In everyday language, this talk studies the question about the optimal shape and location of a thermometer of a given volume to reconstruct the temperature distribution in an entire room. For random initial conditions, this problem was considered by Privat, Trelat and Zuazua (ARMA, 2015), and for short times we remove both the randomness and geometric assumptions in their article. Analytically, we obtain quantitative estimates for the well-posedness of an inverse problem, in which one determines the solution in the whole domain from its restriction to a subset of given volume. Using a new decomposition of $L^2(\Rd)$ into heat packets from microlocal analysis, we conclude that there exists a unique optimal such subset, that it is semi-analytic and can be approximated numerically by solving a sequence of finite-dimensional optimization problems. (joint with Alden Waters)
 

I will report some recent analytical results on microstructures in low-hysteresis shape memory alloys. The modelling assumption is that the width of the thermal hysteresis is closely related to the minimal energy that is necessary to build a martensitic nucleus in an austenitic matrix. This energy barrier is typically modeled by (singularly perturbed) nonconvex elasticity functionals. In this talk, I will discuss recent results on the resulting variational problems, including stress-free inclusions and microstructures in the case of almost compatible phases. This talk is partly based on joint works with S. Conti, J. Diermeier, M. Klar, and D. Melching.

Networks of interacting oscillators give rise to collective dynamics such as localized frequency synchrony. In networks of neuronal oscillators, for example, the location of frequency synchrony could encode information. We discuss some recent persistence results for certain dynamically invariant sets called weak chimeras, which show localized frequency synchrony of oscillators. We then explore how the network structure and interaction allows for dynamic switching of the spatial location of frequency synchrony: these dynamics are induced by stable heteroclinic connections between weak chimeras. Part of this work is joined with Peter Ashwin (Exeter).

Transseries arise naturally when solving differential equations around essential singularities. Just like most Taylor series are not convergent, most transseries do not converge to real functions, even when using advanced summation techniques.

On the other hand, we can show that all classical transseries induce analytic functions on the surreal line. In fact, this holds for an even larger (proper) class of series which we call "omega-series".

Omega-series can be composed and differentiated, like LE-series, and they form a differential subfield of surreal numbers equipped with the simplest derivation. This raises once again the question whether all surreal numbers can be also interpreted as functions. Unfortunately, it turns out that the simplest derivation is in fact incompatible with this goal.

This is joint work with A. Berarducci.

Globally Valued Fields, studied jointly with E. Hrushovski, are a formalism for fields in which the sum formula for valuations holds, such as number fields or function fields of curves. They form an elementary class (in continuous first order logic), and model-theoretic questions regarding this class give rise to difficult yet fascinating geometric questions.
I intend to present « Lyon school » approach to studying GVFs. This consists of reducing as much as possible to local considerations, among other things via the "fullness" axiom.
 

I will review the notion of classifying topos of a first-order (geometric) theory and explain the central role enjoyed by theories of presheaf type (i.e. classified by a presheaf topos) in the context of the topos-theoretic investigation of the model theory of geometric theories. After presenting a few main results and characterizations for theories of presheaf type, I will illustrate the generality of the point of view provided by this class of theories by discussing a topos-theoretic framework unifying and generalizing Fraissé’s construction in model theory and topological Galois theory and leading to an approach to the problem of the independence from l of l-adic cohomology.

The Constantin-Lax-Majda model is a 1d system which shares certain features (related to vortex stretching) with the 3d Euler equation. The model is explicitly solvable and exhibits finite-time blow-up for an open subset of smooth initial data. In 1990s De Gregorio suggested adding a transport term to the system, which is analogous to the transport term in the Euler equation. It turns out the transport term has some regularizing effects, which we will discuss in the lecture.

I will  address the study of decay rates of solutions to dissipative equations. The characterization of these rates will first be given for a wide class of linear systems by the decay character, which is a number associated to the initial datum that describes the behavior of the datum near the origin in frequency space. The understanding of the behavior of the linear  combined with the decay character and the Fourier Splitting method is then used to obtain some  upper and lower bounds for decay of solutions to appropriate dissipative nonlinear equations, both in the incompressible and compressible case.