Analysis Seminar
2005/2006 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.
2004/2005 Seminar in Nonlinear Analysis and Dynamical Systems
2003/2004 Seminars
2003/2004
Seminar in Nonlinear Analysis and Dynamical Systems
2003/2004 Working Seminar in Mathematical Physics
2002/2003 Seminars
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2000/2001 Seminars
1999/2000 Seminars