Analysis, Geometry, and Dynamical Systems Seminar

Past sessions

Virtual knot cobordism

Virtual knot theory is a stabilized version of the classical knot theory of thickened surfaces. This talk will discuss how ideas from classical knot cobordism can be generalized to virtual knot theory and how Rasmussen’s invariants giving lower bounds for the 4-ball genus of classical knots can be generalized to virtual knots.

Shadowing: Right results from wrong numbers

Numerical simulations are indispensable in the investigation of specific dynamical systems. Unfortunately, computer arithmetic with real numbers is necessarily inexact. Since chaotic dynamical systems amplify small errors at an exponential rate, the results of most simulations of such systems are therefore unreliable. In this talk, we will describe the method of shadowing for extracting mathematically rigorous results from erroneous numerical computations. The talk will be illustrated with computer simulations.

Huygens synchronization of two clocks

The synchronization of two pendulum clocks hanging from a wall was first observed by Huygens during the XVII century. This type of synchronization is observed in other areas, and is fundamentally different from the problem of two clocks hanging from a moveable base. We present a model explaining the phase opposition synchronization of two pendulum clocks in those conditions. The predicted behaviour is observed experimentally, validating the model. Joint with L. Melo.

Topological equivalences for one-parameter bifurcations of maps

Homeomorphisms allowing us to prove topological equivalences between one-parameter families of maps undergoing the same bifurcation are constructed in this work. This construction provides a solution for a classical problem in bifurcation theory that was set out three decades ago and remained unexpectedly unpublished until now.

Monads in a tricategory, a monoidal approach to bicategorical structures and generalized multicategories

The talk will revolve around three topics: In one of them we consider monads within a tricategory and their formal theory. In the second, we introduce a new framework for the theory of generalized multicategories. In the third, we describe a monoidal approach to bicategorical structures, which also provides a bridge between the two other topics.

A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations

In this talk I will discuss some recent results obtained with D. Kinderlehrer (Carnegie Mellon Univ.) and X. Xu (Purdue Univ.) for the Poisson-Nernst-Planck equations

\begin{cases}
\partial_t u=\Delta u^m +\operatorname{div}(u\nabla (U+\Psi)),\\
\partial_t v=\Delta v^m +\operatorname{div}(v\nabla (V-\Psi)),\\
-\Delta \Psi =u-v.
\end{cases}
\label{eq:PNP}
The unknowns $u,v\geq 0$ represent the density of some positively and negatively charged particles, $U,V$ are prescribed confining potentials, $\Psi=(-\Delta)^{-1}(u-v)$ is the (nonlocal) self-induced electrostatic potential, and $m\geq 1$ a fixed non-linear diffusion exponent. We show that \eqref{eq:PNP} is the gradient flow of a certain energy functional in the metric space $\left(\mathcal{P}({\mathbb R}^d),\mathcal{W}_2\right)$ of Borel probability measures endowed with the quadratic Wasserstein-Rubinstein-Kantorovich distance $\mathcal{W}_2$. The gradient flow approach in $\left(\mathcal{P}({\mathbb R}^d),\mathcal{W}_2\right)$ is closely related to the theory of optimal mass transport and has successfully been employed for several scalar PDEs (Fokker-Planck, Porous Media, Keller-Segel, thin film...) We exploit this variational structure in order to semi-discretize (in time) the system and construct approximate solutions by means of the DeGiorgi minimizing movement. Sending the time step $h\downarrow 0$ we retrieve global weak solutions for initial data with low integrability and without regularity assumptions. In addition to energy monotonicity we also recover some regularity and new $L^p$ estimates. The proof deals with linear and nonlinear diffusions ($m=1$ and $m\gt 1$) in a unified energetic framework.

Homogenization of functionals with linear growth in the context of $\mathcal{A}$-quasiconvexity

This work deals with the homogenization of functionals with linear growth in the context of $\mathcal{A}$-quasiconvexity. A representation theorem is proved, where the new integrand function is obtained by solving a cell problem where the  coupling between homogenization and the $\mathcal{A}$-free condition plays a crucial role. This result extends some previous work to the linear case, thus allowing for concentration effects. Joint work with J. Matias and P. M. Santos.

The Choquet boundary of amenable nonselfadjoint operator algebras

For any compact and Hausdorff set $X$, the Choquet boundary of  an amenable function algebra $A$ in the algebra $C(X)$ of continuous and complex functions on $X$ is exactly $X$. The problem is lifted to non commutative cases where $A$ is an operator algebra and $X$ is replaced by irreducible representations.

Scalar field cosmology and dynamical systems

In this talk I'll discuss Einstein's field equations and how Killing and homothetic Killing vectors in cosmology in combination with physical first principles, such as general covariance and scale invariance, induce a hierarchical solution space structure, where simpler models act as building blocks for more complicated ones (note that similar considerations are equally applicable when it comes to modified gravity theories). To illustrate the consequences of these quite general aspects, I will consider several examples that will furthermore shed light on a variety of heuristic concepts such as attractor and tracker solutions. I'll focus on flat FLRW cosmology with a source that consists of: a scalar field representation of a modified Chaplygin gas; monomial scalar fields and perfect fluids; inverse power-law scalar fields and perfect fluids. I'll show how physical principles can be used to obtain regular dynamical systems on compact state spaces with a hierarchy of invariant subsets and how local and global dynamical systems techniques then subsequently make it possible to obtain a global understanding of the associated solution spaces and their properties.

Chemical and metabolic networks: sense and sensitivity

For the deceptively innocent case of monomolecular reactions, only, we embark on a systematic mathematical analysis of the steady state response to perturbations of reaction rates. We make sense of this response in terms of the sensitivity of (experimentally accessible) concentrations and (invisible) reaction fluxes. Our function-free approach does not require numerical input. Based on the directed graph structure of the monomolecular reaction network, only, we derive which steady state concentrations and reaction fluxes are sensitive to a rate change, and which are not. Moreover, we establish a transitivity property for the sensitivity of reaction fluxes. The results and concepts are motivated by — and of experimental relevance to — specific metabolic networks in biology, including the ubiquitous tricarboxylic citric acid cycle.

This is joint work with Atsushi Mochizuki (RIKEN Tokyo).

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.

16/04/2013, 15:00 — 16:00 — Room P3.10, Mathematics BuildingFlorin Radulescu, Università di Roma - Tor Vergata

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.