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Mathematical Analysis

Department or Institution involved

Geometrical Analysis and Partial Differential Equations

Topic description
Our research falls in Mathematical Analysis and  Partial Differential Equations, through the geometric theory of functions, singulars integrals and transport phenomena.
The project will aim at contributing significant progress in at least one of the following three subjects, which are part of the main interests of the research project of the our group.

1. Geometric Measure Theory and connection with Calderón-Zygmund theory. Understand better the relation between rectifiability and harmonic measure.
2. Study boundary regularity of vortex patches for various transport equations. Determine the geometric structure of the flow associated to a non-smooth vector field, and analyze its properties from the point of view of Geometric Function Theory. Explore potential applications to fluid mechanics, with especial attention to planar, inviscid, non-perfect fluids.
3. Probabilistic behavior in Geometric Function Theory. Study the role of Central Limit Theorem, Function Lusin Area and Function Quadratic Marcinkiewicz in this setting.

Project supervisor & hosting group

Dr.  Xavier Tolsa

The hosting group is widely recognized for its achievements in the area of mathematical analysis and partial differential equations
The main relevant projects associated with data science are:
ERC Advanced Grant, FP7-320501. Study of several problems in connection with the David-Semmes's conjecture about Riesz transforms and rectifiability;  relations with the behavior of the harmonic measure and the elliptical measure, and with the quasiconformal mappings.
FP7-PEOPLE-2013-ITN: 607643. Non-perfect fluids and connections to Geometric Function Theory. Describe what is the smoothness of the flow map arising from a vector field with borderline sobolev regularity.
MTM2013-44304-P. Study various issues on quadratic functions and rectifiability, and other problems of geometric analysis and possible relations with Partial Differential Equations and  problems with free boundary.
MTM2013-44699-P. Nonlinear transport equation and the dynamics of patch solutions. Give sufficient conditions on the initial domain such that the skeleton for the agregation equation is rectifiable. The agregation equation arises in many applications including materials science, granular flow biological swarms, vortex densities in superconductors and bacterial chemotaxis. One also considers, construction of new  V-states for the 2-D Euler equation without using bifurcation theory.
MTM2014-51824-P. Probabilistic behavior in Geometric Function Theory. Versions of the Central Limit Theorem are studied in two classical contexts that come from the influential works of A. Calderon, E. Stein, A. Zygmund, and others. The first context is the behavior of a harmonic function border in a half space in terms of the size of its function Lusin area. The second is the behavior of incremental ratios of a function defined in Euclidean space in terms of its function Quadratic Marcinkiewicz.
MTM2016-81703-ERC. Geometric and analytic methods in Partial Differential Equations.

Planned secondments
Our group has close collaborations with researchers at other institutions.
Depending on the precise topic of research, we propose some stays of one or two months in some of these centers: U. of Toulouse (P. Thomas, S. Petermichl), U. of Helsinki (K.Astala, E. Saksman, P. Mattila), U. of Edinburg (J. Azzam), U. of Birmingham (M.C. Reguera), ICMAT-Madrid (D. Córdoba, D. Faraco), Paris (P. Santambrogio) and TSS- Transport Simulation Systems (company at Barcelona).

Candidate’s profile  
A PhD in mathematics, preferably in the areas of mathematical analysis or in partial differential equations.

Basic background on Analysis and PDE’s is required. Any more specific background in the topics described in the section “Topic description” will be evaluated very positively.

If you are an eligible candidate interested in applying, please do contact pr.sphere@uab.cat to get you in contact with the Hosting Group.