# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

### Structure stability of pullback attractors

Attractors help us to understand the dynamics of a system thanks to the asymptotic information given about the solutions. But any mathematical object needs a robust definition, that is, needs consistence under perturbations, because any physical model might experiment any kind of natural or artificial changes. Our aim in this talk is to show how a gradient-like pullback attractor has a stable structure under small perturbations and show some nontrivial examples where the perturbation can be found in the time dependent operator or in the nonlinearity.

### Negative Bendixson-Dulac criteria in several dimensions? A chemostat example

The negative criterium of Bendixson and Dulac states that area contracting planar vector fields cannot possess periodic orbits. A physics student asked in class why this would not generalize to three dimensions, obviously, using the curl.

We give a higher-dimensional example, arising from chemostats, where a suitable differential form excludes periodicity. This is joint work with Sze-Bi Hsu.

### Generalized Hamiltonian systems and applications to nonholonomic mechanics

Over twenty years ago Alan Weinstein proposed to formulate mechanics in the framework of Lie algebroids. The use of Lie algebroids in mechanics has recently permitted two unexpected applications: the development of a Hamilton--Jacobi theory and a simple reduced scheme for nonholonomic mechanical systems.

### A geometric definition of the Aubry-Mather set

Given an optical Hamiltonian $H:{T}^{*}M\to {ℝ}^{m}$ on the cotangent bundle of a compact manifold $M$ without boundary its dynamics can be studied with the use of the Lagrangian action functional. Using this approach, Mather defined an important compact invariant set, that he called the Aubry set, and which has the distinguished property of being contained in a Lipschitz Lagrangian graph. In the present seminar, we will study it from a geometric point of view and give the lines of new proofs of some of its most important properties: dynamical invariance, symplectic invariance, and the graph property (which is tautological for the definition we use).

### Geometric properties of elliptic PDEs arising from a variational problem

We will discuss some rigidity, monotonicity and symmetry properties for elliptic PDEs in relation with minimal surfaces and dynamical systems. We will discuss the relation between a problem posed by De Giorgi for the Allen-Cahn equation and a question raised by Bangert in a variational context studied by Moser and related to the Aubry-Mather Theory.

### Convergence of equilibria for thin elastic plates under physical growth conditions

The asymptotic behaviour of the equilibrium configurations of a thin elastic plate is studied, as the thickness $h$ of the plate goes to zero. More precisely, it is shown that critical points of the functional ${E}^{h}$ describing the elastic energy of the thin plate converge to critical points of the $\Gamma$-limit of ${E}^{h}$, in the von Kármán regime. This is proved under the physical assumption that the elastic energy blows up in the case of total compression.

### Quadratic growth BSDE and applications to pricing and hedging of derivatives based on non-tradable underlyings.

We consider Forward-Backward stochastic differential equations with generators that grow quadratically in the control variable (qgFBSDE). This type of equations is used to solve a utility maximization problem in incomplete markets tackling the problem of pricing a financial derivative based on non-traded underlyings such as weather derivatives. We provide new differentiability results for qgFBSDE that in turn allow closed form formulas for the hedging strategy. We present numerical results.

### Virtual Knot Theory and Oriented Extensions of the Jones Polynomial

The original Jones polynomial for classical knots depends only weakly on the orientation of the knot or link to which it is applied. Careful examination of the orientations assigned to states in an oriented bracket polynomial model for the Jones polynomial reveals that there is much more topological information available from orientation when the knot or link is embedded in a thickened surface or regarded as a virtual link. This talk will discuss a large scale generalization that we call the extended bracket polynomial (taking polynomial and graphical values) and the arrow polynomial (joint work with Heather Dye) taking polynomial values with infinitely many new variables. The arrow polynomial is related to the Miyazawa polynomials for virtual knots. The talk will discuss applications of these new invariants to finding virtual crossing number and genus of virtual knots, and we shall discuss extensions of Khovanov homology related to these invariants.

### Equilibrium configurations of epitaxially strained elastic films: existence, regularity, and qualitative properties of solutions

We consider a variational model used in the physical literature to describe the equilibrium configurations of an elastic film epitaxially deposited on a flat rigid substratum, when a lattice mismatch is present between the two materials. After specifying the functional set-up, in which the minimization problem can be properly formulated, we study the regularity and several qualitative properties of equilibrium configurations; that is, of local and global minimizers of the energy functional.

### Entropy, horseshoes and homoclinic trajectories on trees, graphs and on dendrites

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