# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

### Homogenization and multiobjective optimization - computational challenges

I will present two recent projects related to front propagation and optimal control.

The first of these (joint with A. Oberman and R. Takei) deals with 2-scale and 3-scale computations in geometric optics. We propose a new and efficient method to homogenize first-order Hamilton-Jacobi PDEs. Unlike the prior cell-problem methods, our algorithm is based on homogenizing the related Finsler metric. We illustrate by computing the effective velocity profiles for a number of periodic and "random" composite materials.

The second project (joint with A. Kumar) deals with multiple criteria for optimality (e.g., fastest versus shortest trajectories) and optimality under integral constraints. We show that an augmented PDE on a higher-dimensional domain describes all Pareto-optimal trajectories. Our numerical method uses the causality of this PDE to approximate its discontinuous viscosity solution efficiently. The method is illustrated by problems in robotic navigation (e.g., minimizing the path length and exposure to an enemy observer simultaneously).

### A topos approach to quantum theory

I will report on work with Chris Isham on the application of topos theory to physics. The Kochen-Specker theorem shows that a naive realist description of quantum systems is impossible. This can be understood as the inapplicability of Boolean logic to quantum systems. In order to arrive at a more realist description, one can use the internal logic of a certain topos of presheaves. The choice of this topos is directly motivated from the Kochen-Specker theorem. I will show which structures within this topos are of physical significance and how propositions about physical quantities are assigned truth-values. The relation with constructive Gel'fand duality is sketched.

### Iterated function systems, classical and quantum

An iterated function system (IFS) is a collection of maps that are chosen and iterated according to some probability. The geometric and measurable structure of the limit set is usually very interesting and studied with the use of the well known technique of transfer operator. In this talk we will review some results about this problem and its quantum version, where the maps are unitary operators acting on some suitable Hilbert space.

### Type ${\mathrm{II}}_{1}$ Von Neumann Representations for Hecke Operators on Maass Forms and Inequalities for their Eigenvalues

We prove that classical Hecke operators on Maass forms are a special case of completely positive maps on II${}_{1}$ factors, associated to a pair of isomorphic subfactors. This representation induces several matrix inequalities on the eigenvalues of the Hecke operators on Maass forms. These inequalities are the consequence of a "double" action of the Hecke algebra which can be seen only in the type II${}_{1}$ representation. The classical Hecke operators are then a "diagonal" of this double action.

### Theoretical models of cooperation with ODEs: from prebiotic evolution to complex ecosystems

The dynamics of cooperation can be studied using mean field models given by nonlinear ordinary differential equations (ODEs). Such a dynamics is modeled considering heterocatalytic feedback loops, with density-dependent enhancement of reproduction rates between cooperating replicators. These models are used to study the dynamics of catalytic networks named hypercycles which have a particular graph architecture and which has been suggested to be of importance in earlier stages of prebiotic evolution. In this talk we will introduce the mathematical formalism used for the dynamical study of such networks, focusing on the origin of life problem and on the dynamics of ecological systems. We will analyze several low-dimensional catalytic networks providing their stability and bifurcation scenarios. We will also discuss some interesting dynamical phenomenon associated to the bifurcations implying the transition from survival to extinction phases. We will then also compare the results obtained with the ODEs for these kind of networks with other computational tools typical of complex systems analysis.

### Logarithmic dimension bounds for the maximal function along a polynomial curve

Let $ℳ$ denote the maximal function along the polynomial curve $\left({\gamma }_{1}t,\dots ,{\gamma }_{d}{t}^{d}\right)$: $ℳ\left(f\right)\left(x\right)=\underset{r>0}{\mathrm{sup}}\frac{1}{2r}{\int }_{\mid t\mid \le r}\mid f\left({x}_{1}-{\gamma }_{1}t,\dots ,{x}_{d}-{\gamma }_{d}{t}^{d}\right)\mid \mathrm{dt}.$We show that the ${L}^{2}$ norm of this operator grows at most logarithmically with the parameter $d$: $\parallel ℳf{\parallel }_{{L}^{2}\left({ℝ}^{d}\right)}\le c\mathrm{log}d\parallel f{\parallel }_{{L}^{2}\left({ℝ}^{d}\right)},$where $c>0$ is an absolute constant. The proof depends on the explicit construction of a "parabolic" semigroup of operators which is a mixture of stable semigroups.

### The subtle convergence of Wilkinson's method

Wilkinson's iteration is frequently used to compute eigenvalues of symmetric matrices. Decades of experience led to believing that the algorithm performed extremely fast, Indeed, recently Nicolau Saldanha (PUC, Rio de Janeiro), Ricardo Leite (UFES) and I proved that this is so, for matrices whose spectrum does not contain three eigenvalues forming an arithmetic progression. Things may go slightly slower otherwise. The argument uses techniques from the theory of completely integrable systems, a new class of inverse variables for tridiagonal matrices and the construction of some Lyapunov functions. The counter-examples arise from an unexpected property related to the iteration of a discontinuous function on the plane.

### Stationary Markov Chains on $\left[0,1\right]$ that maximize a Potential: Unicity and a L.D.P.

Given a continuous potential $A:\left[0,1{\right]}^{ℕ}\to ℝ$, defined in the Bernoulli space $\Omega =\left[0,1{\right]}^{ℕ}$, where $\left[0,1\right]=\left\{x\phantom{\rule{thinmathspace}{0ex}}\mid \phantom{\rule{thinmathspace}{0ex}}0\le x\le 1\right\}\subset ℝ$, and supposing that $A\left({x}_{1},{x}_{2},{x}_{3},...\right)=A\left({x}_{1},{x}_{2}\right)$ depends only on the two first coordinates, we are interested in finding stationary Markov probabilities ${\mu }_{\infty }$ on $\left[0,1{\right]}^{ℕ}$ that maximize the value $\int A\left(x,y\right)d\mu$, among all stationary Markov probabilities $\mu$ on $\Omega =\left[0,1{\right]}^{ℕ}$. This problem corresponds in Statistical Mechanics to the zero temperature case for the interaction described by the potential $A$. The main purpose is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities ${\mu }_{\beta }$ which weakly converges to ${\mu }_{\infty }$. The probabilities ${\mu }_{\beta }$ are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure in Thermodynamic Formalism. Considering that $A$ depends only on the first two coordinates, much of the work can be done using measures defined in the space $\left[0,1{\right]}^{2}$. We also show that, in the sense of Mañé, the maximizing probability is unique. It means that, if we perturb the observable $A\left(x,y\right)$ by adding a second term that depends only on the variable $x$, and could be seen as a magnetic term, then generically we have the unicity of the maximizing probability. If we consider $\sigma$-invariant measures in $\left[0,1{\right]}^{ℕ}$, where $\sigma$ is the shift map $\sigma \left(\left({x}_{1},{x}_{2},{x}_{3},...\right)\right)=\left({x}_{2},{x}_{3},...\right)$, then stationary Markov probabilities are a special case of $\sigma$-invariant measures. We show that the maximizing problem, now performed in a much larger class, still attains its maximum in a stationary Markov probability. The unicity of maximizing measures still remains, but only after projection on $\left[0,1{\right]}^{2}$.

### Large deviation principle for the Mather measure

We present the rate function and a Large Deviation Principle for the Entropy Penalized Method when the Mather measure is unique. More explicitly, under some natural assumptions about the Lagrangian $L\left(x,v\right)$, $x$ in the torus ${𝕋}^{N}$, there exists a sequence of measures ${\mu }_{\epsilon ,h}$ converging to the Mather measure $\mu$, when $\epsilon ,h\to 0$. We show a LDP of the kind ${\mathrm{lim}}_{\epsilon ,h\to 0}\epsilon \mathrm{ln}{\mu }_{\epsilon ,h}\left(A\right),$ where $A\subset {𝕋}^{N}×{ℝ}^{N}$. The measures ${\mu }_{\epsilon ,h}$ minimizes the entropy penalized problem: $\mathrm{min}\left\{{\int }_{{𝕋}^{N}×{ℝ}^{N}}L\left(x,v\right)d\mu \left(x,v\right)+\epsilon S\left[\mu \right]\right\},$where the entropy $S$ is given by $S\left[\mu \right]={\int }_{{𝕋}^{N}×{ℝ}^{N}}\mu \left(x,v\right)\mathrm{ln}\frac{\mu \left(x,v\right)}{{\int }_{{ℝ}^{N}}\mu \left(x,w\right)\mathrm{dw}}\mathrm{dxdv}.$and the minimization is performed over the space of probability densities on ${𝕋}^{N}×{ℝ}^{N}$ that satisfy the holonomic constrain. We also show some dynamical properties of the discrete time Aubry-Mather problem.

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