# Seminars & Events for 2007-2008

##### Counting Circles and Other Things

Take a circle of radius one and inscribe in it two circles that are tangent to each other. Now add another circle into the original one so that it's tangent to all three. If we repeat this process over and over, we get an old picture known as the Apollonian circle packing. What radii will we get from one such packing?

##### Logarithm laws for horocycles

In joint work with G. Margulis, we prove a logarithm law for unipotent flows on the space of unimodular lattices in $R^n$.

##### Sequences of Hyperbolic $3$-Manifolds with Unfaithful Markings

Let $\Gamma$ be a finitely generated group. To every representation $\rho : \Gamma\to Isom (BH3)$ with discrete and torsion-free image there corresponds a hyperbolic $3$-manifold $M_\rho = BH3 / \rho (\Gamma)$. I will present some new results linking the pointwise convergence of a sequence of such representations with Gromov-Hausdorff convergence of the corresponding quotient manifolds.

##### On dimensionality of mean structure from a single data matrix

This presentation compares various total return world stock indices based on daily data. Due to diversification these indices are noticeably similar. A diversification theorem identifies any diversified portfolio as a proxy for the growth optimal portfolio.

##### A priori estimates for special Lagrangian equations

We discuss recent a priori interior Hessian estimates for solutions of the special Lagrangian equation, when the equation has phase at least a certain value, or when the solution is convex. These equations include the sigma-2 equation in dimension three. The gradient graph of any solution is a minimizing Lagrangian surface.

##### Mathematical and Computational Challenges in Shear Stiffness Imaging of Tissue: Can cancerous and benign lesions be distinguished?

For centuries doctors have palpated tissue to detect abnormalities. We target imaging the stiffness the doctor feels in the palpation exam, including imaging deeper than what can be felt in this exam and distinguishing between benign and cancerous lesions. Current applications include breast and prostate cancer. Current experimentalists with whom we collaborate are: Dr.

##### Modelling high dimensional daily volatilities based on high-frequency data

It is increasingly popular in financial economics to estimate volatilities of asset returns by the methods based on realized volatility and bipower realized volatility from high-frequency data. However the most available methods are not directly relevant when the number of assets involved is large, due to the lack of accuracy in estimating high dimensional matrices.

##### Localisation in the Anderson tight binding model with several particles

The Anderson model (which will celebrate its 50th anniversary in 2008) is among most popular topics in the random matrix and operator theory. However, so far the attention here was concentrated on single-particle models, where the random external potential is either IID or has a rapid decay of spatial correlations.

##### Real singular Del Pezzo surfaces and rationally connected threefolds

Recent results on classification of real algebraic threefolds will be described. Let W -> X be a real smooth projective threefold fibred by rational curves. J. Kollár proved that if the set of real points W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces.

##### 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.

##### The Moment Map and Delzant Polytopes

In symplectic geometry, the moment map (or momentum map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum.

##### Trigonometric sums and continued fractions with even partial quotients

I will talk about the geometric features ("curlicues") of quadratic trigonometric sums and discuss how the renormalization of such sums is connected with continued fraction expansions with even partial quotients. I will also explain a recent renewal-time limit theorem for the sequence of denominators generated by such expansions.

##### Points Surrounding the Origin

##### Length Spectrum of a Flat Metric

I'll discuss joint work-in-progress with M. Duchin and K. Rafi on the geometry of flat structures on surfaces via the lengths of its closed geodesics.

##### Cohen-Lenstra heuristics and the negative Pell equation

For a squarefree integer $d$ we ask, if the negative Pell equation $x2-dy2 = -1$ is solvable over the integers. By easy considerations we see that in this case $d>0$ and that all odd prime divisors of $d$ are congruent to 1 modulo 4. Now we call a $d$ special, if it satisfies those two conditions.

##### The action gap and periodic orbits of Hamiltonian systems

The action and index spectra of a Hamiltonian diffeomorphism and their behavior under iterations carry important information about the periodic orbits of the diffeomorphism.

##### Pricing American Contingent Claims by Stochastic Linear Programming

In this talk I consider pricing of American contingent claims (ACC) as well as theirspecial cases, in a multi-period, discrete time, discrete state space setting. Determining the buyer's price for ACC's requires solving an integer program unlike European contingent claims for which solving a linear program is sufficient.

##### Wallcrossing for K-theoretic Donaldson invariants and computations for rational surfaces

Let $(X,H)$ be a polarized algebraic surface. Let $M=M^H_X(c_1,c_2)$ be the moduli spaces of $H$-semistable rank 2 sheaves on $X$ with Chern classes $c_1, c_2$. K-theoretic Donaldson invariants of $X$ are holomorphic Euler characteristics of determinant line bundles on $M$. These invariants are subject to wallcrossing when $H$ varies.

##### Photon localization and Dicke superradiance : a crossover to small world networks

We study photon localization in a gas of cold atoms, using a Dicke Hamiltonian that accounts for photon mediated atomic dipolar interactions. The photon escape rates are obtained from a new class of random matrices. A scaling behavior is observed for photons escape rates as a function of disorder and system size.

##### 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.