# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

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### 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.

### 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.

### Transport and the involution kernel in Ergodic Theory

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