cvgmt Seminarshttp://cvgmt.sns.it/seminars/en-usMon, 25 Mar 2019 16:30:33 +0000Intrinsic regular surfaces of low co-dimension in Heisenberg groupshttp://cvgmt.sns.it/seminar/682/2019-04-03: F. Corni.<p>In Heisenberg groups, and, more in general, in Carnot groups, equipped with their Carnot- Carathodory metric, the analogous of regular (Euclidean) surfaces of low co-dimension k can be considered G-regular surfaces (H-regular if G = Hn), i.e. level sets of continuously Pansu- differentiable functions f : G −→ Rk whose differential is subjective. If it is possible to split the group G in the product of two suitable homogeneous complementary subgroups M and H, a G-regular surfaces can be locally seen as an uniformly intrinsic differentiable graphs, defined by a unique continuous function φ acting between M and H. Moreover, it turns out that any one co-dimensional H-regular surface locally defines an implicit function φ, which is of class C1 with respect to a suitable non linear vector field ∇φ expressed in terms of the function φ itself.We extend some of these results characterizing uniformly intrinsic differentiable functions φ acting between two complementary subgroups with target space horizontal of dimension k, with 1 ≤ k ≤ n, in terms of the Euclidean regularity of its components with respect to a family of non linear vector fields {∇φj }j=1,...,k. Eventually, we show how the area of the intrinsic graph of φ can be computed through the component of the matrix identifying the intrinsic differential of φ.</p>http://cvgmt.sns.it/seminar/682/Optimal transport planning with a non linear costhttp://cvgmt.sns.it/seminar/683/2019-04-03: <a href="/person/52/">G. Bouchitté</a>.<p>In optimal mass transport theory, many problems can be written in the Monge-Kantorovich form$ \inf\{ \int_{X\times Y} c(x,y) \, d\gamma \ :\ \gamma\in \Pi(\mu,\nu)\}\ \quad (1)$where $\mu,\nu$ are given probability measures on $X,Y$ and $c:X\times Y \to [0,+\infty[$ is a cost function. Here the competitors are probability measures $\gamma$ on $X\times Y$with marginals $\mu$ and $\nu$ respectively (transport plans).Let us recall that if an optimal transport plan $\gamma \in \Pi(\mu, \nu)$is carried by the graph of a map $T:X\to Y$ i.e. if $ <\gamma, \varphi(x,y)> = \int_X \varphi(x,Tx)\, d\mu \quad,\quad T^\sharp \mu= \nu\ ,$then $T$ solves the original Monge problem:\ $ \inf\{ \int_X c(x,Tx) \, d\mu\ :\ T^\sharp \mu= \nu \}.$</p><p>\bigskipHere we are interested in a different case. Indeedin some applications to economy or in probability theory, it can be interesting to favour optimal plans whichare non associated to a single valued transport map $T(x)$. The idea is then to consider, instead of $T(x)$, the family of conditional probabilities $\gamma^x$ such that$ <\gamma, \varphi(x,y)> = \int_X (\int_X \varphi(x,y) d\gamma^x(y))\, d\mu \ ,$and to incorporate in problem $(1)$ an additional cost over $\gamma^x$ as follows$ \inf \left\{ \int_{X\times X} c(x,y) \, d\gamma + \int_X H(x, \gamma^x) \, d\mu\ :\ \gamma\in \Pi(\mu,\nu)\right\}\ ,\quad (2) $being $H:(x,p) \in X\times \mathcal{P}(X) \to [0,+\infty]$ a suitable non linear function.</p><p>In this talk I will describe some results concerning problem $(2)$ (existence, duality principle, optimality conditions)and focus on specific examples where $X=Y$ and $X$ is a convex compact subset of $\Rbb^d$.We will consider in particular the case where $H(x, p)= - \text{var} (p)$ or where $H(x,\cdot)$ is the indicator ofa constraint on the barycenter of $p$ (martingale transport).</p><p>This is from a joined work with Thierry Champion and J.J. Alibert.</p>http://cvgmt.sns.it/seminar/683/The Bernstein problem for higher codimensionhttp://cvgmt.sns.it/seminar/681/2019-05-23: J. Jost.<p>Time and room to be confirmed</p>http://cvgmt.sns.it/seminar/681/