# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

### 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.

### Virtual knot cobordism

Virtual knot theory is a stabilized version of the classical knot theory of thickened surfaces. This talk will discuss how ideas from classical knot cobordism can be generalized to virtual knot theory and how Rasmussen’s invariants giving lower bounds for the 4-ball genus of classical knots can be generalized to virtual knots.

### Shadowing: Right results from wrong numbers

Numerical simulations are indispensable in the investigation of specific dynamical systems. Unfortunately, computer arithmetic with real numbers is necessarily inexact. Since chaotic dynamical systems amplify small errors at an exponential rate, the results of most simulations of such systems are therefore unreliable. In this talk, we will describe the method of shadowing for extracting mathematically rigorous results from erroneous numerical computations. The talk will be illustrated with computer simulations.

### Huygens synchronization of two clocks

The synchronization of two pendulum clocks hanging from a wall was first observed by Huygens during the XVII century. This type of synchronization is observed in other areas, and is fundamentally different from the problem of two clocks hanging from a moveable base. We present a model explaining the phase opposition synchronization of two pendulum clocks in those conditions. The predicted behaviour is observed experimentally, validating the model. Joint with L. Melo.

### Topological equivalences for one-parameter bifurcations of maps

Homeomorphisms allowing us to prove topological equivalences between one-parameter families of maps undergoing the same bifurcation are constructed in this work. This construction provides a solution for a classical problem in bifurcation theory that was set out three decades ago and remained unexpectedly unpublished until now.

### Monads in a tricategory, a monoidal approach to bicategorical structures and generalized multicategories

The talk will revolve around three topics: In one of them we consider monads within a tricategory and their formal theory. In the second, we introduce a new framework for the theory of generalized multicategories. In the third, we describe a monoidal approach to bicategorical structures, which also provides a bridge between the two other topics.

### A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations

In this talk I will discuss some recent results obtained with D. Kinderlehrer (Carnegie Mellon Univ.) and X. Xu (Purdue Univ.) for the Poisson-Nernst-Planck equations

\begin{cases}
\partial_t u=\Delta u^m +\operatorname{div}(u\nabla (U+\Psi)),\\
\partial_t v=\Delta v^m +\operatorname{div}(v\nabla (V-\Psi)),\\
-\Delta \Psi =u-v.
\end{cases}
\label{eq:PNP}
The unknowns $u,v\geq 0$ represent the density of some positively and negatively charged particles, $U,V$ are prescribed confining potentials, $\Psi=(-\Delta)^{-1}(u-v)$ is the (nonlocal) self-induced electrostatic potential, and $m\geq 1$ a fixed non-linear diffusion exponent. We show that \eqref{eq:PNP} is the gradient flow of a certain energy functional in the metric space $\left(\mathcal{P}({\mathbb R}^d),\mathcal{W}_2\right)$ of Borel probability measures endowed with the quadratic Wasserstein-Rubinstein-Kantorovich distance $\mathcal{W}_2$. The gradient flow approach in $\left(\mathcal{P}({\mathbb R}^d),\mathcal{W}_2\right)$ is closely related to the theory of optimal mass transport and has successfully been employed for several scalar PDEs (Fokker-Planck, Porous Media, Keller-Segel, thin film...) We exploit this variational structure in order to semi-discretize (in time) the system and construct approximate solutions by means of the DeGiorgi minimizing movement. Sending the time step $h\downarrow 0$ we retrieve global weak solutions for initial data with low integrability and without regularity assumptions. In addition to energy monotonicity we also recover some regularity and new $L^p$ estimates. The proof deals with linear and nonlinear diffusions ($m=1$ and $m\gt 1$) in a unified energetic framework.

### Homogenization of functionals with linear growth in the context of $\mathcal{A}$-quasiconvexity

This work deals with the homogenization of functionals with linear growth in the context of $\mathcal{A}$-quasiconvexity. A representation theorem is proved, where the new integrand function is obtained by solving a cell problem where the  coupling between homogenization and the $\mathcal{A}$-free condition plays a crucial role. This result extends some previous work to the linear case, thus allowing for concentration effects. Joint work with J. Matias and P. M. Santos.

### The Choquet boundary of amenable nonselfadjoint operator algebras

For any compact and Hausdorff set $X$, the Choquet boundary of  an amenable function algebra $A$ in the algebra $C(X)$ of continuous and complex functions on $X$ is exactly $X$. The problem is lifted to non commutative cases where $A$ is an operator algebra and $X$ is replaced by irreducible representations.

### Scalar field cosmology and dynamical systems

In this talk I'll discuss Einstein's field equations and how Killing and homothetic Killing vectors in cosmology in combination with physical first principles, such as general covariance and scale invariance, induce a hierarchical solution space structure, where simpler models act as building blocks for more complicated ones (note that similar considerations are equally applicable when it comes to modified gravity theories). To illustrate the consequences of these quite general aspects, I will consider several examples that will furthermore shed light on a variety of heuristic concepts such as attractor and tracker solutions. I'll focus on flat FLRW cosmology with a source that consists of: a scalar field representation of a modified Chaplygin gas; monomial scalar fields and perfect fluids; inverse power-law scalar fields and perfect fluids. I'll show how physical principles can be used to obtain regular dynamical systems on compact state spaces with a hierarchy of invariant subsets and how local and global dynamical systems techniques then subsequently make it possible to obtain a global understanding of the associated solution spaces and their properties.

### Chemical and metabolic networks: sense and sensitivity

For the deceptively innocent case of monomolecular reactions, only, we embark on a systematic mathematical analysis of the steady state response to perturbations of reaction rates. We make sense of this response in terms of the sensitivity of (experimentally accessible) concentrations and (invisible) reaction fluxes. Our function-free approach does not require numerical input. Based on the directed graph structure of the monomolecular reaction network, only, we derive which steady state concentrations and reaction fluxes are sensitive to a rate change, and which are not. Moreover, we establish a transitivity property for the sensitivity of reaction fluxes. The results and concepts are motivated by — and of experimental relevance to — specific metabolic networks in biology, including the ubiquitous tricarboxylic citric acid cycle.

This is joint work with Atsushi Mochizuki (RIKEN Tokyo).

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