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<h1>cvgmt weekly bulletin</h1>
<p><i>Weekly bulletin for <a href="http://cvgmt.sns.it">http://cvgmt.sns.it</a></i></p>
<h2>Summary</h2>
<p><b>Seminars by:</b> <a href='/seminar/682/'>Corni</a>, <a href='/seminar/683/'>Bouchitté</a></p>
<p><b>New papers by:</b> <a href='/person/1259/' >Trevisan</a>, <a href='/person/3/' >Ambrosio</a>, <a href='/person/2701/' >Glaudo</a>, <a href='/person/39/' >Spector</a></p>
<p><b>Modified papers by:</b> <a >Schikorra</a>, <a href='/person/2053/'>Ponce</a>, <a href='/person/39/'>Spector</a>, <a >Spector</a>, <a href='/person/2701/'>Glaudo</a>, <a >Shieh</a>, <a href='/person/3/'>Ambrosio</a>, <a >Rodiac</a>, <a >Valdinoci</a>, <a href='/person/1176/'>Cinti</a>, <a href='/person/135/'>Catino</a>, <a >Miraglio</a></p>
<h2>Events next week</h2>
<ul>
<li><p>
<b><a href='/event/515/'>Ohio River Analysis Meeting 2019</a></b><br />
Sat 30 March 2019 - Sun 31 March 2019<br />
University of Cincinnati, Cincinnati, OH, USA<br /></p></li>
<li><p>
<b><a href='/event/521/'>Optimal Transport and Geometric Analysis</a></b><br />
Mon 1 April 2019 - Fri 5 April 2019<br />
Venice<br /></p></li>
<li><p>
<b><a href='/event/533/'>The 2nd ITS-CUNY Symposium on Nonlinear Problems in Geometry</a></b><br />
Wed 3 April 2019 - Thu 4 April 2019<br />
The City University of New York<br /></p></li>
</ul>
<h2>Seminars next week</h2>
<h3>Wed 3 April 2019</h3><ul>
<li><p>
Francesca Corni: <b><a href='/seminar/682/'>Intrinsic regular surfaces of low co-dimension in Heisenberg groups</a></b><br />
Sala Seminari (Dipartimento di Matematica di Pisa), 17:00<br /> </p>
<div style="font-size: smaller;"><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>
</div> </li>
<li><p>
Guy Bouchitté (Université du Sud Toulon-Var): <b><a href='/seminar/683/'>Optimal transport planning with a non linear cost</a></b><br />
Sala Seminari (Dipartimento di Matematica di Pisa), 18:00<br /> </p>
<div style="font-size: smaller;"><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>
\bigskip
Here we are interested in a different case. Indeed
in some applications to economy or in probability theory, it can be interesting to favour optimal plans which
are 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 of
a constraint on the barycenter of $p$ (martingale transport).
</p>
<p>This is from a joined work with Thierry Champion and J.J. Alibert.</p>
</div> </li>
</ul>
<h2>New Papers</h2>
<p><b> Spector:</b> <a href='/paper/4264/'>New Directions in Harmonic Analysis on $L^1$</a></p>
<p><b> Ambrosio, Glaudo, Trevisan:</b> <a href='/paper/4265/'>On the optimal map in the 2-dimensional random matching problem</a></p>
<h2>Modified Papers</h2>
<p><b> Schikorra, Shieh, Spector:</b> <a href='/paper/3145/'>Regularity for a fractional $p$-Laplace equation</a></p>
<p><b> Shieh, Spector:</b> <a href='/paper/3237/'>On a new class of fractional partial differential equations II</a></p>
<p><b> Cinti, Miraglio, Valdinoci:</b> <a href='/paper/3598/'>One-dimensional symmetry for the solutions of a three-dimensional water wave problem</a></p>
<p><b> Glaudo:</b> <a href='/paper/3750/'>On the c-concavity with Respect to the Quadratic Cost on a Manifold</a></p>
<p><b> Spector, Spector:</b> <a href='/paper/3851/'>Uniqueness of Equilibrium with Sufficiently Small Strains in Finite Elasticity</a></p>
<p><b> Catino:</b> <a href='/paper/3948/'>Rigidity of positively curved shrinking Ricci solitons in dimension four</a></p>
<p><b> Ambrosio, Ponce, Rodiac:</b> <a href='/paper/4066/'>Critical weak-$L^{p}$ differentiability of singular integrals</a></p>
<h2>Open Positions</h2>
<p><a href="/position/286/">21 PhD fellowships in Paris 2019/20 - Fondation Science Mathémathiques Paris</a> (deadline: Mon 1 April 2019)</p>
<p><a href="/position/397/">5-9 Postdoc Positions in Jyväskylä</a> (deadline: Mon 1 April 2019)</p>
<p><a href="/position/398/">PhD Position at the University of Sussex</a> (deadline: Fri 31 May 2019)</p>
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