A real analytic function can always be continued holomorphically to some domain. However, the holomorphic continuations of definable functions in an o-minimal structure may not be definable. I will present joint work with P. Speissegger in which we study holomorphic continuations of functions definable in two o-minimal expansions of the real field. I will also discuss how to apply these results to the complex Gamma function and Riemann zeta function.[-]

This talk is focused on the Radon-Carleman Problem, dealing with computing and/or estimating the essential norm and/or the Fredholm radius of singular integral operators of double layer type associated with elliptic partial dfferential operators, on function spaces naturally intervening in the formulation of boundary value problems for the said operator in a given domain. The main goal is to monitor how the geometry of the domain affects the complexity of this type of study and to present a series of results in increasingly more irregular settings, culminating with that of uniformly rectifiable domains.
This is based on joint work with Dorina Mitrea and Marius Mitrea from Baylor University, which has recently appeared in volume V of our Geometric Harmonic Analysis research monograph series in Developments in Mathematics, Springer.[-]

The main question that we will investigate in this talk is: what does the spectrum of a quantum channel typically looks like? We will see that a wide class of random quantum channels generically exhibit a large spectral gap between their first and second largest eigenvalues. This is in close analogy with what is observed classically, i.e. for the spectral gap of transition matrices associated to random graphs. In both the classical and quantum settings, results of this kind are interesting because they provide examples of so-called expanders, i.e. dynamics that are converging fast to equilibrium despite their low connectivity. We will also present implications in terms of typical decay of correlations in 1D many-body quantum systems. If time allows, we will say a few words about ongoing investigations of the full spectral distribution of random quantum channels. This talk will be based on: arXiv:1906.11682 (with D. Perez-Garcia), arXiv:2302.07772 (with P. Youssef) and arXiv:2311.12368 (with P. Oliveira Santos and P. Youssef).[-]

In the ISem, we have encountered sectorial operators $A$ on a Hilbert space $H$. In Lecture 6 we have defined the exponential $\mathrm{e}^{-t A}$ for $t>0$ if the sectoriality angle of $A$ is smaller than $\frac{\pi}{2}$, the so-defined family $\left(\mathrm{e}^{-t A}\right)_{t>0}$ is called the semigroup associated with $A$. In Proposition 6.6 it was shown that the semigroup yields the solution to the abstract Cauchy problem$$\begin{aligned}\partial_{t} u(t)+A u(t) & =0, \quad(t>0) \\u(0+) & =u_{0}\end{aligned}$$by setting $u(t):=\mathrm{e}^{-t A} u_{0}$. In the same way, one can solve the equation$$\begin{align*}\partial_{t} u(t)+A u(t) & =f(t), \quad(t>0) \tag{2.1}\\u(0+) & =0\end{align*}$$by computing the convolution of $\mathrm{e}^{-t A}$ with $f$; that is,$$u(t):=\int_{0}^{t}e^{-(t-s)A}f(s)ds.$$One can now show that sectoriality of $A$ yields the maximal $L_{2}$-regularity of (2.1); that is, if $f\in L_{2}(0,\infty ;H)$ then the sodefined solution $u$ satisfies $u\in H^{1}(0,\infty ;H)$ or equivalently (due to (2.1)) $Au\in L_{2}(0,\infty ;H)$. It is the main object of this project to generalise this result to operators on Banach spaces $X$.
As we will see, sectoriality is not enough to ensure maximal regularity of (2.1). In fact, some stronger property is needed, namely $\mathscr{R}$-sectoriality, which in the Hilbert space case is equivalent to sectoriality. Moreover, the goal to prove such a result for all Banach spaces turns out to be too ambitious, so we will restrict our attention to so-called UMD spaces (sometimes also called $\mathscr{HT}$-spaces to reflect their relation to the Hilbert transform). This class of Banach spaces turns out to be suited for the application of techniques from Fourier analysis, which will be one of the main tools to prove our goal, which can be formulated as:

Maximal regularity of (2.1) in a UMD space is equivalent to $\mathscr{R}$-sectoriality of $A$.

The main source for this project will be [1], where our main result can be found in Theorem 4.4. Moreover, we will have a look at elliptic operators in divergence form, now on $L_{p}(\mathbb{R^{n}})$ and not on $L_{2}(\mathbb{R^{n}})$, and study the $\mathscr{R}$-sectoriality of those operators. If time permits, we can continue the study of elliptic operators, now on half-spaces and on domains.[-]

We investigate the mean-field limit of large networks of interacting biological neurons. The neurons are represented by the so-called integrate and fire models that follow the membrane potential of each neuron and captures individual spikes. However we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. We are able to extend the concept of extended graphons, introduced in Jabin-Poyato-Soler, by introducing a novel notion of discrete observables in the system. This is a joint work with D. Zhou.[-]

We investigate the mean-field limit of large networks of interacting biological neurons. The neurons are represented by the so-called integrate and fire models that follow the membrane potential of each neuron and captures individual spikes. However we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. We are able to extend the concept of extended graphons, introduced in Jabin-Poyato-Soler, by introducing a novel notion of discrete observables in the system. This is a joint work with D. Zhou.[-]

The mean curvature flow (MCF) describes the evolution of a hypersurface in time, where the velocity at each point is given by its mean curvature vector (i.e., the unit normal vector multiplied by the mean curvature). When initiated with a sphere in Rn, the MCF will shrink it homothetically to a point in finite time. In this talk, we introduce an adaptation of the mean convex MCF within the Heisenberg group setting. Our initial objective was to explore potential connections between this flow and the Heisenberg isoperimetric problem. Wewill discuss the existence and uniqueness of solutions and prove that the Pansu sphere does not evolve homothetically under the MCF. This work is based on joint research with Gaia Bombardieri and Mattia Fogagnolo.[-]

Simulation of conditioned diffusion processes is an essential tool in inference for stochastic processes, data imputation, generative modelling, and geometric statistics. Whilst simulating diffusion bridge processes is already difficult on Euclidean spaces, when considering diffusion processes on Riemannian manifolds the geometry brings in further complications. In even higher generality, advancing from Riemannian to sub-Riemannian geometries introduces hypoellipticity, and the possibility of finding appropriate explicit approximations for the score, the logarithmic gradient of the density, of the diffusion process is removed. We handle these challenges and construct a method for bridge simulation on sub-Riemannian manifolds by demonstrating how recent progress in machine learning can be modified to allow for training of score approximators on sub-Riemannian manifolds. Since gradients dependent on the horizontal distribution, we generalise the usual notion of denoising loss to work with non-holonomic frames using a stochastic Taylor expansion, and we demonstrate the resulting scheme both explicitly on the Heisenberg group and more generally using adapted coordinates. Joint work with Erlend Grong (Bergen) and Stefan Sommer (Copenhagen).[-]

In this talk we will don't speak about Joseph-Louis Lagrange (1736-1813) but about Lagrange's reception at the nineteenth Century. "Who read Lagrange at this Times?", "Why and How?", "What does it mean being a mathematician or doing mathematics at this Century" are some of the questions of our conference. We will give some elements of answers and the case Lagrange will be a pretext in order to explain what are doing historians of mathematics: searching archives and – thanks to a methodology – trying to understand, read and write the Past.
Lagrange - mathematical press - complete works - bibliographic index of mathematical sciences (1894-1912) - Liouville - Boussinesq - Terquem[-]

I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close together (super-exponentially in their lengths). Applications include resolution of some conjectures of Furstenberg on the dimension of sumsets and, together with work of Shmerkin, progress on the absolute continuity of Bernoulli convolutions. The main new ingredient is a statement in additive combinatorics concerning the structure of measures whose entropy does not grow very much under convolution. If time permits I will discuss the analogous results in higher dimensions.[-]