## Analysis Seminar

Seminars are usually held on Mondays or Fridays at Concordia or at McGill
For suggestions, questions etc. please contact Dmitry Jakobson (jakobson@math.mcgill.ca) or Galia Dafni (gdafni@mathstat.concordia.ca)

## FALL 2004

Alexei Kokotov (Concordia) will give a series of three talks on Tau-functions and determinants of Laplacians

Friday, September 24, 11:30am
Concordia LB 540 (Library building, 5th floor)
Alexei Kokotov (Concordia)
I. Preliminaries.
Abstract: This talk will be devoted to a brief review of some basic facts from analysis on compact Riemann surfaces:
1) Basic holomorphic objects on a Riemann surface (Bergman bidifferential, prime-form, projective connections, etc.)
2) Laplace operators in smooth and singular metrics and their determinants. Polyakov formula for variation of the determinant of Laplacian within a conformal class of the metric.
3) Branched coverings and variational formulas of the Rauch type.

Friday, October 1, 13:00
Concordia LB 540
Alexei Kokotov (Concordia)
II. Extremal surfaces for the determinant of the Laplacian
Abstract: In this talk we review some results on extremal problems on Riemann surfaces and report the results of numeric experiments in genus 2 case. The plan of this talk runs as follows:
1) Ricci flow and Osgood-Phillips-Sarnak theorem
2) Zograf-Tahhtajan-Fay formula for the variation of the determinant of the Laplacian in the Poincar\'e metric
3) Whittaker conjecture on the uniformization of hyperelliptic curves.
4) Numeric experiments in genus 2 case

Friday, October 8, 13:00
Concordia LB 540
Alexei Kokotov (Concordia)
III. Determinants of Laplacians for Strebel metrics of finite volume
Abstract: In this talk we compute determinants of Laplacians in flat metrics with conical singularities on a Riemann surface. We shall consider the case of Strebel metrics of finite volume: such a metric is defined as the modulus of a meromorphic quadratic differential having at most simple poles. Strebel metrics have conical singularities at the zeros and poles of the meromorphic differential. To get the formulas for the derminants we develop
1) formalism of tau-functions on spaces of quadratic and Abelian differentials,
2) technique of analytic surgery for metrics with conical singularities.
3) variational formulas

Friday, October 15, 13:00-13:45
Concordia LB 540
Alexei Kokotov (Concordia)
Conclusion of A. Kokotov's series of talks.

Joint Analysis/Applied Mathematics seminar
Monday, October 4, 4:15pm
Concordia LB 540
Susan Friedlander (Univ of Illinois-Chicago)
Blow up in a 3-D "toy" model for the Euler equations
Abstract: We present a 3-D vector dyadic model given in terms of an infinite system of nonlinearly coupled ODE. This toy model is inspired by approximations to the fluid equations studied by Dinaburg and Sinai. The model has structural similarities with the Euler equations and it mimics certain important properties of the fluid equations, namely conservation of energy and divergence free velocity. We prove that for certain families of initial data blow-up occurs in the model system in the sense that the H^s, s > 3/2 , norm becomes unbounded in finite time. This is joint work with Natasa Pavlovic.
Monday, October 18, 2:30pm
McGill, Burnside 920
Anthony Quas
Critical rates in nonconventional ergodic averaging.
Abstract: We consider a number of examples of non-conventional ergodic type averages and describe the maximal rates of divergence of the averages. In particular, we show that averaging along the dyadic sequence of times 2^n is in a precise sense the worst possible ergodic average. Along the way, we present a simple construction of a counterexample to a conjecture of Khintchine's.

Friday, November 12, 2:00-3:00pm
McGill, Burnside 920
Manfred Einsiedler (Princeton)
Measure rigidity for the Cartan action on higher rank locally symmetric spaces
Abstract: Furstenberg showed that a closed subset of the circle group that is invariant under squaring and cubing must be finite or the whole circle. He also asked if a similar statement for invariant measures is true. This question is still open, the best available result [Rudolph] assumes positive entropy of the invariant measure. Margulis conjecture on Cartan invariant sets and measures is an analogue of the above problem for locally homogeneous spaces. These are especially interesting in light of possible applications to number theory, e.g. Littlewood's conjecture. Recent joint work with Katok and Lindenstrauss has lead to a generalization of Rudolph's theorem to SL(3,R)/SL(3,Z).

Monday, November 29, 4:00-5:00pm
McGill, Burnside 1214
Nikolai Nadirashvili (CNRS and Chicago)
Complete and Proper Minimal Immesrions

Friday, December 10, 11:00-12:00pm
McGill, Burnside 920
Akos Magyar (Georgia Tech)
A Ramsey type result for lattice points
Abstract: We show that a subset of positive density of the n-dimensional integer lattice contains a "copy" of every k-dimensional simplex which satisfy the obvious necessary conditions, if n>k(k+1). This is a discrete analogue of a result of Bourgain proved for measurable subsets of the n-dimensional Euclidean space.

## WINTER 2005

Friday, January 14, 2:30-3:30pm
McGill, Burnside 920
Artem Zvavitch (Kent State University)
The Busemann-Petty problem for arbitrary measures
Abstract: The Busemann-Petty problem asks whether symmetric convex bodies in R^n with smaller (n-1)-dimensional volume of central hyperplane sections necessarily have smaller n-dimensional volume. Clearly, the Busemann-Petty problem is a triviality for n=2 and the answer is yes''. Minkowski's theorem shows that an origin-symmetric star-shaped body is uniquely determined by the volume of its hyperplane sections. In view of this fact it is quite surprising that the answer to the original Busemann Petty problem can be negative. Indeed, it is affirmative if n<=4 and negative if n>=5. In this talk we will present a generalization of the Busemann-Petty problem to essentially arbitrary measure in place of the volume. We also present applications of the latter result by proving several inequalities concerning the measure of sections of convex symmetric bodies in R^n.
Friday, January 28, 2:30-3:30pm
McGill, Burnside 920
Emily Dryden (McGill and CRM)
Inverse Spectral Problems on Hyperbolic Orbisurfaces
Abstract: Historically, inverse spectral theory has been concerned with the relationship between the geometry and the spectrum of compact Riemannian manifolds, where "spectrum" means the eigenvalue spectrum of the Laplace operator as it acts on smooth functions on a manifold M. We broaden this study to orbifolds, and more specifically to hyperbolic orbisurfaces. Using an appropriate version of the Selberg Trace Formula, we explain the relationship among the Laplace spectrum, the length spectrum, and the singular points in a hyperbolic orbisurface. We then discuss the cardinality of isospectral sets of hyperbolic orbisurfaces.
Friday, February 4, 4-5pm
CRM-ISM Colloquium, UQAM, 200, rue Sherbrooke O., salle SH-3420
Stephan De Bievre (Lille)
Chaos quantique: au-dela du theoreme de Shnirelman
Abstract: Le chaos quantique est l'etude semi-classique de systhmes (spectraux) quantiques dont la limite classique est un systhme dynamique hamiltonien chaotique. Une question centrale (parmi beaucoup d'autres) est la comprehension du comportement des fonctions propres de ses systhmes dans la limite semi-classique. J'expliquerai cette problematique a l'aide d'exemples, puis je passerai en revue quelques résultats recents plus spécifiques pour les systhmes dynamiques discrets, les "applications quantiques" comme le chat d'Arnold et ses perturbations non-lineaires.
Wednesday, February 9, 3:30pm
Concordia Department seminar, Concordia, Rm. LB 540 (Library building)
Alina Stancu (UdeM)
On a Planar Crystalline Flow
Abstract: Crystalline flows have been defined in the '80's to model the evolution of piecewise linear interfaces separating a two-phase two-dimensional system. The evolution of such an interface, G, is described by a system of ordinary differential equations. A crystalline flow can also be viewed as reducing a boundary energy of the closed domain W with boundary G, when the energy density is continuous, but not differentiable. In this talk, we will discuss the uniqueness of the minimizer for such an energy. The proof relies on the asymptotic behavior of solutions to the system of ODE mentioned above, where the convexity of solutions plays an important role.
Sasha Shnirelman (Concordia) will give a series of four one-hour talks on Microglobal Analysis

Friday, February 11, 1:30-2:45pm
Concordia, LB LB 559-6 (Library building)
Thursday, February 17, time TBA
Concordia, Room TBA
Abstract: In this talk I am discussing the following general problem. Consider a nonlinear (pseudo)differential operator mapping one space of sufficiently smooth functions (say, the Sobolev space $H^s$ with $s$ large enough) into another space (for example, $H^{s-m}$, where $m$ is the order of the operator). What are the global geometrical properties of the mapping of the functional space defined by this operator? It turns out that such operator has a rigid geometric structure; it is QUASIRULED. This is a consequence of the fact that all such operators, if considered in the space of sufficiently regular functions, are QUASILINEAR. As a result, we can develop a natural degree theory of such operators and apply it to many interesting problems such as some nonlinear boundary problems for the holomorphic functions. These ideas are further applied to the study of the flows of ideal incompressible fluid based on the group of volume preserving diffeomorphisms. Recently Ebin, Misiolek and Preston proved that the geodesic exponential map on the group of 2-d diffeomorphisms is Fredholm, solving the 35 years old problem. Our approach explains this result very naturally. It is connected with the accurate description of the evolution of singularities for the 2-d Euler equations. It turns out that not only is the exponential map Fredholm, but it is a quasiruled Fredholm map.
The detailed contents of the talk:
1. Quasilinear and quasiruled maps; Fredholm quasiruled maps, their degree; example - the Nonlinear Riemann-Hilbert Problem.
2. Paraproduct and paracomposition; global linearization formula (Bony-Alinhac); microlocal measures and microlocal scalar products; evolution of weak singularities of 2-d Euler equations; integrals and Liapunov functions associated with singularities.
3. Analysis of the geodesic exponential map on the group of volume preserving diffeomorphisms; in 2-d case this map is smooth, Fredholm and quasiruled.

Friday, February 18, 2:30-3:30pm
Burnside 920
Victor Ivrii (Toronto)
Spectral Asymptotics for 2-dimensional Schroedinger Operator with Strong Degenerating Magnetic Field
Abstract: For 2-dimensional Schroedinger operator with the strong but degenerating magnetic field sharp spectral asymptotics are derived These asymptotics can contain fast oscillation terms produced by short periodic trajectories of the related classical dynamics
Monday, March 7, 2:30-3:30pm
Burnside 1120
P. Poulin (McGill)
Molchanov-Vinberg Laplacian
Abstract: It is well known that the Green's function of the standard discrete Laplacian on a lattice exhibits a pathological behavior in dimension greater than 2. Molchanov and Vainberg suggested an alternative to the usual Laplacian and conjectured that a polynomial decay holds for its Green's function. In this talk, I will present a proof of this conjecture.
Friday, March 11, 2:30-3:30pm
McGill, Burnside 920
Felix Finster (Regensburg)
Weighted L^2-estimates of the Witten spinor in asymptotically flat manifolds
Abstract: After a short introduction to asymptotically flat manifolds and a review of Witten's proof of the positive mass theorem, we consider weighted $L^2$-estimates of the Witten spinor. We present estimates which do not depend on the isoperimetric constant, but which instead take into account the interior geometry only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactification of the manifold.

Joint Applied Mathematics/Analysis seminar
Monday, March 14, 2:30-3:30pm

Burnside 1205
John Mallet-Paret (Brown)
Dynamics of Lattice Dynamical Systems
Abstract: We discuss recent results in the theory of lattice differential equations. Such equations are continuous-time infinite-dimensional dynamical systems (that is, infinite systems of ODE's) which possess a discrete spatial structure modeled on a lattice, for example on $Z^d$. As we see, even for rather simply constructed systems a rich variety of dynamical phenomena are present. Of particular interest are sponaneously generated patterns (for example stripes or checks), spatial chaos, and traveling front solutions between equilibria which may either be spatially homogeneous or which exhibit regular patterns. Also of interest are the effects of anisotropy of the lattice, in particular propagation failure of fronts, and the effect of random imperfections in the lattice.

Friday, March 18, 2:30-3:30pm
Burnside 920
M. Levitin (Heriot Watt)
Variational approach to spectral problems: two unusual examples.

Thursday, March 24, 10:30-11:30am
Burnside 1120, McGill
Lior Silberman (Princeton)
An Equivariant Microlocal Lift on Locally Symmetric Spaces

Friday, April 22, 2pm-3pm
Room LB 559-6, Library building, Concordia
E. Pujals (Toronto)
The Lorenz Attractor revisited
Abstract: The article "Deterministic non periodic flow", published by Lorenz nearly four decades ago in the Journal of Atmospheric Sciences, raised a number of mathematical questions that are some leitmotivs for the mayor developments the field of Dynamical systems has been going through. I will try to explain some of these developments and related questions, and some results answering them.
Friday, April 29, 11:00-15:30
ANALYSIS DAY
SCHEDULE:
All talks will be held in Room 5340 at Centre de Recherches Mathematiques, Universite de Montreal, Pavillon Andre-Aisenstadt, 2920 Chemin de la tour, Montreal
11:00-12:00 Tomasz Kaczynski (Sherbrooke)
Homological approach to detection of interesting dynamics
Abstract: I will give an overview of methods based on Conley and fixed point indices for proving interesting features of dynamical systems such as the existence of invariant sets or the existence of periodic orbits of specific periods in specific neighborhoods. As many mathematical statements, the ones we discuss contain hypotheses that are not easily verified in practice. I will describe a homology calculus that enables a computer-assisted rigorous verification of empirically based hypotheses.
12:00-13:30 Lunch break
13:30-14:20 Alexey Kokotov (Concordia)
Some extremal problems for curves of genus 0, 1 and 2
Abstract: Osgood, Phillips and Sarnak proved that the (normalized) determinant of laplacian on elliptic surface is less or equal to $\sqrt{3}/2|\eta(1/2+i\sqrt{3}/2)|^4$, where $\eta$ is Dedekind's eta-function. We shall discuss some analogs of this result in genera 0 and 2.
14:30-15:30 Fedor Nazarov (Michigan State)
On the minimum of one fancy functional
Abstract: Let $\mu$ be a symmetric positive measure on $[-1,1]$ of total mass $1$. We shall find the minimal value of $\int_R |t|\cdot|\widehat\mu(t)|\,dt$.

Monday, May 2, 2:00-3:00pm
Burnside 920
D. Jakobson (McGill)
Lower bounds for pointwise error term in Weyl's law
Abstract: I will discuss our joint work with I. Polterovich on estimates from below for the spectral function of the Laplacian on compact Rimannian manifolds. I will formulate a general result that holds for arbitrary manifolds, then try to indicate how methods of thermodynamic formalism for hyperbolic flows can be applied to improve the bounds for negatively-curved manifolds. This is work in progress.