Math Calendar
Speaker: Francesco Caravenna (U Milano-Bicocca)
Title: Noise Sensitivity and Critical 2D Directed Polymers
Abstract: We investigate the concept of noise sensitivity for functionals of independent random variables, which refers to the property that a small perturbation in the underlying randomness leads to an asymptotically independent functional. We extend classical noise sensitivity criteria beyond the Boolean setting, deriving quantitative estimates with optimal rates.
As an application, we consider the model of directed polymers in random environments, which describes a random walk interacting with a random medium. In the critical dimension d=2 and under a critical rescaling of the noise strength, the partition function of the model is known to converge to a universal limit, the Stochastic Heat Flow. We show that in this regime, the partition function exhibits noise sensitivity.
(Based on joint work with Anna Donadini)
Location and time: JKH 2-3, Room 220 at 11-12.45
Afternoon session:
Speaker: Zakhar Kabluchko (U Muenster)
Title: Beta-type random polytopes and related objects in stochastic geometry
Abstract: A random point in a d-dimensional unit ball is said to have a beta distribution if its density is proportional to (1-|x|^2)^{beta}, where beta>-1 is a parameter. For beta=0 we recover the uniform distribution on the unit ball, the limiting case beta -> -1 corresponds to the uniform distribution on the unit sphere, while the case beta -> infty corresponds to the standard Gaussian distribution. Let X_1,..., X_n be independent random points in the d-dimensional unit ball such that X_i follows a beta distribution with parameter beta_i. Their convex hull [X_1,...,X_n] is called a beta polytope (with parameters n, d, beta_1,...,beta_n). We shall review results on the expected number of k-dimensional faces, expected volume (and other geometric functionals) of beta polytopes and two closely related classes of polytopes called beta' and the beta* polytopes. Several objects in stochastic geometry such as the typical cell of the Poisson-Voronoi tessellation or the zero cell of the homogeneous Poisson hyperplane tessellation (in Euclidean space or on the sphere) are related to beta' polytopes, while their analogues in the hyperbolic space are related to beta* polytopes. This allows for explicit computations for these objects.
Location and time: JKH 2-3, Room 220, 14.15-16.00
I will report on proven parts of the conjecture, to appear soon in my doctoral thesis. Special attention is drawn to a semi-orthogonal sequence obtained from Fourier-Mukai transforms which embed the derived category of X into the one of its Hilbert scheme of three points. The proof of fully-faithfulness of these Fourier-Mukai transforms leads back to the geometry of Hilbert schemes of points, in particular to normal bundle computations on Grassmannian bundles.
| Using (almost) toric fibrations and their visible Lagrangians we can construct many novel and interesting examples of Lagrangian submanifolds of symplectic 4 manifolds. Naturally, one can ask whether visible Lagrangians are all the Lagrangians that exist, or, in other words, how faithful the pictures coming from almost toric fibrations are. I will answer this question for Klein bottles in (S2xS2,\omega_\lambda), i.e. the product of two spheres where the first factor has area 1 and the other factor has area \lambda. In particular, I will first construct a visible Lagrangian Klein bottle when \lambda<2. Then I will show that no Lagrangian Klein bottles exist otherwise. The key input for obstructing the existence of the Klein bottles is Luttinger surgery along with techniques of (compact) pseudoholomorpic curves and Seiberg-Witten theory. This is joint work with J. Evans. |
Fractional calculus is a powerful tool in mathematical modeling, that gets more attention of applied mathematicians and natural scientists since the past decades. Dynamical systems involving fractional order derivatives are able to incorporate the so-called ‘memory effects’, and due to the nonlocal nature of fractional differential operators they are usually used in modeling of the flows through porous media (e.g., the groundwater flows), sub- and super-diffusion processes etc. Additionally, a large choice of fractional derivatives and variations in their order gives more flexibility in comparison to the classical integer-order models. This broad range of applications motivates development of analytical and numerical techniques for analysis and approximation of solutions to the fractional initial and boundary values problems (IVPs and BVPs). However, the nonlinear nature of the studied systems and the nonlocality of their fractional operators lead to major challenges in the field. In my talk I will present some recent results on analysis and approximation of solutions of the nonlinear fractional BVPs with periodic boundary constraints. I will show how coupling between the BVP and an equivalent IVP can lead not only to existence and uniqueness results, but also to an explicit expression for their approximate solutions. Finally, I will highlight some behavioral differences between solutions of the periodic BVP of the integer and of the fractional orders.
Abstract: I will discuss recent work with Skorobogatov establishing the Hasse principle for a broad class of degree 4 del Pezzo surfaces, conditional on finiteness of Tate--Shafarevich groups of abelian surfaces. A corollary of this work is that the Hasse principle holds for smooth complete intersections of two quadrics in P^n for n\geq 5, conditional on the same conjecture. This was previously known by work of Wittenberg assuming both finiteness of Tate--Shafarevich groups of elliptic curves and Schinzel's hypothesis (H).
I will also discuss forthcoming work with Lyczak which, again under the Tate--Shafarevich conjecture, shows that the Brauer--Manin obstruction explains all failures of the Hasse principle for certain degree 4 del Pezzo surfaces about which nothing was known previously.
Title:
Extremes in dynamical systems: max-stable and max-semistable laws
Abstract:
Extreme value theory for chaotic, deterministic dynamical systems is a rapidly expanding area of research. Given a dynamical system and a real-valued observable defined on its state space, extreme value theory studies the limit probabilistic laws for asymptotically large values attained by the observable along orbits of the system. Under suitable mixing conditions the extreme value laws are the same as those for stochastic processes of i.i.d. random variables.
Max-stable laws typically arise for probability distributions with regularly varying tails. However, in the context of dynamical systems, where the underlying invariant measure can be irregular, max-semistable distributions also have a natural place in studying extremal behaviour. In this talk I will first discuss a family of autoregressive processes with marginal distributions resembling the Cantor function. The resulting extreme value law can be proven to be a max-semistable distribution. Alternatively, we can describe the autoregressive process in terms of an iterated map with an invariant measure. Further examples of extreme value laws in dynamical systems are discussed as well.