Alessandro Carlotto

Alessandro Carlotto

Department of Mathematics
Stanford University
450 Serra Mall, Building 380
Stanford, CA 94305

Office: 380J
E-mail Address: carlotto [at] stanford [dot] edu



About
I am a Ph.D. candidate at Stanford University where I work under the direction of Richard Schoen.
During the second half of 2013, I was also a member of the Mathematical Sciences Research Institute (MSRI) in Berkeley for the semester dedicated to Mathematical General Relativity.
Starting on September 1st 2014, I will join the faculty of Imperial College in London.
Starting on September 1st 2015, I will join the faculty of ETH in Zurich.

From 2005 to 2010 I was a student at Scuola Normale Superiore where I obtained my Laurea in Matematica in 2008 defending a thesis entitled "A mass-transportation approach to some functional inequalities" (Advisor: Luigi Ambrosio). In 2010 I obtained my Laurea Specialistica and my Diploma di Licenza with the thesis "A class of existence results for the singular Liouville equation" developed under the direction of Andrea Malchiodi.

In 2011 I have been awarded the Franco Tricerri Prize during the XIX plenary congress of the Unione Matematica Italiana.

I have had visiting positions and have delivered invited lectures at a number of institutions, including Princeton University, ETH, Columbia University, Imperial College, University of Warwick, International Center for Theoretical Physics (ICTP).

I am a member of the American Mathematical Society, European Mathematical Society and Unione Matematica Italiana.

Research
I work on Geometric Variational Problems mostly arising in the contexts of General Relativity, Conformal Geometry, Optimal Transportation and String Theory. Some more specific topics are: complete minimal surfaces in asymptotically flat spaces, marginally outer-trapped surfaces and large-scale structure of initial data sets for the Einstein field equations, scalar curvature flexibility phenomena and gluing constructions, curvature prescription problems on singular spaces, optimal embeddings and conformal Willmore functional, gradient flows in metric spaces (with special emphasis on Wasserstein spaces).

In recent joint work with Otis Chodosh and Yanir Rubinstein, we adapted the Adams-Simon theory to the parabolic setting and made use of a suitable Lojasiewicz-Simon inequality in order to give an essentially complete description of the rate of convergence of the Yamabe flow. We constructed flows that converge exactly at a polynomial rate: those seem to be the first examples of slowly converging geometric flows.

Below are the links to some of my scientific works, but please notice that the arXiv version might slightly differ from the published version.

  • Rigidity and flexibility phenomena in General Relativity, (PhD thesis, in preparation)
  • Localizing solutions of the Einstein constraint equations, (with Richard Schoen, in preparation)
  • A new generation of uniformization theorems, (with Andrea Malchiodi, in preparation)
  • Rigidity of stable marginally outer trapped surfaces in initial data sets, pdf (preprint arXiv 1404.0358)
  • Slowly converging Yamabe flows, pdf (with Otis Chodosh and Yanir Rubinstein, preprint arXiv 1401.3738)
  • Rigidity of stable minimal hypersurfaces in asymptotically flat spaces, pdf (preprint arXiv 1403.6459)
  • On the solvability of singular Liouville equations on compact surfaces of arbitrary genus, pdf (Transactions of the American Mathematical Society , volume 366 (2014), issue 3, 1237-1256)
  • Existence of generalized totally umbilic 2-spheres in perturbed 3-spheres, pdf (with Andrea Mondino, I.M.R.N., to appear)
  • Weighted barycentric sets and singular Liouville equations on compact surfaces , pdf (with Andrea Malchiodi, Journal of Functional Analysis, volume 262 (2012), issue 2, 409-450)
  • Lecture notes on elliptic partial differential equations, pdf (with Luigi Ambrosio and Annalisa Massaccesi, preliminary version)
  • A class of existence results for the singular Liouville equation, pdf (with Andrea Malchiodi, Comptes Rendus de l'Académie des Sciences, volume 349 (2011), issues 3-4, 161-166)
  • A mass-transportation approach to some functional inequalities, pdf (tesi di Laurea in Matematica, defended on September 26th 2008)

  • Teaching
    During the summer 2013 I have been the assistant of Brian White for the graduate class Introduction to minimal surface theory held at the IAS (PCMI). The lecture notes can be found here here, while the problem sets I assigned are here set1, set2, set3. Some of these problems are very basic, while others are motivated by relatively recent research results.

    During the winter quarter 2012-2013 I was the teaching assistant of Yasha Eliashberg for the honors class Math 52H, Multilinear Algebra, Differential Forms and Stokes' Theorem.

    During the spring quarter 2013-2014 I am going to be the teaching assistant of Yasha Eliashberg for the honors class Math 53H, which will be devoted to a rigorous introduction to Ordinary Differential Equations and Dynamical Systems.

    Other
    During this academic year, I am organizing (jointly with Otis Chodosh and Christopher Henderson) the SAGS Elliptic Regularity seminar. A schedule can be found here.

    During the academic year 2011-2012 I organized (jointly with Khoa Lu Nguyen) the Faculty Area Research Seminar at the Department of Mathematics of Stanford University.



    Last updated on April 2nd 2014.