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17/06/2019, 13:30 — 14:30 — Room P4.35, Mathematics Building

Pietro Caputo, *Università Roma Tre*

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Mixing time of the adjacent walk on the simplex

By viewing the $N$-simplex as the set of positions of $N-1$ ordered particles on the unit interval, the adjacent walk is the continuous time Markov chain obtained by updating independently at rate $1$ the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and mixing time and show that the total variation distance to the uniform distribution displays a cutoff phenomenon. The results are extended to a family of log-concave distributions obtained by replacing the uniform sampling by a symmetric Beta distribution. This is joint work with Cyril Labbe' and Hubert Lacoin.

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04/06/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Brian Hall, *University of Notre Dame*

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Large-$N$ Segal-Bargmann transform with application to random matrices

I will describe the Segal-Bargmann transform for compact Liegroups, with emphasis on the case of the unitary group $U(N)$. In this case, the transform is a unitary map from the space of $L^2$ functions on $U(N)$ to the space of $L^2$ holomorphic functions on the "complexified" group $\operatorname{GL}(N;\mathbb{C})$. I will then discuss what happens in the limit as $N$ tends to infinity. Finally, I will describe an application to the eigenvalues of random matrices in $\operatorname{GL}(N;\mathbb{C})$. The talk will be self-contained and have lots of pictures.

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30/05/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Simão Correia, *Faculdade de Ciências, Universidade de Lisboa*

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Critical well-posedness for the modified Korteweg-de Vries equation and self-similar dynamics

We consider the modified Korteweg-de Vries equation over $\mathbb{R}$ $$ u_t + u_{xxx}=\pm(u^3)_x. $$ This equation arises, for example, in the theory of water waves and vortex filaments in fluid dynamics. A particular class of solutions to (mKdV) are those which do not change under scaling transformations, the so-called *self-similar* solutions. Self-similar solutions blow-up when $t\to 0$ and determine the asymptotic behaviour of the evolution problem at $t=+\infty$. The known local well-posedness results for the (mKdV) fail when one considers critical spaces, where the norm is scaling-invariant. This means that self-similar solutions lie outside of the scope of these results. Consequently, the dynamics of (mKdV) around self-similar solutions are currently unknown. In this talk, we will show existence and uniqueness of solutions to the (mKdV) lying on a critical space which includes both regular and self-similar solutions. Afterwards, we present several results regarding global existence, asymptotic behaviour at $t=+\infty$ and blow-up phenomena at $t=0$. This is joint work with Raphaël Côte and Luis Vega.

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28/05/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Diogo Arsénio, *New York University Abu Dhabi*

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Recent progress on the mathematical theory of plasmas

The incompressible Navier–Stokes–Maxwell system is a classical model describing the evolution of a plasma (i.e. an electrically conducting fluid). Although small smooth solutions to this system (in the spirit of Fujita–Kato) are known to exist, the existence of large weak solutions (in the spirit of Leray) in the energy space remains unknown. This defect can be attributed to the difficulty of coupling the Navier–Stokes equations with a hyperbolic system. In this talk, we will describe recent results aiming at building solutions to Navier–Stokes–Maxwell systems in large functional spaces. In particular, we will show, for any initial data with finite energy, how a smallness condition on the electromagnetic field alone is sufficient to grant the existence of global solutions.

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21/05/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Cédric Bernardin, *University of Nice Sophia-Antipolis*

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Microscopic models for multicomponents SPDE’s with a KPZ flavor

The usual KPZ equation is the scaling limit of weakly asymmetric microscopic models with one conserved quantity. In this talk I will present some weakly asymmetric microscopic models with several conserved quantities for which it is possible to derive macroscopic SPDEs with a KPZ flavor.

Joint work with R. Ahmed, T. Funaki, P. Gonçalves, S. Sethuraman and M. Simon.

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14/05/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Conrado Costa, *Leiden University*

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Random walks in cooling random environments: stable and unstable behaviors under regular diverging cooling maps

Random Walks in Cooling Random Environments (RWCRE), a model introduced by L. Avena, F. den Hollander, is a dynamic version of Random Walk in Random Environment (RWRE) in which the environment is fully resampled along a sequence of deterministic times, called refreshing times. In this talk I will consider effects of the ressampling map on the fluctuations associated with the annealed law and the Large Deviation principle under the quenched measure. I conclude clarifying the paradox of different fluctuations and identical LDP for RWCRE and RWRE. This is a joint work with L. Avena, Y. Chino, and F. den Hollander.

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16/04/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Phillipo Lappicy, *ICMC, Universidade de São Paulo e CAMGSD-IST, Universidade de Lisboa*

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A nonautonomous Chafee-Infante attractor: a connection matrix approach

The goal of this talk is to present the construction of the global attractor for a genuine nonautonomous variant of the Chafee-Infante parabolic equation in one spatial dimension. In particular, the attractor consists of asymptotic profiles (which correspond to the equilibria in the autonomous counterpart) and heteroclinic solutions between those profiles. We prove the existence of heteroclinic connections between periodic and almost periodic asymptotic profiles, yielding the same connection structure as the well-known Chafee-Infante attractor. This work is still an ongoing project with Alexandre N. Carvalho (ICMC - Universidade de São Paulo).

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02/04/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Clement Erignoux, *Università Roma Tre*

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Hydrodynamics for a non-ergodic facilitated exclusion process

The Entropy Method introduced by Guo, Papanicolaou and Varadhan (1988) has been used with great sucess to derive the scaling hydrodynamic behavior of wide ranges of conserved lattice gases (CLG). It requires to estimate the entropy of the measure of the studied process w.r.t. some good, usually product measure. In this talk, I will present an exclusion model inspired by a model introduced by Gonçalves, Landim, Toninelli (2008), with a dynamical constraint, where a particle at site $x$ can only jump to $x+\delta$ iff site $x-\delta$ is occupied as well. I will give some insight on the different microscopic and macroscopic situations that can occur for this model, and briefly describe the steps to derive the hydrodynamic limit for this model by adapting the Entropy Method to non-product reference measures. I will also expand on the challenges and question raised by this model and on some of its nice mapping features. Joint work with O. Blondel, M. Sasada, and M. Simon.

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26/03/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Ofer Busani, *University of Bristol*

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Transversal fluctuations in last passage percolation

In Last Passage Percolation(LPP) we assign i.i.d Exponential weights on the lattice points of the first quadrant of $\mathbb{Z}^2$. We then look for the up-right path going from $(0,0)$ to $(n,n)$ that collects the most weights along the way. One is then often interested in questions regarding (1) the total weight collected along the maximal path, and (2) the behavior of the maximal path. It is known that this path's fluctuations around the diagonal is of order $n^{2/3}$. The proof, however, is only given in the context of integrable probability theory where one relies on some algebraic properties satisfied by the Exponential Distribution. We give a probabilistic proof for this phenomenon where the main novelty is the probabilistic proof for the lower bound. Joint work with Marton Balazs and Timo Seppalainen

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19/03/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Nicola Vassena, *Free University Berlin*

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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.

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12/02/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Federico Sau, *Delft University*

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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.

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18/12/2018, 11:00 — 12:00 — Room P3.10, Mathematics Building

Benito Frazão Pires, *Universidade de São Paulo*

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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.

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17/12/2018, 15:00 — 16:00 — Room P4.35, Mathematics Building

Daniel Gonçalves, *Universidade Federal de Santa Catarina, Brasil*

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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).

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09/10/2018, 15:00 — 16:00 — Room P3.10, Mathematics Building

Franco Severo, *Institut des Hautes Études Scientifiques*

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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.

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28/08/2018, 15:00 — 16:00 — Room P3.10, Mathematics Building

Marco Morandotti, *TUM, Munique*

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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.

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04/07/2018, 15:15 — 16:15 — Room P4.35, Mathematics Building

Rajesh Kumar, *BITS Pilani, India*

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Convergence analysis of finite volume scheme for solving coagulation-fragmentation equations

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04/07/2018, 14:00 — 15:00 — Room P4.35, Mathematics Building

Ankik K. Giri, *IIT Roorkee, India*

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Recent developments in the theory of coagulation-fragmentation models

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19/06/2018, 11:00 — 12:00 — Room P4.35, Mathematics Building

Marcel de Jeu, *Leiden University*

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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.

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29/05/2018, 15:00 — 16:00 — Room P3.10, Mathematics Building

Elvira Zappale, *Università degli Studi di Salerno*

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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.

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12/04/2018, 11:30 — 12:30 — Room P3.10, Mathematics Building

Anastasiia Panchuk, *Academia Nacional das Ciências de Kiev*

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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.

- 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.
- 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).
- 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.