22/11/19  Seminario  15:45  16:45  1101 D'Antoni  Guido Lido  Tor Vergata e Leiden  The Poincaré torsor and the quadratic Chabauty method.
Let C be a curve of genus g>1 defined over the rationals and let J be its Jacobian. Faltings's theorem states that C has only finitely many rational points, but in practice there is no general procedure to provably compute the set C(Q). When the rank of J(Q) is smaller than g, we can use Chabauty's method: embedding C in J the set C(Q) is a subset of the intersection of C(Qp) and the closure of J(Q) inside the padic manifold J(Qp); since this intersection is finite and computable up to finite precision we can use it to compute C(Q). Minhyong Kim has generalized this method inspecting the (Qpprounipotent etale) fundamental group of C and his ideas have been made effective in some new cases by Balakrishnan, Dogra, Muller, Tuitman and Vonk: their "quadratic Chabauty method" works when the rank of
J(Q) is smaller than g+s1 (with s the rank of the NeronSeveri group of J). In this seminar we describe a reinterpretation of the quadratic Chabauty method that does not need the fundamental group of C but uses only some geometry over the integers and the Gmtorsor associated to the Poincaré bundle over J. This work is in collaboration and under the supervision of Bas Exidhoven. 
22/11/19  Seminario  14:30  15:30  1101 D'Antoni  Ivan PENKOV  Jacobs University, Bremen  Some older and some recent results on the indvarieties G/P for the indgroups G = GL(∞) , O(∞) , Sp(∞)
About 15 years ago, Dimitrov and I worked out the flag realizations of the homogeneous indvarieties GL(∞)/P for arbitrary splitting parabolic indsubgroups P. An essential difference from the finitedimensional case is that we have to work with generalized flags, not with usual flags. Generalized flags are chains of subspaces which have more interesting linear orders. In the first part of the talk, I will recall our results with Dimitrov. In the second part, I will explain(without proof) two recent results. The first one (joint with A. Tikhomirov) is a purely algebraicgeometric construction of the indvarieties of generalized flags. The second one (joint with L.Fresse) answers the following question: on which multiple indvarieties of generalized flags, i.e. direct products of indvarieties of generalized flags, does GL(∞) act with finitely many orbits?
N.B.: this talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006 
19/11/19  Seminario  16:00  17:00  1101 D'Antoni  Yuri Kozitski  Lublin  Stochastic evolution of an interacting particle system in the
continuum: an analytic approach.
An infinite system of point particles placed in the continuum is
studied. Its constituents perform random jumps with mutual repulsion, and
this evolution is described by constructing path measures of certain kind
on the space of possible trajectories of the system. 
15/11/19  Seminario  09:30  12:30  1201 Dal Passo  Giovanni Peccati  University of Luxembourg  MINICORSO: The MalliavinStein Method
I will provide a selfcontained introduction to a collection
of probabilistic techniques developed in the last decade, focussing on
quantitative limit theorems for nonlinear functionals of Gaussian
fields, obtained by combining two techniques: (i) the Malliavin calculus
of variations, and (ii) Stein's method for probabilistic approximations.
I will develop in full detail at least one geometric application related
to the structure of level sets of Gaussian fields, and I will try to
point out a number of further directions of research, connected e.g. to
concentration estimates, entropic bounds, discrete geometric structures
and the analysis of Boolean functions. 
14/11/19  Seminario  14:00  17:00  1201 Dal Passo  Giovanni Peccati  University of Luxembourg  MINICORSO: The MalliavinStein Method
I will provide a selfcontained introduction to a collection
of probabilistic techniques developed in the last decade, focussing on
quantitative limit theorems for nonlinear functionals of Gaussian
fields, obtained by combining two techniques: (i) the Malliavin calculus
of variations, and (ii) Stein's method for probabilistic approximations.
I will develop in full detail at least one geometric application related
to the structure of level sets of Gaussian fields, and I will try to
point out a number of further directions of research, connected e.g. to
concentration estimates, entropic bounds, discrete geometric structures
and the analysis of Boolean functions. 
13/11/19  Seminario  16:00  17:00  1201 Dal Passo  Robin Hillier  Lancaster University  Roots of completely positive maps
We introduce the concept of completely positive roots of completely positive maps on operator algebras. We do this in different forms: as asymptotic roots, proper discrete roots and as continuous oneparameter semigroups of roots. We present structural and general existence and nonexistence results, some special examples in settings where we understand the situation better, and several challenging open problems. Our study is closely related to Elfving's embedding problem in classical probability and the divisibility problem of quantum channels. 
13/11/19  Seminario  15:00  16:00  1200 Biblioteca Storica  Ernesto Mistretta  Università di Padova  Holomorphic symmetric differentials, abelian variesties, complex tori, and parallelizable compact complex manifods
We review some recent constructions obtained with S. Urbinati on positivity, base loci, and Iitaka fibrations for higher rank vector bundles, then use some of these to obtain a biholomorphic and a bimeromorphic characterization of abelian varieties.
Then we see how to extend some of these results to complex tori and to compact complex parallelizable manifolds. 
12/11/19  Seminario  16:00  17:00  1101 D'Antoni  David Marti'Pete  Polish Academy of Science  On the connectivity of the escaping set in the punctured plane
A function $f$ is a transcendental selfmap of the punctured plane if $f:mathbb{C}^* omathbb{C}^*$ is a holomorphic function,
$mathbb{C}^*=mathbb{C}setminus{0}$, and both $0$ and $infty$ are essential singularities of $f$. For such maps, the escaping set $I(f)$ consists of the points whose orbit accumulates to a subset of ${0,infty}$.
We will look at the connectivity of $I(f)$ and show that either $I(f)$ is connected, or has infinitely many components.
We also proved that $I(f)cup {0,infty}$ is either connected, or has exactly two components,
one containing $0$ and the other $infty$. This gives a trichotomy regarding the connectivity of the sets $I(f)$ and
$I(f)cup {0,infty}$, and we will give examples of functions for which each case arises.
To give an example of a transcendental selfmap $f$ of $C^*$ for which $I(f)$ is connected, we adapted the socalled
spider's web structure due to Rippon and Stallard to the punctured plane. Finally, whereas Baker domains of transcendental entire
functions are simply connected, we showed that Baker domains can be doubly connected in $C^*$ by constructing the first such example. We also proved that if $f$ has a doubly connected Baker domain, then its closure contains both $0$ and $infty$, and hence
$I(f)cup{0,infty}$ is connected in this case. This is a joint work with Vasiliki Evdoridou (Open University) and Dave Sixsmith (University of Liverpool). 
12/11/19  Seminario  14:45  15:45  1201 Dal Passo  Gabriele Mondello  Sapienza, Università di Roma  Minimizing immersions of a hyperbolic surface in a hyperbolic 3manifold
Let (S,h) be a closed hyperbolic surface and M be a hyperbolic 3manifold. We consider incompressible maps from S to M that are critical points of an energy functional F which is homogeneous of degree 1. We prove the uniqueness of smooth minimizing maps from (S,h) to M in a given homotopy class. When (S,h) is fixed, smooth minimizing maps from (S,h) are described by a simple holomorphic datum on S: a complex selfadjoint Codazzi tensor of determinant 1. The space of admissible data is smooth and naturally equipped with a complex structure, for which the monodromy map taking a data to the monodromy representation of the image is holomorphic. Minimizing maps are in this way reminiscent of shearbend coordinates, with the complexification of F analoguous to the complex length. This is joint work with Francesco Bonsante and JeanMarc Schlenker. 
08/11/19  Seminario  15:45  16:45  1101 D'Antoni  Antonio Trusiani  Tor Vergata e Chalmers University  Complex MongeAmpère equations and metric geometry of potentials with prescribed singularities on compact Kähler manifolds.
Given (X,L) a compact Kähler manifold, the study of (degenerate) complex Monge Ampère equations arises in several questions such as the search of KählerEinstein metrics. On the set E1(X, L,T), consisting basically of potentials slightly more singular than T (the prescribed singularities), it is possible to define functionals whose critical points solve the equations. We will show that E1(X,L,T) has a natural complete metric topology associated to a distance d, and that it can be seen as limit of (E1(X,L,Tk),d) either
in a GromovHausdorff sense and in the category of metric spaces when the singularities increase. Moreover we will prove that, given a family A of nested prescribed singularities, X_A can be equipped of a natural complete distance dA which extends the distances d, and that the MongeAmpère map MA : X_A > Y_A becomes a homeomorphism, where Y_A is a certain set of positive Borel measures equipped with a strong topology. This helps to study the stability of solutions with prescribed singularities of degenerate complex MongeAmpère equations when the measures and/or the prescribed singularities
change. Some examples in the case (X,L) polarized projective manifold, with algebraic singularities will be given and, time permitting, we will see some applications to the (log )KählerEinstein setting.
