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)

Alexei Kokotov (Concordia) will give a series of three talks on

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.

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

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

Concordia LB 540

**Alexei Kokotov** (Concordia)

Conclusion of A. Kokotov's series of talks.

Concordia LB 540

Blow up in a 3-D "toy" model for the Euler equations

McGill, Burnside 920

Critical rates in nonconventional ergodic averaging.

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).

McGill, Burnside 1214

**Nikolai Nadirashvili** (CNRS and Chicago)

Complete and Proper Minimal Immesrions

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.

McGill, Burnside 920

The Busemann-Petty problem for arbitrary measures

McGill, Burnside 920

Inverse Spectral Problems on Hyperbolic Orbisurfaces

CRM-ISM Colloquium, UQAM, 200, rue Sherbrooke O., salle SH-3420

Chaos quantique: au-dela du theoreme de Shnirelman

Concordia Department seminar, Concordia, Rm. LB 540 (Library building)

On a Planar Crystalline Flow

Concordia, LB LB 559-6 (Library building)

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.

Burnside 920

Spectral Asymptotics for 2-dimensional Schroedinger Operator with Strong Degenerating Magnetic Field

Burnside 1120

Molchanov-Vinberg Laplacian

McGill, Burnside 920

Weighted L^2-estimates of the Witten spinor in asymptotically flat manifolds

Monday, March 14, 2:30-3:30pm

Burnside 1205

Dynamics of Lattice Dynamical Systems

Burnside 920

Variational approach to spectral problems: two unusual examples.

Burnside 1120, McGill

An Equivariant Microlocal Lift on Locally Symmetric Spaces

Room LB 559-6, Library building, Concordia

The Lorenz Attractor revisited

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

Homological approach to detection of interesting dynamics

Some extremal problems for curves of genus 0, 1 and 2

On the minimum of one fancy functional

Burnside 920

Lower bounds for pointwise error term in Weyl's law

2004/2005 Seminar in Nonlinear Analysis and Dynamical Systems

2003/2004 Seminar in Nonlinear Analysis and Dynamical Systems

2003/2004 Working Seminar in Mathematical Physics