Organizers: Giovanni Catino and Fabio Cipriani

**Luigi Vezzoni**, Università degli Studi di Torino,

*The Quantitative Alexandrov Theorem in Space forms*, Tuesday, November 27, 2018, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
The talk focuses on a recent generalization of a classical result of Alexandrov. The celebrated Alexandrov's Soap Bubble Theorem states that the spheres are the only closed (i.e. compact and without boundary) constant mean curvature hypersurfaces embedded in the Euclidean space. The theorem has been generalized to the hyperbolic space and to the hemisphere and to a large class of curvature operators. The main result of the talk is a quantitative version of Alexandrov's theorem which I've obtained in collaboration with Giulio Ciraolo and Alberto Roncoroni by using a quantitative study of the method of the moving planes. The theorem implies a new pinching Theorem for hypersurfaces in space forms.
**Alessandro Zilio**, Université Paris Diderot,

*Predator-prey model with competition, the emergence of territoriality*, Tuesday, October 30, 2018, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
I will present a series of works in collaboration with Henri Berestycki (PSL), dealing with systems of predators interacting with a single prey. The system is linked to the Lotka-Volterra model of interaction with diffusion, but unlike more classic works, we are interested in studying the case where competition between predators is very strong: in this context, the original domain is partitioned in different sub-territories occupied by different predators. The question that we ask is under which conditions the predators segregate in packs and whether there is a benefit to the hostility between the packs. More specifically, we study the stationary states of the system, the stability of the solutions and the bifurcation diagram, and the asymptotic properties of the system when the intensity of the competition becomes infinite.
**Shmuel Zamir**, The Hebrew University,

*Strategic Use of Seller Information in Private-Value First-Price Auctions*, Monday, October 22, 2018, time 11:00, Politecnico di Milano, Dipartimento di Matematica, Sala del Consiglio 7° piano

**Abstract:****Abstract:**
In the framework of a private-value-first-price auction, we consider the seller as a player in a game with the buyers in which he has private information about their realized valuations. We ask whether the seller can benefit by using his private information strategically. We find that in fact, depending upon his information, set of signals, and commitment power the seller may indeed increase his revenue by strategic transmission of his information. For example, in the case of two buyers with values distributed independently and uniformly on [0,1], a seller informed of the private values of the buyers, can achieve a revenue close to 1/2 by sending verifiable messages (compared to 1/3 in the standard auction), and this is the largest revenue that can be obtained with any signalling strategy.
**Lorenzo Toniazzi**, University of Warwick,

*Caputo Evolution Equations with time-nonlocal initial condition*, Tuesday, October 09, 2018, time 15:15, Aula Seminari 3° piano

**Abstract:****Abstract:**
Consider the Caputo evolution equation (EE) $\partial_t^\beta u =\Delta u$ with initial condition $\phi$ on $\{0\}\times\mathbb R^d$, $\beta\in(0,1)$. As it is well known, the solution reads $u(t,x)=\mathbf E_x[\phi(B_{E_t})]$. Here $B_t$ is a Brownian motion and the independent time-change $E_t$ is an inverse $\beta$-stable subordinator. The fractional kinetic $B_{E_t}$ is a popular model for subdiffusion \cite{Meerschaert2012}, with remarkable universality properties \cite{BC11,Hai18}.\\
We substitute the Caputo fractional derivative $\partial_t^\beta$ with the Marchaud derivative. This results in a natural extension of the Caputo EE featuring a \emph{time-nonlocal initial condition} $\phi$ on $(-\infty,0]\times\mathbb R^d$. We derive the new stochastic representation for the solution, namely $u(t,x)=\mathbf E_x[\phi(-W_t,B_{E_t})]$. This stochastic representation has a pleasing interpretation due to the non-obvious presence of $W_t$, elucidating the notion of time-nonlocal initial conditions. Here $W_t$ denotes the waiting/trapping time of the fractional kinetic $B_{E_t}$. We discuss classical-wellposedness \cite{T18}, and time permitting weak-wellposedness \cite{DYZ17,DTZ18} for the respective extensions of Caputo-type EEs (such as in \cite{chen,HKT17}).
Bibliography:
Barlow, \u Cern\'y (2011). Probability theory and related fields, 149.3-4: 639-673.
Chen, Kim, Kumagai, Wang (2017). arXiv:1708.05863.
Du, Toniazzi, Zhou (2018). Preprint. Submitted in Sept. 2018.
Du, Yang, Zhou (2017). Discrete and continuous dynamical systems series B, Vol 22, n. 2.
Hairer, Iyer, Koralov, Novikov, Pajor-Gyulai (2018). The Annals of Probability, 46(2), 897-955.
Hern\'andez-Hern\'andez, Kolokoltsov, Toniazzi (2017). Chaos, Solitons \& Fractals, 102, 184-196.
Meerschaert, Sikorskii (2012). De Gruyter Studies in Mathematics, Book 43.
Toniazzi (2018). To appear in: Journal of Mathematical Analysis and Applications. arXiv:1805.02464.
**Luca Ratti**, Politecnico di Milano,

*An inverse boundary value problem arising from cardiac electrophisiology*, Tuesday, September 18, 2018, time 15:15, Politecnico di Milano, Dipartimento di Matematica, Aula Seminari 3° Piano

**Abstract:****Abstract:**
The cardiac electrical activity can be comprehensively described throughout the monodomain model, consisting of a semilinear parabolic equation coupled with a nonlinear ordinary differential equation.
In my talk, I will introduce the inverse problem of identifying conductivity inhomogeneities in the monodomain system, taking advantage of data acquired on the boundary of the domain. Due to the complexity of the task, I will first tackle the stationary counterpart of the problem, regarding which it is possible to formulate well-posedness results both for the forward and for the inverse problem, and to rigorously introduce reconstruction procedures. Similar results are then generalized to the full complexity of the original model.
Throughout the presentation, I will focus on the problem of localizing small size inhomogeneities, as well as arbitrarily large ones, by means of the constraint optimization of a suitable misfit functional. The first task is achieved by relying on an asymptotic expansion of the boundary voltage with respect to the size of the inclusion, and employing tools from the topological optimization theory. The second issue is analyzed by means of the regularization theory of inverse problems and introducing a convenient relaxation of the optimization problem. The theoretical results are supported by numerical experiments, which are exhaustively reported.
This is a joint work with Elena Beretta, Cristina Cerutti, Cecilia Cavaterra, Andrea Manzoni and Marco Verani.
**Zhe Zhou**, Chinese Academy of Sciences, Beijing,

*Rotation number of the linear Schrödinger equation with discontinuous almost periodic potentials*, Thursday, September 13, 2018, time 15:00, Aula Seminari 3° piano

**Abstract:****Abstract:**
In this talk, based on the celebrated paper [R. Johnson and J. Moser, Comm. Math. Phys., 1982], we will recover the rotation numbers of the Schrödinger equation. The essential elements in the proof are positive homogeneity and almost periodicity. From this point of view, the concept of rotation numbers may be introduced in the case of discontinuous potentials. Moreover, we will show the optimal estimate of rotation numbers in such case.