# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

### Introduction to sensitivity of chemical reaction networks

This talk is an introductory overview of my research topic: Sensitivity of Networks.

We address the following questions: How does a dynamical network respond to perturbations of equilibrium - qualitatively? How does a perturbation of a targeted component spread in the network? What is the sign of the response?

In more detail, we consider general systems of differential equations inspired from chemical reaction networks: $dx/dt = S r(x)$. Here, $x$ might be interpreted as the vector of the concentrations of chemicals, $S$ is the stoichiometric matrix and $r(x)$ is the vector of reaction functions, which we consider as positive given parameters. Abstractly - for a given directed network: the vector $x$ represents the vertices, the matrix $S$ is the incidence matrix and the vector $r(x)$ refers to the directed arrows.

Sensitivity studies the response of equilibrium solutions to perturbations of reaction rate functions, using the network structure as ONLY data. We give here an introduction of the results and techniques developed through this structural approach.

### Self-duality for conservative interacting particle systems

In this talk, we will sketch some recent developments about the notion of duality for conservative interacting particle systems. In particular, we will show the simplification that arises in presence of self-duality when considering hydrodynamic limits in a dynamic disorder (joint work with F. Redig and E. Saada). We will find all particle systems which admit a special form of self-duality (joint work with F. Redig) and, in conclusion, we will use the spectral point of view of this notion to address some open questions.

### Symbolic dynamics of piecewise contractions

A map $f:[0,1]\to [0,1]$ is a piecewise contraction if locally $f$ contracts distance, i.e., if there exist $0<\lambda<1$ and a partition of $[0,1]$ into intervals $I_1,I_2,\ldots,I_n$ such that $\left\vert f(x)-f(y)\right\vert \le\lambda \vert x-y\vert$ for all $x,y\in I_i$ $(1\le i\le n)$. Piecewise contractions describe the dynamics of many systems such as traffic control systems, queueing systems, outer billiards and Cherry flows. Here I am interested in the symbolic dynamics of such maps. More precisely, we say that an infinite word $i_0 i_1 i_2\ldots$ over the alphabet $\mathcal{A}=\{1,2,\ldots,n\}$ is the natural coding of $x\in [0,1]$ if $f^k(x)\in I_{i_k}$ for all $k\ge 0$. The aim of this talk is to provide a complete classification of the words that appear as natural codings of injective piecewise contractions.

### Inﬁnite alphabet ultragraph edge shift spaces: relations to $C^\ast$-algebras and chaos

We explain the notion of ultragraphs, which generalize directed graphs, and use this combinatorial object to deﬁne a notion of (one-sided) edge shift spaces (which, in the ﬁnite case, coincides with the edge shift space of a graph). We then go on to show that these shift spaces have some nice properties, as for example metrizability and basis of compact open sets. We examine shift morphisms between these shift spaces: we give an idea how to show that if two (possibly inﬁnite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph $C^\ast$-algebras are isomorphic. Finally we describe Li-Yorke chaoticity associated to these shifts and remark that the results obtained mimic the results for shifts of finite type over finite alphabets (what is not the case for infinite alphabet shift spaces with the product topology).

### Existence of phase transition for percolation on general graphs

The first step in the study of percolation on a graph $G$ is proving that its critical point $p_c(G)$ for the emergence of an infinite cluster is nontrivial, that is, $p_c(G)\lt 1$. In this talk we prove that, if the isoperimetric dimension of a graph $G$ (with bounded degree) is strictly larger than $4$, then $p_c(G)\lt 1$. This settles a conjecture of Benjamini and Schramm saying that $p_c(G)\lt 1$ for any transitive graph with super-linear growth.

The proof proceeds by first proving the existence of an infinite cluster for percolation with certain random edge-parameters induced by the Gaussian Free Field (GFF). Then we integrate out the randomness in the environment by using a multi-scale decomposition of the GFF.

Joint work with Hugo Duminil-Copin, Subhajit Goswami, Aran Raoufi and Ariel Yadin.

### Spatially inhomogeneous evolutionary games

We study an interaction model of a large population of players based on an evolutionary game, which describes the dynamical process of how the distribution of strategies changes in time according to their individual success.

Differently from spatially homogeneous dynamical games, we assume that the population of players is distributed over a state space and that they are each endowed with probability distributions of pure strategies, which they draw at random to evolve their states. Simultaneously, the mixed strategies evolve according to a replicator dynamics, modeling the success of pure strategies according to a payoff functional.

We establish existence, uniqueness, and stability of Lagrangian and Eulerian solutions of this dynamical game by using methods of ODE and optimal transport on Banach spaces.

### Banach lattice algebra representations in harmonic analysis

If $G$ is a locally compact group, then natural spaces such as $L^1(G)$ or $M(G)$ carry more structure than just that of a Banach algebra. They are also vector lattices, so that they are, in fact, Banach lattice algebras. Therefore, if they act by convolution on, say, $L^p(G)$, it is a meaningful question to ask if the corresponding map into the Banach lattice algebra $L_r(L^p(G))$ of regular operators on $L^p(G)$ is not only an algebra homomorphism, but also a lattice homomorphism. Analogous questions can be asked in similar situations, such as the left regular representation of $M(G)$.

In this lecture, we shall give an overview of what is known in this direction, and which approaches are available. The rule of thumb, based on an underlying general principle, seems to be that the answer is affirmative whenever the question is meaningful.

This is joint work with Garth Dales and David Kok.

### Optimal design problems for energies with nonstandard growth

Some recent results dealing with optimal design problems for energies which describe composite materials, mixed materials and Ogden ones will be presented.

### A piecewise linear map with two discontinuities: bifurcation structures in the chaotic domain

In the current work we consider a one-dimensional piecewise linear map with two discontinuity points and describe different bifurcation structures observed in its parameter space. The structures associated with periodic orbits have been extensively studied before (see, e.g., Sushko et al., 2015 or Tramontana et al., 2012, 2015). By contrast, here we mainly focus on the regions associated with robust multiband chaotic attractors. It is shown that besides the standard bandcount adding and bandcount incrementing bifurcation structures, occurring in maps with only one discontinuity, there also exist peculiar bandcount adding and bandcount incrementing structures involving all three partitions. Moreover, the map's three partitions may generate intriguing bistability phenomena.

1. Sushko I., Tramontana F., Westerhoff, F. and Avrutin V. (2015): Symmetry breaking in a bull and bear financial market model. Chaos, Solitons and Fractals, 79, 57-72.
2. Tramontana, F., Gardini L., Avrutin V. and Schanz M. (2012): Period Adding in Piecewise Linear Maps with Two Discontinuities. International Journal of Bifurcation & Chaos, 22(3) (2012) 1250068 (1-30).
3. Tramontana, F., Westerhoff, F. and Gardini, L. (2015): A simple financial market model with chartists and fundamentalists: market entry levels and discontinuities. Mathematics and Computers in Simulation, Vol. 108, 16-40.

### Invariance principle for a slowed random walk driven by symmetric exclusion

Joint work with Milton Jara.

We establish an invariance principle for a random walk driven by the symmetric exclusion process in one dimension. The walk has a drift to the left (resp. right) when it sits on a particle (resp. hole). The environment starts from equilibrium and is speeded up with respect to the walker. After a suitable space-time rescaling, the random walk converges to a sum of a Brownian motion and a Gaussian process with stationary increments, independent of the Brownian motion. Our main tool in the proof is an estimate on the relative entropy between the law of the environment process and the equilibrium measure of the exclusion process.

### Geometric and combinatoric structures in stationary Markov chains

We study the combinatoric and geometric structure of stationary non-reversible Markov chains defined on graphs, in particular in applications we focus on non-equilibrium states for interacting particle systems from a microscopic viewpoint. We discuss two different discrete geometric constructions. We apply both of them to determine non reversible transition rates corresponding to a fixed invariant measure. The first one uses the equivalence of this problem with the construction of divergence free flows on the transition graph. The second construction is a functional discrete Hodge decomposition for translational covariant discrete vector fields. This decomposition is unique and constructive. The stationary condition can be interpreted as an orthogonality condition with respect to an harmonic discrete vector field. This decomposition applied to the instantaneous current of any interacting particle system on a finite torus tell us that it can be canonically decomposed in a gradient part, a circulation term and an harmonic component. All the three components can be computed and are associated with functions on the configuration space.

### A random particle system and nonentropy solutions of the Burgers equation on the circle

We consider a particle system which is equivalent to a process valued on the space of nonentropy solutions of the inviscid Burgers equation. Such solutions are conjectured to be relevant for the study of the KPZ fixed point. We prove ergodicity and obtain some properties of the stationary measure.

Joint work with C.-E. Bréhier (Lyon) and M. Mariani (Rome).

### Percolation on the stationary distribution of the voter model on $\mathbb{Z}^d$

The voter model on ${\mathbb Z}^d$ is a particle system that serves as a rough model for changes of opinions among social agents or, alternatively, competition between biological species occupying space. When the model is considered in dimension 3 or higher, its set of (extremal) stationary distributions is equal to a family of measures $\mu_\alpha$, for $\alpha$ between 0 and 1. A configuration sampled from $\mu_\alpha$ is a field of 0's and 1's on ${\mathbb Z}^d$ in which the density of 1's is $\alpha$. We consider such a configuration from the point of view of site percolation on ${\mathbb Z}^d$. We prove that in dimensions 5 and higher, the probability of existence of an infinite percolation cluster exhibits a phase transition in $\alpha$. If the voter model is allowed to have long range, we prove the same result for dimensions 3 and higher. Joint work with Balázs Ráth.

### Absence of eigenvalues of Schrödinger operators with complex potentials

We prove that the spectrum of Schrödinger operators in three dimensions is purely continuous and coincides with the non-negative semiaxis for all potentials satisfying a form-subordinate smallness condition. By developing the method of multipliers, we also establish the absence of point spectrum for electromagnetic Schrödinger operators in all dimensions under various alternative hypotheses, still allowing complex-valued potentials with critical singularities. This is joint work with Luca Fanelli and Luis Vega.

### Asymptotic behavior of the exclusion process with slow boundary

The system of interacting particles that will be presented (the exclusion process with slow boundary) arouses considerable interest in its applicability, for modeling mass transfer between reservoirs with different densities. But it also arouses interest in its theoretical part because of its non-triviality, for example: the invariant measure is given through matrices of Ansatz, see Derrida. Another interesting theoretical aspect, which will be the main focus of this talk, is the behavior of particles density (hydrodynamic limit) is given by the heat equation with boundary conditions. These boundary conditions have phase transition, which depends on how slow the behavior at the border is. More specifically, if the boundary has transfer rate of the order of $N^{- a}$, where $N$ is the scale parameter and $a$ is a fixed non-negative real number, then we get for $a$ in $[0,1)$, Dirichlet boundary conditions, for $a\gt 1$ Neumann boundary conditions and the critical case is when $a=1$, which has Robin boundary conditions. In this talk, in addition to the hydrodynamic limit, other results for the scale limits of this model will be presented, such as fluctuations and large deviations.

### C*-algebra valued numerical range for adjointable operators and some applications

Let $A$ be C*-algebra and $E$ a Hilbert C*-module over $A$. For an adjointable operator $T$ on $E$, we define an $A$-valued numerical range $W(T)$ of $T$. We derive properties of $W(T)$ which are the analogs of the classical numerical range for operators acting on Hilbert space, including the Toeplitz-Hausdorff theorem and the equality between the numerical radius and the operator norm for normal adjointable operators.

### Some loci of rational cubic fourfolds

We shall report on joint work with Michele Bolognesi and Giovanni Staglianò on the irreducible divisor $\mathcal C_{14}$ inside the moduli space of smooth cubic hypersurfaces in $\mathbb P^5$. A general point of $\mathcal C_{14}$ is, by definition, a smooth cubic fourfold containing a smooth quartic rational normal scroll (or, equivalently, a smooth quintic del Pezzo surfaces) so that it is rational. We shall prove that every cubic fourfold contained in $\mathcal C_{14}$ is rational.

In passing we shall review and put in modern terms some ideas of Fano, yielding a geometric insight to some known results on cubic fourfolds, e.g. the Beauville-Donagi isomorphism, and discuss also the connections of our results with the recent examples about the bad behavior of rationality in smooth families of fourfolds.

### On (special versions of) the Hartshorne Conjecture on Complete Intersections

We shall present some general techniques for studying projective embedded manifolds uniruled by lines, based on the Hilbert scheme of lines passing through a general point of the manifold and contained in it. The main applications will be the proofs of Hartshorne Conjecture for quadratic manifolds, of the classification of quadratic Hartshorne varieties, of the classification of Severi varieties. Our approach will show many connections between these problems, which were overlooked before, and also a uniform way of solving them. If time allows, we shall also discuss some open problems including the Barth-Ionescu Conjecture.

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