**Eric Carlen **(carlen@math.rutgers.edu)

### A Kac model with exclusion

We consider a one dimension Kac model with conservation of energy and an exclusion rule. This process bears some resemblance to Kac models for Fermions in which the exclusion represents the effects of the Pauli exclusion principle. However, the “non-quantized” exclusion rule here, with only a lower bound on the gaps, introduces interesting novel features, and a detailed notion of Kac’s chaos is required to derive an evolution equation for the evolution of rescaled empirical measures for the process, as we show here.

This is joint work with Bernt Wennberg.

**João Gouveia **(jgouveia@mat.uc.pt)

### Applications of slack matrices: a biased survey

A slack matrix is a natural matrix associated to a polytope, that combines the descriptions by facets and by vertices into a single object. This object was introduced in 1991 by Yannakakis, that used it to characterize the extension complexity of a polytope: a measure of how amenable the polytope is to be represented via a lifted linear program. In the last 10 years, these object have seen renewed interest not only in terms of complexity measures but also as a way of characterizing algebraically some combinatorial properties of polytopes and polyhedral cones. In this talk we will give a very brief bird's eye view of the topic highlighting some recent developments and ongoing work.

**Jorge Milhazes de Freitas **(jmfreita@fc.up.pt)

### Enriched functional limit theorems for chaotic dynamics and heavy tailed observables

We consider stochastic processes arising from chaotic systems by evaluating an heavy tailed observable function along the orbits of the system. We prove the convergence of a normalised sum process to a Lévy process with excursions, designed to describe the oscillations observed during the clusters of extremal observations. The applications to specific systems include both hyperbolic and non-uniformly expanding systems.

**Hugo Tavares **(hugo.n.tavares@tecnico.ulisboa.pt)

### Free Boundary Problems with long-range interaction

In this talk we consider a class of variational shape optimisation problems for densities that repel each other at a certain distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional minimised in the class of functions that attain some H^1 boundary conditions, subject to the constraint that the supports of different densities are at a certain fixed distance. We show a connection with solutions to variational elliptic systems with nonlocal competing interactions, investigate the optimal regularity of the solutions (and prove uniform estimates with respect to the distance parameter), prove a free-boundary condition and derive some preliminary results characterising the free boundary. The talk is based in the following works:

[1] N. Soave, H. Tavares, S. Terracini, A. Zilio, Variational problems with long-range interaction, Arch. Rational Mech. Anal.228(2018), 743–772.

[2] Nicola Soave, Hugo Tavares and Alessandro Zilio, Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius, arXiv:2106.03661.

**Daniel Graça **(dgraca@ualg.pt)

### Characterizing complexity bounds with differential equations

Standard computational complexity classes, such as P, NP, PSPACE or EXPTIME are defined using discrete computational models. In this talk we will show how several of these classes, such as P, PSPACE or EXPTIME, can be characterized using a purely continuous model of computation consisting of polynomial ordinary differential equations and which is equivalent to Shannon's General Purpose Analog Computer. This talk describes joint work with Olivier Bournez, Riccardo Gozzi, and Amaury Pouly.

**Giosuè Muratore **(muratore.g.e@gmail.com)

### Gromov-Witten invariants and enumeration of curves

The introduction of Gromov-Witten invariants has led to great improvements in enumerative geometry. In this talk, we will introduce this beautiful subject. Also, we will see how GW invariants can be used to solve classical problems. If time permits, we will see some recent applications.

**Ivana Ljubić **(ivana.ljubic@essec.edu)

### Lower Bounds for Ramsey Numbers on Circulant Graphs: A Bilevel Optimization Approach

We address the problem of finding lower bounds for small Ramsey numbers R(m,n) using circulant graphs.

Our constructive approach is based on finding feasible colorings of circulant graphs using Integer Programming (IP) techniques.

First we show how to model the problem as a Stackelberg game and, using the tools of bilevel optimization, we transform it into a single-level IP problem with an exponential number of constraints. Using related results from graph theory, we provide strengthening valid inequalities for whose separation we develop a tailored branch-and-bound algorithm.

With our new method, we improve 17 best-known lower bounds for R(3,n) where n ranges between 26 and 46.

To the best of our knowledge, this is a first IP-based approach to tackle this very challenging combinatorial optimization problem.

Joint work with: F. Furini and P. San Segundo

**Teresa Faria **(teresa.faria@fc.ul.pt)

### Global dymanics for non-autonomous Nicholson systems

The original Nicholson’s blowflies equation was introduced by Gurney et al. (Nature, 1980), and since then many generalizations have been studied. Here, we consider a Nicholson system with patch structure and multiple discrete delays, where all the coefficients and delay functions are continuous and nonnegative. Under conditions that guarantee the permanence of the system, explicit uniform lower and upper bounds for all solutions are given, which are used to establish sufficient conditions for the global asymptotic stability of all solutions. For the case of periodic systems, a criterion for the global attractivity of a positive periodic solution is derived. The results apply to more general Nicholson systems with distributed delay.