| DATE |
SPEAKER |
TOPIC |
| Feb 6 |
Bill Goldman
University of Maryland |
Projective geometry on manifolds
Rich classes of geometric structures on manifolds
are defined by coordinate atlases taking values
in a fixed homogeneous space. The existence and
classification of such structures leads to
a moduli space, which itself is modelled on
the algebraic variety of representations of the fundamental
group in the automorphism group of the geometry.
Topological symmetries lead to actions of mapping class groups
on the moduli spaces, whose dynamics reflects
the topology and the geometry. This talk will present
various examples of the general classification problem,
in dimensions 2 and 3. |
| Feb 13 |
Erez Lapid
Hebrew University |
Volume of polytopes, operator analogues, and Arthur's trace formula
There are two ways (among many others) to compute the volume of a (convex)
polytope. One using a formula of Brion and another using an argument of
P. McMullen and R. Schneider.
The ensuing identity suggests a non-commutative generalization which we
can currently prove for Coxeter zonotopes (e.g. a permutahedron).
This algebraic equality plays a role in Arthur's trace formula.
This has applications to spectral asymptotics of locally symmetric spaces.
No prior knowledge of these subjects is assumed.
Joint work with Tobias Finis and Werner Muller
|
| Feb 20 |
Francois Labourie
Universite de PARIS-SUD |
Finite and infinite representations of surface groups and cross ratios
In this talk, I explain how cross ratios -- special functions of four
arguments -- on the boundary at infinity of surfaces groups describe both
finite (in $SL(n,R)$) and infinite (in a group related to diffeomorphisms
of the circle) dimensional representations of surface groups.
If time permits, I will explain what are the generalisation of McShane's
identity in that context as well as some conjectural picture relating
these representations to complex analysis.
|
| Feb 27 |
Jeff Lagarias
University of Michigan |
Integral Apollonian Circle Packings
Apollonian circle packings are infinite packings of circles, constructed recursively from a
initial configuration of four mutually touching circles by adding circles externally tangent
to triples of such circles. Configurations of four mutually touching circles were studied
by Descartes in 1643. If the initial four circles have integer curvatures, so do all the
circles in the packing. If in addition the circles have rational centers so do all the circles
in the packing. Why? This talk describes results in geometry, group theory and number
theory arising from such packings. (This is joint work with Ron Graham, Colin Mallows,
Allan Wilks, and Catherine Yan.)
|
| Mar 5 |
Natasa Sesum
Columbia University |
The harmonic mean curvature flow of
a 2-dimensional hypersurface
The harmonic mean curvature flow is the flow that moves a hypersurface
embedded in R^3 by the speed given by a ratio of the Gauss and the mean
curvature of the given surface in the direction of its normal. It is a
fully nonlinear, weakly parabolic equation, degenerate at the points at
which our hypersurface changes its convexity and fast diffusion when the mean curvature
tends to zero. We prove a short time existence of such a flow in a
nonconvex case. We also prove that if the mean curvature does not go to zero, the flow
becomes strictly convex at some time and shrinks to a round point.
This is an example of a curvature flow in higher dimension (besides the
curve shortening flow in a plane which has been known for a long time)
that exhibits a nice shrinking property in a spherical manner in a finite
time, even if we start evolving nonconvex hypersurfaces. In that sense
this flow behaves better that the mean curvature flow that in the most
nonconvex cases develops singularities before shrinking to a point.
|
| Mar 12 |
Artur Avila
CNRS/IMPA/CMI |
Chaoticity of the Teichm\"uller flow
A non-zero Abelian differential on a compact Riemann surface determines an
atlas, outside the singularities, whose coordinate changes are
translations. The vertical flow with respect to this translation
structure generalizes the genus one notion of rational and irrational
flows on tori.
A fundamental tool in the understanding of the dynamics of vertical flows
is the Teichm\"uller flow (acting on the moduli space of Abelian
differentials), regarded as a renormalization operator. The chaotic
nature of the dynamics of the Teichm\"uller flow has been a much
researched topic, and currently it is known that it displays exponential
decay of correlations. (This is equivalent to the spectral gap for the
ambient $SL(2,\mathbb{R})$ action, a very familiar result in genus $1$.) Even
much weaker aspects of the chaoticity of the Teichm\"uller flow however
can be exploited in the description of the dynamics of typical vertical
flows. Two such results are the proofs of the Kontsevich-Zorich
conjecture and of weak mixing for interval exchange transformations.
|
| Mar 26 |
No Colloquium
this week |
|
| Apr 2 |
Nalini Anantharaman
Ecole Polytechnique |
Entropy and the localization of eigenfunctions
We study the behaviour of the eigenfunctions of the laplacian, on a compact
negatively curved manifold, and for large eigenvalues.
The Quantum Unique Ergodicity conjecture predicts that the probability
measures defined by these eigenfunctions should converge weakly to the
Riemannian volume. We prove an entropy lower bound on these probability
measures, which shows for instance that it is difficult for them to
concentrate on closed geodesics.
|
| Apr 9 |
Marc Levine
Northeastern University |
Algebraic cobordism: applications and perspectives
We will survey our theory, with F. Morel, of algebraic cobordism. This is the algebraic analog of complex cobordism, and may be viewed as a refinement of the Chow ring, replacing algebraic cycles with algebraic manifolds. We will discuss its relation with the Chow ring and the Grothendieck group of coherent sheaves, with applications to Riemann-Roch and degree formulas (used in the proof of the Bloch-Kato conjecture). With R. Pandharipande, we have given a simple description of the relations defining algebraic cobordism, the so-called double point cobordism; we will discuss applications this has had to Donaldson-Thomas theory. Finally, we will discuss the relation of our geometric theory with a more sophisticated version defined using motivic homotopy theory.
|
| Apr 16 |
Roma Bezrukavnikov
MIT and IAS |
Characters of finite Chevalley groups and categorification
An important branch of representation theory studies representations of reductive groups over finite fields, such as GL(n,F_q), Sp(2n,F_q) etc. A deep theory due mostly to Lusztig and Shoji provides a classification of irreducible representations and a formula for their characters in terms of certain algebro-geometric objects called character sheaves.
In a joint work with M. Finkelberg and V. Ostrik we establish some new nice features of the geometric objects, motivated by an attempt to find a conceptual explanation for the beautiful but somewhat mysterious results of Lusztig and Shoji.
|
| Apr 23 |
Jim Bryan
UBC |
The classical and quantum geometry of polyhedral singularities and their resolutions
Abstract: Let G be a finite subgroup of SO(3). Such groups admit an ADE classification: they are the cyclic groups, the dihedral groups, and the symmetries of the platonic solids. The singularity C^3/G has a natural Calabi-Yau resolution Y given by Nakamura's G-Hilbert scheme. The classical geometry of Y (its cohomology) can be described in terms of the representation theory of G. The quantum geometry of Y (its quantum cohomology) can be described in terms of R, the ADE root system associated to G. This leads to an interesting family of algebra structures on the affine root lattice of R. Other aspects of the "quantum geometry" of Y and C^3/G (namely their Gromov-Witten and Donaldson-Thomas theories) are also governed by the root system R. One nice application is an attractive formula for the number of colored boxes piled in the corner of a room --- generalizing the classical formula of MacMahon for the case of uncolored boxes
|
| Apr 30 |
Stanislav Shvartsman
Princeton |
Paint-by-numbers: pattern formation in two-dimensional sheets of cells
One of the basic mechanisms responsible for the formation of three-dimensional organs relies on the regulated folding of epithelia
(two-dimensional
sheets of cells). This process is driven by the spatially nonuniform and dynamic distribution of multiple chemical components (products of gene
expression) across the epithelia that prepare for folding. Some of the key questions in this class of biological problems are related to the total
number of involved genes, the diversity and dynamics of their expression patterns, and the mechanisms of pattern formation. I will present the
results of our experimental and computational work that explores these questions during the formation of an elaborate three-dimensional structure
(the fruit fly eggshell). I will also try to discuss the mathematical problems associated with the combinatorial construction of complex
two-dimensional patterns from a small number of building blocks and dynamics of piecewise linear models of epithelial patter
ning.
|