 2002
Speaker:  Theodore Voronov (University of Manchester (UMIST))
Title: Differential operators, brackets and connections
Abstract: I am going to describe remarkable relations between differential operators and bracket structures. It is known that an operator of the second order acting on functions defines a "bracket", i.e. a symmetric bilinear operation on functions satisfying the Leibniz rule (basically, the polarized principal symbol). This works on ordinary manifolds as well as on supermanifolds. In the super context, this gives a relation between odd Laplacians (or "Batalin-Vilokovisky D-operators") and odd Poisson brackets. I will show how this relation between differential operators and brackets can be made 1-1 if one considers the algebra of densities instead of the algebra of functions. This construction implicitly involves "generalized" connections: notice that every differential operator of the second order acting on functions encodes in its coefficients an "upper connection" in the bundle of volume forms (basically, the subprincipal symbol). An extension of these ideas to operators of higher order would lead to homotopy algebras.
   This is a joint work with Hovhannes Khudaverdian.
 

October 4, 2002
Speaker:  Harold G. Donnelly (Purdue University)
Title: Bounds for eigenfunctions of the Laplacian on compact Riemannian manifolds
Abstract: Suppose that φ is an eigenfunction of  -Δ  with eigenvalue  λ≠0. It is proved that

||φ||_∞  ≤   c₁λ^(n-1)/4||φ||₂ ,

where n is the dimension of  M and  c₁ depends only upon a bound for the absolute value of the sectional curvature of  M and a lower bound for the injectivity radius of  M. It is then shown that if  M admits an isometric circle action, and the metric is generic, one has exceptional sequences of eigenfunctions satisfying the complementary bounds

||φ_k||_∞  ≥   c₂λ _k^(n-1)/8 ||φ_k|| ² .

 

October 11, 2002
Speaker:  Stanislav Dubrovskiy (Northeastern University)
Title: Moduli space of symmetric connections
Abstract: We are interested in local differential invariants of a symmetric connection, under smooth coordinate changes. We consider the action of origin-preserving diffeomorphisms on a space of jets of connections and calculate dimensions of moduli spaces in generic case. We show that the corresponding Poincarè series is a rational function. This confirms one more time the 1894' finitness claim of Tresse, stated for any "natural" differential-geometric structure.
 

October 18, 2002
Speaker:  Marina Ville (CNRS and Boston University)
Title: Milnor numbers of minimal surfaces in 4-manifolds
Abstract: When a sequence of smooth embedded complex curves (C_n)  in  CP^2  degenerates to a branched curve C₀, we lose topology  ( g(C_n)>g(C₀) ) and gain singularity. Milnor gave a precise meaning to this assertion. There is a quantity - now called the Milnor number - we can compute on the branch points of  C₀  which tells us how much topology we have lost going from  C_n  to  C₀. 
   So we ask: is there anything even remotely similar if the  C_n's and  C₀  are more general surfaces (e.g. minimal surfaces) in a 4-manifold? Can we define a Milnor number for a sequence of minimal surfaces? It turns out that we have to define not one, but two Milnor numbers (the "tangent" and the "normal" one); these numbers coincide in the complex case.
   We will define these Milnor numbers, give explain geometric and topological interpretations and show how they give a partial answer to our question above.
 

October 25, 2002
Speaker:  Mikhail Shubin (Northeastern University)
Title: A new family of necessary and sufficient discreteness of spectrum conditions for Schrödinger operators
Abstract: I will explain new criteria of discreteness of spectrum for the Schrödinger operators with semi-bounded below potentials. They extend a well known result by A.Molchanov (1953) who was the first to formulate a necessary and sufficient condition for the discreteness of spectrum in terms of the Wiener capacity.
   We provide a new family of such conditions which depend on a functional parameter describing "negligible" sets.
   (This is a joint work with V.Maz'ya.)
 

November 1, 2002
Speaker:  Roland Duduchava (Razmadze Mathematical Institute (Tbilisi, Georgia))
Title: Mathematical theory of cracks: the Wiener-Hopf method
Abstract:  A crack in elastic media is modelled by the Neumann boundary value problem for a homogeneous second order partial differential equation with constant coefficients. We use potentials and the Wiener-Hopf method to obtain a full asymptotic expansion of the solution. We show that these asymptotics do not contain logarithmic terms.
 

November 8, 2002
Speaker:  Mihaela Iftime (Northeastern University)
Title: On cylindrically symmetric solutions of Einstein's field equation
Abstract: I will present a stationary cylindrically symmetric solution of Einstein's equation with dust and positive cosmological constant. The solution approaches Einstein static universe on the axis of rotation.
 

November 15, 2002
Speaker:  Eric Wang (Northeastern University)
Title: Associative cones and integrable system
Abstract: Associative 3-manifolds play an important role in calibrated geometry of G₂manifolds. We study associative cones in R⁷ with isolated singularity. In particular, we give an integrable system formulation using moving frame. We will discuss some applications.
                   (joint work with Shengli Kong)
 

November 22, 2002
Speaker:  Jeff Viaclovsky (MIT)
Title: Fully nonlinear equations on Riemannian manifolds
Abstract: We define a conformal invariant using maximal volumes, and use this to prove existence of solutions to a class of conformally invariant fully nonlinear second order PDEs.
 

December 6, 2002
Speaker:  Hubert Bray (MIT)
Title: Classification of Prime 3-Manifolds with Yamabe Invariant Greater Than RP³
Abstract: In a joint work with Andre Neves, we classify prime 3-manifolds with Yamabe invariant greater than Y₂, the Yamabe invariant of RP³. The Yamabe invariant of a closed manifold is a smooth topological invariant which is defined geometrically and reduces to the Euler characteristic in dimension 2 by the Gauss Bonnet formula. However, prior to this work, the only known values of the Yamabe invariant for 3-manifolds had been 0 and
                    Y₁ = 6 (2p²)^2/3,
the Yamabe invariant of S³. We increase the number of known values by 50% by proving that the Yamabe invariants of RP³ and RP²xS¹ are equal to Y₂ = Y₁/2^2/3. We also prove that any 3-manifold with Yamabe invariant Y > Y₂  must either be S³ or a connect sum with an S² bundle over S² (S²xS¹ or the 3-dimensional Klein bottle). Hence, the Poincare conjecture for manifolds with Y > Y₂ (which includes an infinite number of 3-manifolds) follows. Also, since the classification of 3-manifolds reduces to the classification of prime 3-manifolds, it is natural to try to make a list of prime 3-manifolds. Ordered by their Yamabe invariants, we conclude that the first five prime 3-manifolds are S³, S²xS¹, K (the 3-dimensional Klein bottle), RP³, and RP² xS¹.
 

January 10, 2003
Speaker:  Dimitri Gourevitch (Université de Valenciennes)
Title: Noncommutative index on the quantum sphere
Abstract: Noncommutative (NC) index was introduced by A.Connes.
Given a NC algebra A, NC index is defined as a pairing
            K₀(A) x K⁰(A)→ K
where K₀(A) is defined by projective A-modules, K⁰(A) is defined by finite-dimensional A-modules, and K is the basic field. In the talk I will introduce quantum orbits whose particular case is a quantum sphere, define a version of the NC index well adapted to the algebras in question and compute NC index on this sphere.
 

January 24, 2003
Speaker:  Alexander Kozhevnikov (University of Haifa)
Title: Random fields estimation and elliptic boundary value problems
Abstract: A basic equation in the random fields estimation theory is solved by reducing it to an elliptic boundary value problem in an external domain. The notions of random fields and estimation theory will be explained during the talk.
 

January 31, 2003
Speaker:  Vladimir Kondratiev (Moscow State University)
Title: Elliptic problems with non-linear boundary conditions on a non-compact part of the boundary
Abstract: We will describe new results about the following problems, concerning solutions of second order linear elliptic equations with non-linear boundary conditions in unbounded domains: Phragmén - Lindelöf type theorems, existence or non-existence of positive solutions.
 

February 7, 2003
Speaker:  Ilya Zakharevich )
Title: Geometry of bi-Poisson structures
Abstract: Nowadays, the theory of "compatible" pairs of Poisson brackets permeates seemingly unrelated domains of math, from the classical method of separation of variables, to quantum cohomology. We discuss the recent progress in the geometry of finite-dimensional Poisson pairs. While answers to some flavors of the classification problem are known (for example, bi-Poisson structures are related to the "most general" settings of the twistor transform), others lead to complicated questions in geometry, complex analysis, and the theory of PDE of the principal type.
 

February 14, 2003
Speaker:  Tatyana Shaposhnikova (Linköping University - Sweden)
Title: Pointwise interpolation inequalities for derivatives and their applications
Abstract: I overview recent results, obtained together with V.Maz'ya, concerning interpolation inequalities for functional and fractional derivatives.
   A typical example is the Landau type inequality on the real line
               |u'(x)|²    ≤  8/3 Mu(x) Mu''(x),
   where the constant 8/3 is best possible and M is the Hardy-Littlewood maximal operator.
   Similar inequalities are used in an elementary proof of a theorem by H.Brezis and P.Mironescu on the continuity of the composition operator in fractional Sobolev spaces.
   New limiting properties of fractional Sobolev spaces initiated recently by Bourgain, Brezis, and Mironescu will be discussed as well.
 

February 21, 2003
Speaker:  Vladimir Mazya (Linköping University - Sweden)
Title: Maximum principles for solutions of elliptic and parabolic systems
Abstract: Maximum principles for solutions of elliptic and parabolic equations of the second order are classical and very important facts of the theory of partial differential equations. Recently Kresin and Maz'ya found a complete algebraic description of elliptic and parabolic systems satisfying the maximum principle. Unexpected phenomena occur when the boundary of a domain has edges and vertices.
   The talk is a survey of these and related results. No advanced knowledge of PDEs is required.
 

February 28, 2003
Speaker:  Vladimir Mazya (Linköping University - Sweden)
Title: Old and new spectral criteria for the Schroedinger operator
Abstract:  The lecture is a survey of the conditions on the potential responsible for various spectral properties of the Schroedinger operator:positivity and strict positivity, semiboundedness, descreteness of the spectrum,form-boundedness, finiteness and descreteness of the negative spectrum, etc.
 

March 7, 2003
Speaker:  Vladimir Mazya (Linköping University - Sweden)
Title: Theory of Sobolev multipliers
Abstract:  By a multiplier acting from one function space S₁ into another, S₂, one means a function which defines a bounded linear mapping of S₁ into S₂ by pointwise multiplication. In particular, multipliers in spaces of differentiable functions arise in various problems of analysis and the theory of differential and integral equations. For example, coefficients of differential operators can be naturally considered as multipliers. The same is true for symbols of pseudo-differential operators. Multipliers also appear in the theory of differential mappings preserving Sobolev spaces. Solutions of boundary value problems can be sought in classes of multipliers. Because of their algebraic properties, multipliers are suitable objects for generalization of basic facts of calculus (theorem on superposition, implicit function theorem etc.). Regardless of the substantiality and the usefulness of multipliers in Sobolev spaces, until recently they attracted relatively little attention. In the present talk I give a survey of principal known results in this area.
 

March 14, 2003
Speaker:  Matvei Libine (University of Massachusetts at Amherst)
Title: Equivariant Forms and Character Formulas
Abstract: This talk is based on my article math.RT/0208024 available on arXiv.org. I will talk about interplay between geometry and representation theory. Namely between the integral localization formula for equivariant forms and the Weyl and Kirillov's character formulas. I will explain the compact group case and then I will move on to recent developments for NONcompact groups.
 

April 4, 2003
Speaker:  Igor Verbitsky (University of Missouri)
Title: The form boundedness problem for the Schrödinger operator and its relativistic countepart  
Abstract: We present necessary and sufficient conditions for the relative form boundedness and compactness of the Schrödinger operator  H = H₀ + V,  where  H₀ = -D   is the Laplacian on the Euclidean space, with an arbitrary real- or complex-valued distributional potential V. Analogous results for the relativistic Schrödinger operator where                         
                            H₀ = (-D +m²)^1/2 -m 
will be discussed. This is joint work with Vladimir Maz'ya.
 

April 11, 2003
Speaker:  Semyon Alesker (Tel Aviv University)
Title: Non-commutative determinants and Monge-Ampere equations.
Abstract: There are various constructions of non-commutative determinants (super, quantum, Gelfand-Retakh...). First we discuss old and new properties of the Diedonne and Moore determinants. Based on these constructions, we introduce a class of plurisubharmonic functions of quaternionic variables and quaternionic Monge-Ampere equations. They are analogous to the classical real and complex cases. Then we discuss the solvability of the Dirichlet problem for them. Some connections to geometry will be mentioned.
 

April 18, 2003
Speaker:  Ari Belenkiy (Bar-Ilan University)
Title: Projective geometry of quantum mechanics, Einstein-Podolsky-Rosen experiment and Bell's correlation
Abstract: Applications of newly developed state vector stochastic reduction on Kahler manifolds (following Lane Hughston) are considered. Looking at the EPR experiment from the point of view of finite dimensional projective geometry might lend support one of the alternatives considered by John Bell.
 

April 25, 2003
Speaker:  Katrin Leschke (Technische Universität Berlin and University of Massachusetts at Amherst)
Title: Sequences of Willmore surfaces in the four-sphere
Abstract: We construct sequences of Willmore surfaces in S⁴ by using a Baecklund transformation of Willmore surfaces.
   For Willmore tori with non-zero normal bundle degree the sequence has to be finite, and we obtain a classification result.
   (This is joint work with Franz Pedit)
 

May 9, 2003
Speaker:  Nadja Kurt (University of Massachusetts at Amherst)
Title: Discrete curves and the Toda Lattice
Abstract: A novel interpretation of the one dimensional Toda lattice hierarchy (and reductions thereof) will be given in terms of flows on discrete curves. Among others the three Poisson structures of the Toda lattice (trihamiltonian structure) will be derived from a canonical structure on closed curves.
 

May 16, 2003
Speaker:  Alex Suciu (Northeastern University)
Title: Free abelian covers and systolic inequalities
Abstract:  I will describe recent work with M. Katz and M. Kreck, on systolic inequalities satisfied by arbitrary Riemannian metrics on a compact, orientable, smooth manifold X. Applying Gromov's filling inequality to the typical fiber of the map from X to its Jacobi torus, we prove an interpolating inequality for two flavors of shortest length invariants of loops. The inequality is a lower bound for the total volume of the manifold. The procedure works, provided X is aspherical, and the lift of the typical fiber of the Jacobi map is non-trivial in the homology of the maximal free abelian cover of X. For nilmanifolds, our ``fiberwise'' inequality typically gives stronger information than the filling inequality for X itself. For 3-manifolds with first Betti number 2, a sufficient condition for our systolic inequality to hold is the non-vanishing of a certain Massey product.
 

May 23, 2003
Speaker:  Maxim Braverman (Northeastern University)
Title: Topological calculation of the phase of the zeta-regularized determinants
Abstract: We show that for a large class of elliptic operators the phase of the zeta-regularized determinant is a topological invariant which can be explicitly calculated. We consider some examples where the phase is related to such classical topological invariants as the degree of the map and the Hopf invariant. Some of our examples were known to physicists. But not only the proofs but the very formulations of the results were not rigorous even by the standards usual for the physical literature.
   (Joint project with A. Abanov)
 

September 5, 2003
Speaker:  Boris Pavlov (V.A. Fock Institute of Physics, St.Petersburg, Russia and University of Auckland, New Zealand)
Title: Modelling of quantum networks
Abstract: A mathematical design of a quantum network is equivalent to the Inverse Scattering Problem for the Schrodinger equation on a composite domain consisting of quantum wells and finite or semi-infinite quantum wires attached to them. We suggest an alternative approach based on using of a solvable model to this difficult problem and to the relevant problems of choice and optimization of the construction and working parameters of the quantum network. We suggest a general principle of construction of quantitatively consistent solvable models for one-particle scattering processes in the network assuming that the transmission of an electron across the wells from one quantum wire to the other happens due to excitation of oscillatory modes in the well. This approach permits to obtain an explicit approximate formula for transmission coefficients based on numerical results on the discrete spectrum of the Schrodinger operator on the quantum wells.
 

September 12, 2003
Speaker:  Theodore Voronov (University of Manchester (UMIST))
Title: Inverse problem of calculus of variations and forms on field space
Abstract: The inverse problem of calculus of variations (in its simplest form) is to find out whether given functions depending on fields and their derivatives are the variational derivatives of some functional. There is a classical Helmholtz--Volterra condition, which is necessary and locally sufficient. We give an alternative criterion in terms of the identical vanishing of the variation of a certain functional on an extended space where the number of independent variables is increased by one, and explain its relation with the Helmholtz--Volterra criterion using the de Rham complex on an infinite-dimensional space of fields.
 

September 12, 2003,  3:30 PM (Note the special time)
Speaker:  Fabio Podesta (University of Florence, Italy)
Title: Cohomogeneity One Kaehler manifolds and new examples of Kaehler-Einstein Cohomogeneity One Kaehler manifolds and new examples of Kaehler-Einstein metrics
Abstract: I will discuss the geometry of compact Kaehler manifolds with vanishing first Betti number and which admit an isometric action of a compact Lie group with codimension one principal orbits. I will also show how to construct new invariant Kaehler-Einstein metrics on some cohomogeneity one compact Kaehler manifolds, when the principal orbits are Levi non-degenerate.
 

September 19, 2003
Speaker:  Gudlaugur Thorbergsson (University of Cologne, Germany)
Title: Isometric actions on symmetric spaces
Abstract:  In the first part of the talk I will review the definitions and basic properties of variationally complete and polar actions on Riemannian manifolds. In the second part I will explain my joint work with Gorodski in which we prove that a variationally complete action on a compact symmetric space is hyperpolar. The converse was already proved by Conlon in 1971.
 

September 26, 2003
Speaker:  Ernst Heintze (University of Augsburg, Germany)
Title: Involutions of Kac-Moody algebras and infinite dimensional symmetric spaces
Abstract: In finite dimensions, compact Lie groups with a biinvariant metric are important examples of Riemannian manifolds. They are in turn special examples of the so called symmetric spaces G/K where K is the fixed point set of an involution on G.
   The closest analogue of a compact Lie group in infinite dimensions is an affine Kac-Moody group and thus of a symmetric space, the quotient of an affine Kac-Moody group by the fixed point set of an involution.
   The purpose of this talks is to outline a new classification of these infinite dimensional symmetric spaces or equivalently of the involutions of affine Kac-Moody algebras. We show in particular that it can be reduced to well known problems in finite dimensions.
 

October 1, 2003,  2PM (Note the special date and time)
Speaker:  Thomas Kappeler (University of Zurich)
Title: Well-posedness of KdV in H^-1(T)
Abstract: In this talk I present recent results on the normal form for the KdV equation on the circle. They are used to show that KdV is well posed on the Sobolev spaces H ^-a(T)   for 0 a  1.
 

October 17, 2003
Speaker:  Maxim Braverman (Northeastern University)
Title: The L2-torsion without the determinant class condition
Abstract:  We define the combinatorial and the analytic L2-torsions of a flat Hilbertian bundle as an element of the determinant line of its extended cohomology. In the case when the bundle is of determinant class, our definitions reduces to the construction of Carey, Farber, and Mathai. In the general case, we show that the ratio of the analytic and the combinatorial L2-torsions is equal to the relative torsion introduced by Carey, Mathai, and Mishchenko. In particular, applying the recent result of Burghelea, Friedlander, and Kappeler we obtain a Cheeger-Muller type theorem stating the equality between the analytic and the combinatorial L2-torsions.
   (Joint work with A. Carey, M. Farber, and V. Mathai)
 

October 24, 2003
Speaker:  Hui Ma (University of Massachusetts at Amherst)
Title: Hamiltonian stationary Lagrangian surfaces in CP²
Abstract: A Lagrangian submanifold in a Kähler manifold is called Hamiltonian stationary if its area is critical with respect to all Hamiltonian deformation. We present a (new) equivalent condition of Hamiltonian stationary Lagrangian surfaces in CP² and show that any nonsuperminimal Hamiltonian stationary Lagrangian torus in CP² can be constructed from a pair of commuting Hamiltonian ODE on a finite dimensional subspace of a certain loop Lie algebra.
   (joint work with Weihuan Chen and Franz Pedit)
 

October 31, 2003
Speaker:  Megan M. Kerr (Wellesley College)
Title: Low-Dimensional Homogeneous Einstein Manifolds
Abstract:  I will describe joint work with Christoph Böhm, investigating the Einstein equation for G-invariant metrics on compact homogeneous spaces. We prove that every compact, simply connected homogeneous space of dimension less or equal than 11 admits a homogeneous Einstein metric. The result is sharp: Wang and Ziller showed that in dimension 12 the compact, simply connected homogeneous space SU(4)/SU(2) does not admit any homogeneous Einstein metrics. (Here SU(2) < Sp(2) < SU(4) and SU(2) is maximal in Sp(2).) Classification results up to dimension seven have been published, but the case-by-case classification gets significantly more difficult as the dimension increases, since the number of spaces to be considered rapidly increases with the dimension. I will also describe an infinite family of 12-dimensional simply connected homogeneous torus bundles which do not admit G-invariant Einstein metrics. These are the first non-existence examples where the isotropy representation has four summands.
 

November 7, 2003
Speaker:  Mikhail Shubin (Northeastern University)
Title: Semiclassical asymptotics and vanishing of quantum Hall conductivity
Abstract: I will explain a new method of obtaining semiclassical asymptotics for magnetic Schroedinger operators with invariant Morse type potentials on covering spaces of compact manifolds. It provides a new existence proof for spectral gaps and also gives an information about the spectral projections, implying vanishing of classes of these projections in $K$-theory for small coupling constant. An important corollary is vanishing of the corresponding higher traces in cyclic cohomology, which in turn implies vanishing of the quantum Hall conductivity for weak magnetic fields.
   This is a joint work with Yu.Kordyukov and V.Mathai
 

December 5, 2003
Speaker:  Martin Magid (Wellesley College)
Title: Time-like isothermic surfaces associated to Grassmannian Systems
Abstract:  C.-L. Terng defined the U/K system for a symmetric space based on a semi-simple U in 1997. This is a non-linear first order system of partial differential equations defined using U/K. This system gives rise to a one-parameter family of flat connections called the Lax connection of the U/K system. I will show that time-like isothermic surfaces in pseudo-riemannian space R^n-j,j are associated to the Grassmannian O(n-j+1,j+1)/O(n-j,j) x O(1,1)-system.
 

January 16, 2004
Speaker:  Ionel Popescu (MIT)
Title: A Probabilistic Approach to Morse Inequalitites
Abstract: Starting with the heat kernel of the Witten Laplacian in terms of Feynman-Kac like integral, combined with a simple Markov property and estimates on the solution to a initial-boundary problem on a ball in Euclidean space we prove Morse inequalities. Based on this argument we will discuss also the case of a Bott-Morse function where the idea is to compare the associated Laplacians with respect to Bismut connection and Levi-Civita connection around the critical submanifolds.
 

February 6, 2004
Speaker:  Victor Roitburd (Rensselaer Polytechnic Institute)
Title: Asymptotic dynamics of non-equilibrium free-boundary problems
Abstract: Free-boundary models provide a convenient and rather accurate description for many phase transition type phenomena, such as freezing/melting or burning. I'll give a brief and elementary introduction to free-boundary problems for the heat equation, and explain how some types of them relate to reaction-diffusion systems. The talk will be mostly concerned with asymptotic dynamics of solutions of a free-boundary problem arising from the so-called solid-state combustion. Numerical experiments reveal a huge variety of dynamical scenarios (some animations will be shown). Nonetheless, it turns out that the possible asymptotic regimes (a global attractor) occupy a compact set in the space of all the regimes, and that its Hausdorff dimension is finite. In some sense the PDE system behaves like a fancy nonlinear oscillator. The proofs are based on classical potential theory estimates. An elementary description of the Hausdorff dimension and of its computation will be given. Results of the talk are obtained in a joint work with Michael Frankel of Indiana University-Purdue University Indianapolis
 

February 20, 2004
Speaker:  Dimitri Yafaev (Université de Rennes 1)
Title: Scattering by magnetic fields
Abstract:  For the magnetic Schrödinger operator we will discuss the definition and spectral properties of the scattering matrix. In particular, the essential spectrum of the scattering matrix can be found in terms of the decay of the magnetic field at infinity. Under appropriate conditions we can also describe singularities of the scattering amplitude (the Schwartz kernel of the scattering matrix).
   An important point of our approach is that we consider the scattering matrix as a pseudo-differential operator on the unit sphere and find an explicit expression of its principal symbol in terms of the vector potential. Another ingredient is an extensive use of a special gauge adapted to a given magnetic field.
 

March 19, 2004
Speaker:  Dan Mangoubi (Technion - Israel Institute of Technology)
Title: Symplectic Aspects of the First Eigenvalue of the Laplacian
Abstract: Let (M, w) be a compact symplectic manifold of  dim > 2. We are interested to know whether we can find a Riemannian metric on M compatible with w and with arbitrary large first positive eigenvalue. L. Polterovich proved that it is the case under some technical condition on M, which is fullfiled for manifolds of the form MxT⁴, where T⁴ is the torus.
   We will discuss ideas which hopefully will let us prove it for any compact (M, w)  of dim>2.
 

October 15, 2004
Speaker:  Maxim Braverman (Northeastern University)
Title: The phase of the determinant of a Dirac type operator and the degree of a map.
Abstract: I will present an an example of a Dirac type operator depending on a map V from a manifold to a sphere, the phase of whose determinant can be calculated in terms of the topological degree of V. An important part of our calculation is the study of the imaginary part of the spectrum of this operator. In this study a new version of the Witten deformation technique is used.
   A special case of our result was suggested by physicists. But not only the proofs but the very formulations of the results were not rigorous even by the standards usual for the physical literature.
   (Joint project with A. Abanov)
 

October 29, 2004
Speaker:  Mikhail Shubin (Northeastern University)
Title: The Miura transform
Abstract: The Miura transform is a non-linear map r ---> r'+r² on functions of one real variable. The importance of this transform lies in its relations with some non-linear partial differential equations, e.g. the famous Korteweg - de Vries equation. I will describe properties of this transform, in particular, recently found description of the image of this transform in some spaces of functions and distributions on R. This description is related with the spectra of the Schr\"odinger operators on R.
   The talk will be based on joint results of T.Kappeler, P.Perry, P.Topalov and the speaker.
 

November 18, 2004,  12:00 (Note the special date and time)
Speaker:  Alexander Turbiner (Institute for Nuclear Sciences - National University of Mexico)
Title: Perturbations of integrable systems and Dyson-Mehta integrals
Abstract: Olshanetsky-Perelomov quantum Hamiltonians are unique both completely integrable and exactly-solvable multidimensional Hamiltonians related to root systems. We will show that they admit algebraic forms being represented as linear differential operators with polynomial coefficients, which allows a Lie-algebraic interpretation of these Hamiltonians. The existence of algebraic form also allows to present a quite general class of perturbations for which one can develop a constructive, `algebraic perturbation theory', where all corrections are found by pure algebraic means. These perturbations can be classified in terms of representation theory. Physically relevant many-body anharmonic oscillators turned out to be among these perturbed problems. Corrections to eigenvalues are given by ratios of generalized Dyson-Mehta integrals, hence they can be found by algebraic means. They are interesting by themselves.
 

December 3, 2004
Speaker:  Gideon Maschler (University of Toronto)
Title: Conformally-Einstein Kähler metrics
Abstract: We describe the classification of Kähler metrics which are conformal to Einstein metrics on manifolds of complex dimension m>2. Included is the case where the Einstein metric is defined only away from the non-empty zero set of the conformal factor, giving rise to examples of asymptotically hyperbolic Einstein metrics. Locally, these metrics are given by a 3-parameter family on the total space of a line bundle over a Kähler-Einstein base. In the global classification, the metrics are extended to an associated projectivized bundle. The allowed Chern numbers for these bundles are parameterized via a discrete subset lying on a family of plane plane algebraic curves. This work is joint with A. Derdzinski.

January 21, 2005
Speaker:  Gabriel Katz (Bennington College)
Title: Morse theory on manifolds with boundary and convexity
Abstract: Classical Morse Theory links singularities of Morse functions with the topology of a closed manifold. The singularities cause an interruption of the gradient flow; and the homology or even the topological type of the manifold can be expressed in terms of such interruptions (i.e. in terms of descending manifolds, attaching maps, spaces of the flow trajectories which connect the singularities).
   On manifolds with boundary an additional source of the flow interruption occurs: it comes from a particular geometry of the boundary, or rather from the failure of the boundary to be convex with respect to the flow. In fact, one can trade the singularities in the interior of the manifold for these new non convexity effects. In our approach, these boundary effects take a central stage, while the singularities remain in the background.
   We will discuss some applications of this philosophy to 3-dimensional manifolds. In particular, we will reformulate the Poincare' Conjecture in terms of the new convexity.
 

February 3, 2005,  1:30PM (Note the special date and time)
Speaker:  Mikhail Agranovich (Moscow Institute of Electronics and Mathematics, Russia)
Title: Spectral problems for second order strongly elliptic systems with spectral parameter in boundary or transmission conditions
Abstract: We consider spectral problems in Rⁿ, n і 3, for a second order strongly elliptic system satisfying some additional conditions. The spectral parameter is contained in the boundary or transmission conditions on a Lipschitz surface S. It is either closed or bounded and non-closed. The aim is to describe spectral properties of the corresponding operators of the Neumann-to-Dirichlet type on S in the simplest Sobolev spaces. In the second case, the Dirichlet and Neumann problems with boundary data on S are also considered.
 

February 11, 2005
Speaker:  Joseph Coffey (New York University)
Title: Failure of parametric H-principle for maps with prescribed Jacobian
Abstract: Let M and N be closed n-dimensional manifolds, and equip N with a volume form σ. Let μ be an exact n-form on M. Arnold then asked the question: When can one find a map  f:M→N  such that  f*σ=μ. In 1973 Eliashberg and Gromov showed that this problem is, in a deep sense, trivial: It satisfies an h-principle, and whenever one can find a bundle map f_bdl:T M→T N which is degree 0 on the base and such that  f_bdl*(σ)=μ  one can homotop this map to a solution f. That is if the naive topological conditions are satisfied on can find a solution. There is no further interesting geometry in the problem.
We show the corresponding parametric h-principle fails- if one considers families of maps inducing μ from σ, one can find interesting topology in the space of solutions which is not predicted by an h-principle. Moreover the homotopy type of such maps is "quantized": for certain families of forms homotopy type remains constant, jumping only at discrete values.
 

February 25, 2005
Speaker:  Michael E. Taylor (University of North Carolina at Chapel Hill)
Title: Scattering length and spectral theory of Schrodinger operators
Abstract: The theory of capacities has played an important role in potential theory for a long time. About 30 years ago, M. Kac began to explore the use of the notion of scattering length of a positive potential, as an analogue of capacity. We will discuss some basic properties of this scattering length, and apply it to study the spectrum of Schrodinger operators with positive potentials. We will obtain variants of some results of Molchanov and Maz'ya and Shubin.
 

March 18, 2005
Speaker:  Jonathan Weitsman (University of California Santa Cruz)
Title: Measures on Banach manifolds and supersymmetric quantum field theory
Abstract: We show how to construct measures on Banach manifolds associated to supersymmetric quantum field theories. As examples of our construction we produce measures corresponding to spaces of maps from a Riemann surface to a semisimple Lie group (the Wess-Zumino-Novikov-Witten model) and to gauge theory in three dimensions. We show that these measures are positive, and that the Wess-Zumino-Novikov-Witten measure where the Riemann surface is P¹ has finite mass. As an application we show that formulas arising from expectations in this measure reproduce the results of Frenkel and Zhu from vertex operator algebras.
 

March 25, 2005
Speaker:  Raphaël Ponge (Ohio State University)
Title: Spectral asymmetry, zeta functions and the noncommutative residue
Abstract:  Motivated by an approach developed by Wodzicki, we look at the spectral asymmetry of (possibly nonselfadjoint) elliptic PsiDO's in terms of theirs zeta functions. Using formulas of Wodzicki we look at the spectral asymmetry of elliptic PsiDO's which are odd in the sense of Kontsevich-Vishik. Our main result implies that the eta function of a selfadjoint elliptic odd PsiDO is regular at every integer point when the dimension and the order have opposite parities (this generalizes a well known result of Branson-Gilkey for Dirac operators) and allows us to relate the spectral asymmetry of a Dirac operator on a Clifford bundle to the Riemannian geometric data. As a consequence, we can express the Einstein-Hilbert action of a Riemanian metric in terms of the difference of two zeta functions of a Dirac operator, hence obtain a new spectral interpretation of this action.
 

April 8, 2005
Speaker:  Oleg Gleizer (UCLA)
Title: TBA
Abstract: TBA

• September 16, 2005
  Speaker:  Maxim Braverman (Northeastern University)
  Title: Refined Analytic Torsion
  Abstract: For an acyclic representation of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to a unitary representation, we define a refinement of the Ray-Singer torsion associated to this representation. This new invariant can be viewed as an analytic counterpart of the refined combinatorial torsion introduced by Turaev.
     The refined analytic torsion is a holomorphic function of the representation of the fundamental group. When the representation is unitary, the absolute value of the refined analytic torsion is equal to the Ray-Singer torsion, while its phase is determined by the eta-invariant. The fact that the Ray-Singer torsion and the eta-invariant can be combined into one holomorphic function allows to use methods of complex analysis to study both invariants. I will present several applications of this method.
     (Joint work with Thomas Kappeler)
   
• January 13, 2006
  Speaker:  Alexander Kushkuley (Clearing Corporation)
  Title: On positive semidefinite approximation of matrces with prescribed block-diagonal structure
  Abstract: Let  sym(Λ)  be an affine plane of symmetric n x n  matrices with fixed positive definite block diagonal submatrix  Λ  and let  P(Λ) Ì  sym(Λ)  be the set of all positive semidefinite matrices in   sym(Λ). Consider the following optimization problem: given  A Î sym(Λ)   find the closest matrix to  A  in  P(Λ) .  The problem belongs to a class of "positive semidefinite completion" problems that are usually solved by methods of convex programming. A related rank reduction problem is to find a positive semidefinite approximation to matrix   A Î sym(Λ)   of rank less or equal than some  0 < k < n . In this paper both problems are studied in a rather straightforward manner, as problems of finding critical points of Euclidian distance function. Besides presenting some algorithms, we observe, that
     
    (a)    the number of critical points of distance function on P(Λ) is allways finite;
    (b)   P(Λ)  is stratified by connected open manifolds P_k(Λ) of matrices of rank exactly  k;
    (c)   tangent plane, normal subspace and shape operator for a given point  S Î P_k(Λ)  Ì sym(Λ)  can be characterized algebraically in terms of  S;
    (d)   if  Λ  is diagonal then total Betti number of the stratified space  П_i=1^k P_i(Λ)    is equal to  2^n-k.
   
• January 20, 2006
  Speaker:  Peter Topalov (Northeastern University)
  Title: Solutions of mKdV in classes of functions unbounded at infinity
  Abstract: Investigation of relation between the Korteweg - de Vries and modified Korteweg - de Vries equations (KdV and mKdV) leads to a new algebro-analytic mechanism which is similar to the Lax L-A pair but includes a first order operator Q instead of the 3rd order operator A. This allows an explicit control of eigenfunctions of the Schr\"odinger operator L when its time-dependent potential satisfies KdV. In particular, we establish global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity.
     (joint work with T. Kappeler, P. Perry, and M. Shubin)
   
• March 24, 2006
  Speaker:  Mihai Stoiciu (Williams College)
  Title: The Distribution of the Eigenvalues of Random CMV Matrices
  Abstract: Recent developments in the theory of orthogonal polynomials on the unit circle have emphasized the importance of CMV matrices; they are the unitary analog of Jacobi matrices. We prove that the asymptotic local statistical distribution of various classes of random CMV matrices is Poisson. This means that, as in the case of random Schrodinger operators, there is no local correlation between the eigenvalues.

• March 31, 2006
  Speaker:  Mikhail Shubin (Northeastern University)
  Title: Crystal lattice vibrations and specific heat at low temperatures
  Abstract: Behavior of the specific heat of a solid at low temperature is a classical subject in solid state physics which dates back to a pioneering work by Einstein (1907) and its refinement by Debye (1912). Using a special quantization of crystal lattices and calculating the asymptotic of the integrated density of states at the bottom of the spectrum, we obtain a rigorous derivation of the classical Debye  T³  law.
     The talk is based on joint work by the speaker and T.Sunada.