# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

Newer session pages: Next 11 10 9 8 7 6 5 4 3 2 1 Newest

### Ergodic theory and the distribution of ${n}^{2}x$ modulo one

I will discuss some recent results on the distribution of certain unipotent orbits on hyperbolic surfaces and their implication on the randomness of the fractional parts of the sequence ${n}^{2}x$.

### Floquet solutions for linear ODEs with quasiperiodic coefficients

We investigate the local reducibility of linear analytic multifrequency cocycles on SL(2,R) using renormalisation. In this way we show that some linear ODEs with coefficients depending quasiperiodically on time can be conjugated to others with constant coefficients. Much in the same way as Floquet theory for periodic coefficients.

### A lower bound to the spectral threshold in curved tubes

Motivated by the theory of quantum waveguides, we consider the Laplacian in curved tubes of arbitrary cross-section rotating together with the Frenet frame along curves in Euclidean spaces of arbitrary dimension, subject to Dirichlet boundary conditions on the cylindrical surface and Neumann conditions at the ends of the tube. We prove that the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Dirichlet Laplacian in a torus determined by the geometry of the tube. This is a joint work with Pavel Exner and Pedro Freitas.

### The $N$-Membranes Problem for Nonlinear Degenerate Systems

We consider the elliptic variational inequality associated with quasilinear nonlinear systems, including those of $p$-Laplacian type, and with the ordering constraint of the $N$-membrane problem. We extend the Lewy-Stampacchia inequalities for the solution, obtaining new regularity results for the derivatives of the solution, including in the case of linear operators ( $p=2$) integrability of second derivatives for all $q>1$. Considering the $N$-membrane problem as coupled $\left(N-1\right)$-obstacle problem we obtain also the corresponding conditions for the stability of the coincidence sets for the variation of external forces.

### Asymptotic properties of the ground-state solutions of singularly perturbed elliptic Hamiltonian systems

We study the shape of the positive solutions of an elliptic system of the form
 $-{\epsilon }^{2}\Delta u+u=g\left(v\right),-{\epsilon }^{2}\Delta v+v=f\left(u\right)$

as the parameter $\epsilon$ goes to zero, namely the appearance of "spike-layer patterns" for the solutions of the system having minimal energy. Here the nonlinear terms are assumed to have superlinear and subcritical growth at infinity and we consider both cases of Neumann and Dirichlet boundary conditions over a given bounded open set $\Omega \subset {R}^{n}$. The proofs combine standard estimates in elliptic equations with new ideas in the calculus of variations (variational methods and Morse theory). This is joint work with J. Yang (to appear in Trans. Amer. Math. Soc.) and with A. Pistoia (to appear in J. Differential Equations).

### Noncommutative topology and quantales

The classical Gelfand representation theorem tells us that any unital (complex) $C$*-algebra is, up to isomorphism, the algebra of continuous complex valued functions on a compact Hausdorff space. The noncommutative analogue of this result is the Gelfand-Naimark theorem, which shows how any $C$*-algebra can be concretely realized as an algebra of bounded operators on some Hilbert space. However, there is a sense in which this noncommutative "analogue" fails to provide a characterization of those operators on the Hilbert space that actually lie in the given $C$*-algebra, and in 1971 Giles and Kummer (and also Akemann, in a related but independent way) introduced a notion of noncommutative topology in terms of which, essentially, every $C$*-algebra becomes an algebra of "continuous functions".

But lacking in their approach is a self-contained (i.e., independent of $C$*-algebras) characterization of what should be meant by such a noncommutative topology, and it was partly in an attempt to answer this question that quantales were proposed by Mulvey in 1983 as possible candidates for such topologies. In this talk I shall focus on the connections between quantales and $C$*-algebras, in particular addressing as an example, if time allows, Connes' noncommutative space of Penrose tilings.

### Topological entropy and homological growth on graphs

We establish a precise relation between the topological entropy and entropies arising from the homological growth and the exponential growth rate of the number of periodic points for a piecewise monotone graph map showing that the first one is the maximum of the latter two. This nontrivially extends a result of Milnor and Thurston on piecewise monotone interval maps. For this purpose we generalize the concept of Milnor-Thurston zeta function involving Lefschetz zeta function.

### Young measures - Basic ideas and recent applications

Young measures have been successfully applied to various mathematical problems in material science and other areas. They are used as a tool to study problems which include different length scales, e.g., when microscopical structures affect the global behavior of a material.
In this talk we will give an introduction to Young measures, filling the abstract definitions with some intuitive ideas. We will apply this concept to equations modelling shape memory alloys (one-dimensional thermoelasticity with nonconvex energy density) and will present a recent existence result (in collaboration with Johannes Zimmer, MPI Leipzig).

### Nonuniformly hyperbolic dynamics and Lorenz attractors

We consider the statistical properties of Lorenz-like strange attractors which arise out of two-dimensional return maps for the flow. The questions we ask are: Does the system admit an ergodic invariant measure which is mixing? How fast is the mixing? Is the measure physical: ie does it describe the statistical behaviour of many points in the basin? To answer these questions we use a "Markov extension" or "Young tower" to reduce the dynamics to that of an expanding map over a hyperbolic base set, with infinitely many branches having variable return times. This talk is based on ongoing work with S. Luzzatto. I hope to make the talk accessible to nonspecialists in ergodic theory.

### Global Rough Solutions for Nonlinear Dispersive Equations

This will be a PDE talk aimed at a general audience, describing some recent work with J. Colliander, G. Staffilani, H. Takaoka, and T. Tao. We will focus on new results which give basic descriptions of the long time behavior of semilinear Schroedinger equations, but some of the techniques employed are applicable to other PDE's. In particular, the talk will describe how so-called almost-conservation laws and interactaction Morawetz-type inequalities help understand the regularity properties of these equations.

### On the role of quadratic oscillations in nonlinear Schrödinger equations

We consider a nonlinear semi-classical Schrödinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. After a short explanation of this result, we investigate the question to know how particular such initial data are. The first step consists in finding a "linearizability condition": when is the nonlinear term relevant in the semi-classical limit? This conditions reduces the problem to the study of a linear equation. For this equation, we use a "profile decomposition" technique, introduced by P. Gerard. This shows that according to the nonlinearity considered, quadratic oscillations are the only relevant initial data, or are not. We also present some applications to a nonlinear superposition principle, and to the study of finite time blow up.

### Semi-free Hamiltonian circle actions on 6-dimensional symplectic manifolds

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