# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

### Reconnection, Vortex Knots and the Fourth Dimension

Vortex knots tend to unravel into collections of unlinked circles by writhe preserving reconnections. We can model this unravelling by examining the world line of the knot, viewing each reconnection as a saddle point transition. The world line is then seen as an oriented cobordism of the knot to a disjoint collection of circles. Cap each circle with a disk (for the mathematics) and the world line becomes an oriented surface in four-space whose genus can be no more than one-half the number of recombinations. Now turn to knot theory between dimensions three and four and find that this genus can be no less than one-half the Murasugi signature of the knot. Thus the number of recombinations needed to unravel a vortex knot $K$ is greater than or equal to the signature of the knot $K$. This talk will review the backgrounds that make the above description intelligible and we will illustrate with video of vortex knots and discuss other bounds related to the Rasmussen invariant. This talk is joint work with William Irvine.

### Unbounded Attractors Under Perturbations

We put forward the recently introduced notion of unbounded attractors. These objects will be addressed in the context of a class of 1-D semilinear parabolic equations. The nonlinearities are assumed to be non-dissipative and, in addition, defined in such a way that the equation possesses unbounded solutions as time goes to infinity. Small autonomous and non-autonomous perturbations of these equations will be treated. This is based on joint work with A. Carvalho and S. Bruschi.

### Representations of graph algebras via branching systems and the Perron-Frobenius Operator

In this talk we show how to obtain representations of graph algebras from branching systems and show how these representations connect to the Perron-Frobenius operator from Ergodic Theory. We will describe how every permutative representation of a graph algebra is unitary equivalent to a representation arising from a branching system. Time permitting we will give an application of the branching systems representations to the converse of the Cuntz-Krieger Uniqueness Theorem for graph algebras.

### Fractional Fick's law for the boundary driven exclusion process with long jumps

A fractional Fick's law and fractional hydrostatics for the one dimensional exclusion process with long jumps in contact with infinite reservoirs at different densities on the left and on the right are derived in this presentation. In the first part it will be described precisely the model studied and the results obtained. In the second one we will recall basic facts on the fractional Laplacian and explain what we mean by stationary solution of a fractional diffusion equation with Dirichlet boundary conditions. In the last part it will be devoted to the comments of the proofs of the results.

### Geometric regularity theory for fully nonlinear elliptic equations

In this talk, we examine the regularity theory for fully nonlinear elliptic equations of the form $F(D^2u)=f(x) \qquad\text{ in } B_1,$ where $F$ is a $(\lambda,\Lambda)$-elliptic operator and $f:B_1\to\mathbb{R}$ is a continuous source term, in appropriate Lebesgue spaces. We recur to a set of tools known as geometric–tangential analysis to produce a priori estimates for the solutions in Sobolev spaces, under minimal (asymptotic) assumptions on the operator $F$. In addition, we discuss regularity in $p$-BMO spaces and the density of $W^{2,p}$- solutions in the class of continuous viscosity solutions. We conclude the talk with the study of a degenerate problem; in this case, we produce a result on the optimal regularity of solutions in Holder spaces.

### Asymptotic behaviour of operators which are similar to normal operators

Let $H$ be a complex Hilbert space and $B(H)$ be the set of all bounded operators on $H$. If $T\in B(H)$ satisfies $$\sup \left\{ \|T^n\| \colon n\in\mathbb{N} \right\} < \infty,$$ then we call it a powerbounded operator, in notation: $T\in \operatorname{PWB}(H)$. It can be easily seen that the set $$H_0 = H_0(T) = \{x\in H \colon \lim_{n\to\infty} \|T^nx\| = 0\}$$ is a closed subspace which will be called the stable subspace of $T$.

Sz.-Nagy's celebrated similarity theorem from the year 1947 states that a bounded operator $T$ is similar to a unitary operator if and only if $T$ is powerbounded and there is a constant $c>0$ such that $$c\|x\| \leq \|T^n x\| \quad \text{and} \quad c\|x\| \leq \|T^{*n} x\|\qquad (n\in\mathbb{N}, x\in H).$$ (And actually, this happens if and only if $T$ is invertible and we have $T, T^{-1} \in \operatorname{PWB}(H)$.)

Motivated by this famous and beautiful theorem, recently I have proved a generalization of the necessity part. In that work I have considered powerbounded operators that are similar to normal operators, and explored their asymptotic behaviour.

If time will permit, I will also show a characterization of injective weighted bilateral shift operators which are similar to normal operators.

### On groups which do not admit faithful Hamiltonian actions on closed symplectic manifolds

The biinvariance, nondiscreteness and separability of the Hofer metric yield restrictions on potential Hamiltonian actions on closed symplectic manifolds. I will prove a general theorem for metric groups and apply it to Hamiltonian actions. The examples will include noncompact semisimple Lie groups (known to Delzant), diffeomorphism groups, and certain automorphisms groups of ordered sets. I will also discuss open problems.

This is a joint work in progress with Assaf Libman and Ben Martin.

### Range additivity, partial orders and some interesting classes of operators on Hilbert spaces

Two operators $A$ and $B$ between Hilbert spaces are said to be range additive if ${\rm Im}\,(A)+{\rm Im}\,(B)={\rm Im}\,(A+B)$. This notion reappears in several instances in the literature, and it carries information about both algebraic and topological properties of the operators $A$ and $B$. It is also strongly connected with the study of different partial orders on the algebra of Hilbert space operators, which originated from the papers of Drazin and Hartwig from 1970s and 1980s, but has recently became popular again. Interestingly, a part of this study has a notable quantum mechanical interpretation when it is restricted to self-adjoint operators, i.e. bounded quantum observables.

In the first part of the talk we will recall some facts concerning closed subspaces and operator ranges in Hilbert spaces, and give a presentation of the previously mentioned notions. After highlighting some existing results and problems, in the second part of the talk we are going to present our approach to these problems, and some of our results. Some of the results that will be presented are from a joint work with G. Fongi and A. Maestripieri.

### The convolution algebra of a topos

I will explain a new construction that attaches an involutive algebra (or a $C^\ast$-algebra) to a topos satisfying certain local separation and local compactness conditions.

The construction is very close to the construction of groupoid $C^\ast$-algebras and generally produces the same results, but it shows that the associated algebra has a very nice universal property in terms of the topos and has some surprising connection with a form of Verdier duality.

I will not assume any knowledge of topos theory, and try as much as possible to give an intuitive understanding of what are toposes.

### Some recent results about dimensional reduction

I will describe some models that I recently studied, about thin structures, modeling rubber-like materials, bending phenomena, and optimal design problems.

### Boundary effects on heat conduction

In this talk I will present a toy model for the heat conduction, which consists of a stochastic dynamics in contact with slowed reservoirs. In this model, particles evolve on the bulk according to nearest neighbor symmetric random walks and at the reservoirs, they can enter or leave the system at a rate which is slower than the rates in the bulk. The main purpose of the talk is to analyze the macroscopic PDE's, governing the space-time evolution of the density of particles and to discuss recent results on the non-equilibrium density fluctuations.

### Reflexivity of bilattices

My talk will be devoted to reflexivity and hyperreflexivity of bilatties.  Bilattices were defined by Shulman in [3]. These structures were studied later in [4] in connection with operator synthesis, in [3] in the context of reflexivity and in [2] in the context of hyperreflexivity. A subspace analogue for a lattice is called a bilattice [4]. Let $\mathcal{H}$ and $\mathcal{K}$ be Hilbert spaces. Denote by $\mathcal{P}(\mathcal{H})$ the lattice of all orthogonal projections on $\mathcal{H}$.  A bilattice is a set $\Sigma\subseteq\mathcal{P}(\mathcal{H})\times\mathcal{P}(\mathcal{K})$ containing pairs $(0,I), (I,0), (0,0)$ and $(P_1\wedge P_2, Q_1\vee Q_2), (P_1\vee P_2, Q_1\wedge Q_2)\in\Sigma$ whenever $(P_1,Q_1), (P_2,Q_2)\in\Sigma$.

Define after [5] $\operatorname{op}\Sigma = \{T\in\mathcal{B}(\mathcal{H},\mathcal{K}) : QTP = 0, \ \forall (P,Q)\in\Sigma\}.$ Then $\operatorname{op}\Sigma$ is always a reflexive subspace and all reflexive subspaces are of this form. The bilattice $\operatorname{bil}\mathcal{S}$ of a subspace $\mathcal{S}\subseteq\mathcal{B}(\mathcal{H},\mathcal{K})$ is defined to be the set $\operatorname{bil}\mathcal{S} = \{(P,Q) : Q\mathcal{S} P = \{0\}\}.$ A bilattice  $\Sigma$ is called reflexive if $\operatorname{bil}\operatorname{op}\Sigma = \Sigma$.

Given a bilattice $\Sigma\subseteq \mathcal{P}(\mathcal{H})\times\mathcal{P}(\mathcal{K})$ and a pair of projections $(P,Q)\in\mathcal{P}(\mathcal{H})\times\mathcal{P}(\mathcal{K})$, let $\alpha((P,Q),\Sigma) = \sup\{\|QTP\| : \|T\|\leq 1, T\in\operatorname{op}\Sigma\}$ and $d((P,Q),\Sigma)=\inf\{\|P-L_1\|+\|Q-L_2\|: (L_1,L_2)\in\Sigma\}.$ A bilattice $\Sigma\subseteq\mathcal{P}(\mathcal{H})\times\mathcal{P}(\mathcal{K})$ is called hyperreflexive if there exists a constant $\kappa > 0$ such that $d((P,Q), \Sigma)\leq \kappa \alpha((P,Q),\Sigma)$, for each pair $(P,Q)\in\mathcal{P}(\mathcal{H})\times\mathcal{P} (\mathcal{K})$.

### References

1. K. R. Davidson and K. J. Harrison, Distance formulae for subspace lattices, J. London Math. Soc. (2) 39 (1989), 309-323.
2. K. Klis-Garlicka, Hyperreflexivity of bilattices, Czech. Math. J. 66(141) (2016), 119-125.
3. K. Klis-Garlicka, Reflexivity of bilattices, Czech. Math. J. 63(138) (2013), 995-1000.
4. V. S. Shulman, in "Nest Algebras" by  K. R. Davidson: a review, Algebra and Analiz, vol.2, no. 3 (1990), 236-255.
5. V. S. Shulman and L. Turowska, Operator synthesis. I. Synthetic sets, bilattices and tensor algebras, J. Functional Analysis, 209 (2004), no. 2, 293-331.

### On asymptotic integration of ordinary differential equations

In this talk we are concerned with asymptotic properties of solutions of a general class of second order nonlinear differential equations
u^{\prime\prime}+f(t,u,u^{\prime})=0.\label{e} Our particular interest is to establish (global) existence of non-trivial asymptotically linear (AL) solutions of  (\ref{e}), that is, solutions satisfying for some real $a\neq0$ $\underset{t\rightarrow+\infty}{\lim}u^{\prime}(t)=\underset{t\to +\infty}{\lim}\dfrac{u(t)}{t}=a,$ as, for instance, solutions with asymptotic representations $u(t)=at+o(t)\qquad\text{or}\quad u(t)=at+b+o(1)$ as $t\rightarrow+\infty.$

We start by discussing the history of the problem which originates from the work of German and Italian mathematicians already in the 19th century and conclude by presenting some of the recent results.

### Discretization of the incompressible Euler Equation: a Lagrangian approach based on semi discrete optimal transport

We approximate the regular solutions of the incompressible Euler equation by flows of ODEs taking values in finite-dimentional spaces. This approach à la Brenier relies on the one hand on Arnold's interpretation of the Euler equation as geodesics in the space of measure-preserving diffeomorphisms, and on the other hand on the semi-discrete Optimal Transport. This approach is naturally associated with a numerical scheme, which will be shown to converge towards regular solutions of the incompressible Euler equation.

### The importance of being just late

Delays are a ubiquitous nuisance in control. Delays increase finite-dimensional phase spaces to become infinite-dimensional. But, are delays all that bad?

Following an idea of Pyragas, we attempt noninvasive and model-independent stabilization of unstable $p$-periodic phenomena $u(t)$ by a friendly delay $\tau$. Our feedback only involves differences $u(t-\tau)-u(t)$. When the time delay $\tau$  is chosen to be an integer multiple $np$ of the minimal period $p$, the difference and the feedback vanish alike: the control strategy becomes noninvasive on the target periodic orbit.

We survey promise and limitations of this idea, including applications and an example of delay control of delay equations.

The results are joint work with P. Hovel, W. Just, I. Schneider, E. Scholl, H.-J. Wunsche, S. Yanchuk, and others. See also http://dynamics.mi.fu-berlin.de/

### A hierarchy of models for the flow of fluids through porous solids

The celebrated equations due to Fick and Darcy are approximations that can be obtained systematically on the basis of numerous assumptions within the context of Mixture Theory; these equations however not having been developed in such a manner by Fick or Darcy.  Relaxing the assumptions made in deriving these equations via mixture theory, selectively, leads to a hierarchy of mathematical models and it can be shown that popular models due to Forchheimer, Brinkman, Biot and many others can be obtained via appropriate approximations to the equations governing the flow of interacting continua.  It is shown that a variety of other generalizations are possible in addition to those that are currently in favor, and these might be appropriate for describing numerous interesting technological applications, e.g., enhanced oil recovery.

### Quantum Enhancements of Biquandle Counting Invariants

Biquandles are algebraic structures related to knots. For every finite biquandle $X$ there is an integer-valued knot and link invariant called the biquandle counting invariant. An enhancement is a stronger invariant which determines but is not determined by the counting invariant. In this talk we will discuss current work on enhancements defined analogously to quantum invariants.

### Numerical range of adjointable operators on a Hilbert $C^\ast$-module and applications

We define and investigate a new numerical range of adjointable operators on a  Hilbert $C^\ast$-module over $C^\ast$-algebra. It is used to characterize  positive  linear maps between the algebras  of adjointable operators.

### The Banach algebra associated with a topological dynamical system

If $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then a Banach algebra $\ell^1(\Sigma)$ of crossed product type is naturally associated with the topological dynamical system $\Sigma(X,\sigma)$. It is a Banach algebra with isometric involution that is not a $C^*$-algebra. If $X$ consists of one point, then $\ell^1(\Sigma)$ is the group algebra of the integers.

The structure of this algebra, which depends on the dynamical properties of $\Sigma$, is more intricate than that of its well-investigated $C^*$-envelope. Its study has been taken up in joint work with Svensson and Tomiyama. In this talk, we will survey the known results about its ideal structure, its algebraically irreducible representations in Banach spaces and the primitive ideal space in the hull-kernel topology.

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