# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

### Mathematics in Portugal — Science and Higher Education Statistics

I will give an overview of the new statistics that are being produced at the Ministério da Educação e Ciência about scientific production and undergraduate degrees in Portugal. The discussion will be focused on, but not limited to, the discipline of Mathematics.

http://infocursos.mec.pt/

### Dynamic Markov-Dubins problem

Andrei Andreyevich Markov proposed in 1889 the problem (solved by Lester Dubins in 1957) of finding the twice continuously differentiable (arc length parametrized) curve with bounded curvature, of minimum length, connecting two unit vectors at two arbitrary points in the plane. We consider the following variant, which we call the "dynamic Markov-Dubins problem" (dMD): to find the time-optimal $C^2$ trajectory connecting two velocity vectors having possibly different norms. The control is given by a force whose norm is bounded. The acceleration may have a tangential component, and corners are allowed, provided the velocity vanishes there. We show that for almost all the two vectors boundary value conditions, the optimization problem has a smooth solution. We suggest some research directions for the dM-D problem on Riemannian manifolds, in particular we would like to know what happens if the underlying geodesic problem is completely integrable. Path planning in robotics and aviation are natural applications, and we also outline a pursuit problem. Finally, we suggest an application to dynamic imaging science. Short time processes (in medicine and biology, in environment sciences, geophysics, even social sciences) can be thought as tangent vectors. This is joint work with Alex Castro (PUC/RJ).

### Open sets of diffeomorphisms with trivial centralizer in the $C^1$ topology

On the torus of dimension $2$, $3$, or $4$, we show that the subset of diffeomorphisms with trivial centralizer in the $C^1$ topology has nonempty interior. We do this by developing two approaches, the fixed point and the odd prime periodic point, to obtain trivial centralizer for an open neighbourhood of Anosov diffeomorphisms arbitrarily near certain irreducible hyperbolic toral automorphisms.

### On extremizers for Fourier restriction inequalities

This talk will focus on extremizers for a family of Fourier restriction inequalities on planar curves. It turns out that, depending on whether or not a certain geometric condition related to the curvature is satisfied, extremizing sequences of nonnegative functions may or may not have a subsequence which converges to an extremizer. We hope to describe the method of proof, which is of concentration compactness flavor, in some detail. Tools include bilinear estimates, a variational calculation, a modification of the usual method of stationary phase and several explicit computations.

### An entropic selection principle of evolutionary theory

The outcome of competition between populations of replicating entities is predicted by evolutionary entropy, a nonequilibrium analogue of the Gibbs-Boltzmann entropy in statistical thermodynamics. This talk will outline the mathematical basis of this selection principle and describe certain applications to the origin of metabolic diseases.

http://www.sciencedirect.com/science/article/pii/S0370157313001191#

### Dilation of operators on $L^p$-spaces

Let $1\lt p \lt\infty$, let $(\Omega,\mu)$ be a measure space and let $T : L^p(\Omega)\to L^p(\Omega)$ be a bounded operator. We say that it admits a dilation (in a loose sense) when there exist another measure space $(\Omega',\mu')$, an invertible operator $U$ on ${L^p}(\Omega')$ such that $\{U^n:n\in\mathbb{Z} \}$ is bounded and two bounded operators $J : L^p(\Omega) \to L^p(\Omega')$ and $Q : L^p(\Omega')\to L^p(\Omega)$ such that $T^n=Q {U^n} J$ for any integer $n\ge 0$. When $p=2$, this property is equivalent to $T$ being similar to a contraction. The main question considered in this talk is to characterize operators with this property when $p\ne 2$. Our results give partial answers and strong connections with functional calculus properties. The talk will include motivation for this dilation question. (Joint work with C. Arhancet.)

### A new method for large time behavior of convex Hamilton-Jacobi equations

We introduce a new method to study the large time behavior for general classes of Hamilton-Jacobi type equations, which include degenerate parabolic equations and weakly coupled systems. We establish the convergence results by using the nonlinear adjoint method and identifying new long time averaging effects. These methods are robust and can easily be adapted to study the large time behavior of related problems.

### Ramanujan-Petersson conjectures and Operator Algebras

Apoio do Instituto Cultural Romeno

### On the asymptotic behaviour of nonlocal nonlinear problems

This lecture deals with nonlocal nonlinear problems. Our main results concern existence, uniqueness and asymptotic behavior of the weak solutions of a nonlinear parabolic equation of reaction-diffusion nonlocal type by an application of the Faedo-Galerkin approximation and Aubin-Lions compactness result. Moreover, we prove continuity with respect to the initial values, the joint continuity of the solution and a result on the existence of the global attractor for the problem $\begin{cases} u_{t}-a(l(u))\Delta u+| u|^{\rho }u=f(u) \phantom{.} \text{ in} \phantom{.} \Omega \times (0,T), \\ u(x,t)=0 \phantom{.} \text{ on }\phantom{.} \partial \Omega \times (0,T), \\ u(x,0)=u_{0}(x) \phantom{.}\text{ in }\phantom{.} \Omega , \end{cases}$ when $0\lt \rho \le 2/(n-2)$ if $n\ge 3$ and $0\lt \rho \lt \infty$ if $n=1,2$, where $u=u(x,t)$ is a real valued function, $\Omega \subset \mathbb{R}^{n}$ is a bounded smooth domain, $n\ge 1$ with regular boundary $\Gamma =\partial \Omega$, $p\geq 2$. Moreover, $a$ and $f$ are continuous functions satisfying some appropriate conditions and $l: L^{2}(\Omega )\to\mathbb{R}$ is a continuous linear form.

### Manifolds on the verge of a regularity breakdown

There are two main stabibility arguments for solutions in dynamical systems: the theory of normal hyperbolicity and the Kolmogorov Arnold Moser theory of perturbations. In recent times, there has been progress in developing versions of the theory that are well suited for computations. The theory does not require that the system is close to integrable, but rather uses geometric identities. The theorems prove that approximate solutions satisfying some non-degeneracy assumptions correspond to a true solution. Furthermore, the proofs lead at the same time to very efficient algorithms. Implementing these algorithms, leads to some conjectural insights on the phenomena that happen at breakdown. They turn out to be remarkably similar to phenomena that were observed in phase transition and "renormalization group" provides a unifying point of view. Nevertheless, many questions remain open.

### Multiscale Geometrical Changes in Continua: Structured Deformations and Some Questions in Analysis

Classes of injective deformations of a continuous body that are stable under composition and under the taking inverses are central to the description of geometrical changes of a continuous body at both the macroscopic and microscopic levels. Classical, simple, and structured deformations are described and assessed from the point of view of continuum mechanics and of variational problems. The natural way of assigning an energy to a structured deformation, through relaxation of an energy assigned to a smaller class of deformations, leads to various alternative, "variationally friendly" notions of structured deformations. Several of these alternative notions are examined from the point of view of approximation by deformations in a smaller class and of relaxation of an initial energy defined on the smaller class of deformations. Questions are raised as to whether or not different notions of structured deformations lead to different relaxed energies for a structured deformation that satisfies the defining requirements of two or more notions. A recent example of an explicit formula for a relaxed energy is used to illustrate these questions.

### Gibbs-Markov structures vs statistical properties in dynamical systems

In a classical approach to dynamical systems one frequently uses certain geometric structures of the system to deduce statistical properties, such as invariant measures with stochastic-like behaviour, large deviations or decay of correlations. Such geometric structures are generally highly non-trivial and thus a natural question is the extent to which this approach can be applied. We show that in many cases stochastic-like behaviour itself implies that the system has certain geometric properties, which are therefore necessary as well as sufficient conditions for the occurrence of the statistical properties under consideration.

### On trilinear oscillatory integrals

We examine a certain class of trilinear integral operators which incorporate oscillatory factors ${e}^{iP}$, where $P$ is a real-valued polynomial, and prove smallness of such integrals in the presence of rapid oscillations. Tools include sublevel set estimates, higher dimensional versions of van der Corput's lemma and corresponding multilinear analogues. This is joint work with Michael Christ.

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