# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

Newer session pages: Next 6 5 4 3 2 1 Newest

### 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.

### Kähler-Ricci flow: infinite and finite dimensional approach

In Kähler geometry, the Kähler-Ricci flow is a powerful tool to study the existence of Kähler-Einstein metrics. I will discuss how one can think of the Kähler-Ricci flow in terms of so-called balanced metrics, introduced by S. K. Donaldson. In the case of toric Fano-Einstein manifolds, we obtain a simple algorithm to compute an approximation of the Kähler-Einstein metric. For manifolds with negative first Chern class, the algorithm is related to the work of H. Tsuji. Finally, our work is related to the problem of G. I. T stability for a smooth projective manifold.

### On well-posedness theory for nonlinear wave equations aside from the ${H}^{s}$-scale

The Cauchy problem for the cubic nonlinear Schrödinger equation ${\mathrm{iu}}_{t}+{u}_{\mathrm{xx}}=\mid u{\mid }^{2}u,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}u\left(x,0\right)={u}_{0}\left(x\right)$ is considered, with emphasis on the case of one space dimension, i.e., $x\in R$. We review the meanwhile classical results of Tsutsumi and Cazenave/Weissler concerning data in the standard Sobolev spaces ${H}^{s}\left({R}^{n}\right)$. Two invariance properties of the equation---scaling and Galilean invariance---lead to certain obstructions to (time local) well-posedness. A related counterexample of Kenig, Ponce, and Vega showing ill-posedness is sketched. As a consequence, in the one-dimensional case the restriction to ${H}^{s}\left(R\right)$-data seems to be inadequate. A generalization of the standard theory due to the author is presented. Finally, we discuss several other canonical dispersive equations, such as KdV, mKdV, and DNLS, for which very similar problems appear.

### Symmetry breaking in Moser-Trudinger inequalities and a Hénon type problem in dimension two

In this talk, we discuss a Hénon type functional with an exponential nonlinearity in dimension two. To study the symmetry properties of maximizers, we establish a link with Moser-Trudinger inequalities and consider the symmetry properties of the extremal functions for these inequalities.

### Spectral computations for self-similar groups

Self-similar groups, also known as automaton groups, have recently been popularized by the Grigorchuk school. They are groups acting on Cantor sets in such a way that their structure mimics the self-similarity of the Cantor set. Many interesting problems have been solved using self-similar groups including Milnor's problem on growth, Day's problem on elementary amenable groups, Hubbard's twisted rabbit problem, to name a few.

Grigorchuk and Zuk used a self-similar action of the lamplighter group to compute the spectral measure for the simple random walk with respect to its self-similar generating set. This led to a counterexample to the strong form of Atiyah's conjecture on ${L}_{2}$-betti numbers. Dicks and Schick generalized the result to wreath products $G$ wr $Z$ with $G$ a finite group and with "analogous" generators using a different technique. We recover Dicks and Schick's results using the original method of Grigorchuk and Zuk. In the process we clarify the relationship between the Kesten spectral measure and a spectral measure introduced by Grigorchuk and Zuk for self-similar groups. This is joint work with Mark Kambites and Pedro V. Silva.

### Transversality in Scalar Reaction-Diffusion Equations on a Circle

We prove that stable and unstable manifolds of hyperbolic periodic orbits for general scalar reaction-diffusion equations on a circle $u_t=u_{xx}+f(x,u,u_x),\ t\in\mathbb{R},\ x\in S^1,$ always intersect transversally. The argument also shows that for a periodic orbit there are no homoclinic connections. The main tool used in the proofs is Matano's zero number theory dealing with the Sturm nodal properties of the solutions.

### On the total disconnectedness of the quotient Aubry set

In Mather's studies on the existence of Arnold diffusion, it turns out that it might be useful to understand certain metric aspects of what is called the quotient Aubry set. In particular, it seems to be interesting to know whether this set has a "small" dimension. We prove that under suitable hypotheses on the Lagrangian, the associated quotient Aubry set, corresponding to a certain cohomology class, is totally disconnected, i.e., every connected component consists of a single point. We will also discuss the relation between this problem and a Morse-Sard-like property for (difference of) critical subsolutions of Hamilton-Jacobi equation.

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