# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

### 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.

### Knotoids and Virtual Knot Theory

Knotoids are open-ended knot diagrams whose endpoints can be in different regions of the diagram. Two knotoids are said to be isotopic if there is a sequence of Reidemeister moves that connects one diagram to the other without moving arcs across endpoints. The definition is due to Turaev. We will discuss three dimensional interpretations of knotoids in terms of projections of open-ended embeddings of intervals into three dimensional space, and we shall discuss a number of invariants of knotoids based on concepts from virtual knot theory. Knotoids are a new branch of classical knot theory and they promise to provide a way to measure the “knottiness” of open interval embeddings in three space. This talk is joint work with Neslihan Gugumcu.

### 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.

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