description

The research in Analysis and Partial Differential Equations at the CRM covers a broad range of topics, from classical function theory in one and several complex variables to the study of Banach spaces and its operators. The interplay between singular operators and geometric function theory has been very succesful. On PDE’s the research is centered around reaction-diffusion and integro-differential equations (regularity and qualitative properties of solutions), population dynamics and biological evolution, as well as several wave problems in mathematical physics and mathematical modelling.

research lines

##### Analysis

The research group in Analysis has a long and consolidated trajectory, with very relevant results, as shown by the many publications in top journals.

The research lines cover different areas of Analysis as Harmonic Analysis, Geometric Function Theory in one complex variable, Geometric measure Theory and PDEs with special focus on the following lines:

- Singular integrals, square functions, and rectifiability.
- Geometric properties of harmonic and elliptic measures.
- Boundary behaviour of analytic functions and its connection to Stochastic Processes

##### Partial Differential Equations (PDEs)

The PDE’s group research deals with applications to Mathematical Biology. Two problems have received the most attention: on the one hand, the rigorous definition and computation of the basic reproduction number in continuously structured populations, as the spectral radius of the next generation operator, and a new method to do this by means of a sequence of models when the latter cannot be defined properly as a bounded operator; on the other hand, the equivalence between the usual formulation of Structured Population Dynamics based on PDEsfor the densities of individuals and the delay formulation based on Volterra type integral equations for the birth rates.

##### Mathematical Modelling and Numerical Analysis

Motivated by problems in different disciplines, mathematical modeling seeks to explain and understand phenomena in nature and technology by means of mathematical language. This is an interdisciplinary field, that uses mathematical concepts for the progress of other sciences, including biology, physics, engineering, business, economics and risk management… This is as well an interdisciplinary topic as tools from different areas in mathematics can be exploited, e.g. differential equations, statistics, data science, numerical analysis, discrete mathematics, algebra and geometry.

members

postdoctoral researchers

PhD Students

**Xavier Cabré**

PhD in Mathematics, adviser: Louis Nirenberg, Courant Institute, New York University, 1994. Kurt Friedrichs Prize, New York University, 1995. Member of the Institute for Advanced Study, Princeton, 1994-95. Habilitation à diriger des recherches, Université Pierre et Marie Curie-Paris VI, 1998. Harrington Faculty Fellow, The University of Texas at Austin, 2001-02. Tenure Associate Professor, The University of Texas at Austin, 2002-03. ICREA Research Professor since 2003 and Catedrático de Matemática Aplicada since 2008, at the Universitat Politècnica de Catalunya. Fellow of the American Mathematical Society, inaugural class, 2012. Editor of the Journal of the European Mathematical Society, 2014-2017. Plenary speaker at the 8th European Congress of Mathematics, 2021.

**Àngel Calsina**

Since 2005 I am professor in the Department of Mathematics of the UAB. Previously I was professor at the University of Girona, and prior to this, assistant professor and associate professor at the UAB. I have been IP of several research projects and I have been advisor of nine doctoral thesis.

**Carme Cascante**

I did my thesis at the Universitat Autònoma de Barcelona. I was a postdoctoral researcher at University of Wisconsin at Madison in 1988-1990 and spent research stays at University of Tsukuba and University of Missouri. Currently, I am professor at University of Barcelona.

I am a member of the editorial board of Butlletí de la Societat Catalana de Matemàtiques. Since 2019, I am the Director of the Barcelona Graduate School of Mathematics, BGSMath.

**Albert Clop**

I graduated in Mathematics in 2001 at Universitat de Barcelona. In 2006, I got a PhD in Mathematics at Universitat Autònoma de Barcelona, under the supervision of Dr. Joan Mateu and Dr. Joan Orobitg. Later I went through 2,5 years of postdoctoral training at University of Helsinki, University of Jyväskylä and Universidad Autónoma de Madrid.

In 2010 I became a “Ramón y Cajal” researcher.

**Gyula Csato**

Associate Professor

Universitat de Barcelona, Facultat de Matemàtiques i Informàtica, Gran Via de les Corts Catalanes, 585, 08007, Barcelona, Spain

+34934021631

and Centre de Recerca Matemàtica, Edifici C, Campus Bellaterra, 08193 Bellaterrra, Spain.

Research interests

General: partial differential equations, calculus of variations, differential geometry.

In particular: differential forms, first order boundary value problems, pullback equation, Jacobian equation, divergence equation, isoperimetric problems, rearrangements and symmetry, analysis on manifolds, Moser-Trudinger inequalities, non-local operators, fractional mean curvature

**Juan Jesús Donaire**

Ph. D., Universitat Autònoma de Barcelona (UAB), Award Date: 19 Nov 1995

Degree, Universitat Autònoma de Barcelona (UAB), Award Date: 1 Jan 1990

**Jordi Marzo**

I did my Ph.D. at the Universitat de Barcelona. I was a PostDoc from 2008 to 2010 at the NTNU-Norway. I’m Associate professor in the Universitat de Barcelona. My research interests are in mathematical analysis in a broad sense. Together with C. Beltrán, U. Etayo and J. Ortega-Cerdà we recently solved an important problem about well conditioned polynomials posed by M. Shub and S. Smale in 1993.

**Albert Mas Blesa**

PhD in Mathematics with European Mention under the direction of Mark Melnikov and Xavier Tolsa at Universitat Autònoma de Barcelona (2011). Postdoctoral research positions at Universidad del País Vasco (2011-2013, scientific collaboration with Luis Vega), at Universitat Autònoma de Barcelona (2014, scientific collaboration with Xavier Tolsa), and at Universitat Politècnica de Catalunya as a “Juan de la Cierva” researcher (2015-2016, scientific collaboration with Xavier Cabré and Joan Solà-Morales). Professor Lector (Assistant Professor) at Universitat de Barcelona (2017-2018). Professor Agregat (Associate Professor) at Universitat Politècnica de Catalunya (since 2019).

**Joan Mateu**

Associate Professor at UPC from 1990-1995. Associate Professor at UAB from 1995 until now. As a researcher I have proposed to solve problems that range from approximation questions for solutions of elliptic equations in different norms, problems on geometric measure theory related to the semiaditivity of analytic capacity, singular integrals, quasi-conformal mappings and in recent years I am mainly interested in problems related to the applications of harmonic analysis in fluid mechanics and in material sciences. In all these fileds I have obtained quite interesting results.

**Artur Nicolau**

**Joan Orobitg**

In 1990 I got the PhD in Mathematics, Universitat Autònoma de Barcelona, under the supervision of Joan Verdera. From 1990 to 1993 I worked at the Universitat Politècina de Catalunya.

Currently I’m Associate Professor at Universitat Autònoma de Barcelona.

Coordinator of the famous Barcelona Analysis Seminar (1995-2003).

**Joaquim Ortega**

I did my thesis in Mathematics at the Universitat Autònoma de Barcelona. I was a postdoc researcher in the University of Wisconsin at Madison in 1995 where I returned as a visiting professor during the course 2004/2005. I worked at the Universitat Autonòma de Barcelona and at the Universitat Politècnica de Catalunya. Currently, I am a professor at the University of Barcelona since 1997.

I have done several long research stays at the Mittag-Leffler Institute, Center for Advanced Studies at Oslo and at the Universities of Goteborg and Trondheim where I have regular collaborators and shorter visits at the Mathematical Research Institute of Oberwolfach and the Simons Center for Geometry and Physics. I am an invited lecturer at the European Congress of Mathematics in Berlin 2016. My main research interest is complex analysis in one and several variables. Particularly the study of the inhomogeneous Cauchy-Riemann equation to tackle problems as the size of the Bergman kernel or the description of zero sets, sampling an interpolating sequences. Other topics of my interest are Dirichlet series, from a point of view of function theory in the infinite-dimensional polydisk and lately random point processes and optimal configuration sets.

**Jordi Pau**

**Laura Prat**

Activitats de caràcter científic o professional anteriors a la situació actual

1. Becaria FPI (UAB) 1998-2001.

2. Ajudant LUC (UB) 2001-2003.

3. Ajudant LUC (UB) 2003-2006.

4. Fulbright Postdoctoral Scholar (UCLA) 2005.

5. Investigadora Juan de la Cierva (UAB) 2006-2009.

6. Professora Lectora (UAB) 2009-2015.

7. Professora Agregada Interina (UAB) 2015-2018.

8. Professora Agregada (UAB) 2018-..

Beques i ajuts personals

1. Beca predoctoral FPI. MEC. 1998-2001.

2. Fons per estada predoctoral de 3 mesos a Jyväskylä. MEC. 2000.

3. Estada postdoctoral de la CNRS a l’Institute Fourier (Grenoble). 3 mesos. 2004.

4. Beca postdoctoral Fulbright a UCLA. 2005.

5. Contracte Juan de la Cierva. MEC.UAB. 2006-2009.

Formació acadèmica

Llicenciada en Matemàtiques (UAB) 1997.

Certificat d’Aptitud Pedagògica (UAB) 1998.

Doctora en Matemàtiques (UAB) 2003.

**Martí Prats**

Currently associate professor (professor lector) at Universitat de Barcelona. I my Ph.D. in UAB under Xavier Tolsa’s tuition, on Calderón-Zygmund operators and Sobolev spaces on domains, with applications to quasiconformal mappings. After that I have been postdoc in UAM with Daniel Faraco and in Aalto Yliopisto with Kari Astala, working on the Calderón inverse problem with techiques of quasiconformal mappings. We could develop a characterization for the sets of conductivities where the inverse mapping is continuous. I also obtained a postdoc position and a JdC-I position in UAB with Xavier Tolsa to work on free boundary problems, reaching remarkable results on the thin one-phase free boundary problem and in the VMO two-phase problem for harmonic measure. Actually I am working in harmonic measure problems with Xavier Tolsa and our Ph.D. student Ignasi Guillén.

**Xavier Ros Oton**

Xavier is ICREA Research Professor and Catedràtic d’Universitat at UB since 2020. Previously, he has been Assistant Professor at Universität Zürich, as well as R. H. Bing Instructor at the University of Texas at Austin. He is a mathematician who works on Partial Differential Equations (PDE). Specifically, he studies the regularity of solutions to elliptic and parabolic PDE, and he is mostly known for his results on free boundary problems and integro-differential equations. He is the PI of an ERC Starting Grant (2019-2024), has received several awards for young mathematicians in Spain, as well as the Scientific Research Award from the Fundación Princesa de Girona in 2019. Furthermore, in 2021 he was awarded the Stampacchia Gold Medal, an international prize awarded every three years in recognition of outstanding contributions to the Calculus of Variations.

**Olli Saari**

I am a Ramon y Cajal research fellow at the UPC. Previously I have held post-doctoral positions at the Unviersity of Bonn (Germany), Aalto University (Finland) and the Mathematical Sciences Research Institute (the US). I obtained my doctoral degree from Aalto University in 2016 under the supervision of Juha Kinnunen.

**Susana Serna**

**Sergey Tikhonov**

I graduated from the Lomonosov Moscow State University in 1999 and obtained the PhD degree in Mathematics from MSU in 2003. From September 2012, I am an ICREA Research Professor at the Centre de Recerca Matemàtica.

2003: PhD in Mathematics, Lomonosov Moscow State University, Moscow.

2004-2006: Marie Curie Fellow at CRM, Barcelona.

2006-2008: Post-doctoral Fellow at the Scuola Normale Superiore, Pisa.

2008-September 2012: ICREA Researcher at CRM, Barcelona.

2009: ISAAC Award.

2012: ICREA Research Professor at CRM, Barcelona.

2013: Humboldt Research Fellowship for Experienced Researchers.

**Xavier Tolsa**

I was born in Barcelona in 1966. First I studied engineering, but later I turned to mathematics. After obtaining my PhD in mathematics in 1998 (UAB), I spent about one year in Goteborg (University of Goteborg – Chalmers) and another year in Paris (Université de Paris-Sud), until I came back to Barcelona (UAB) by means of a “Ramón y Cajal” position. Since 2003 I am an ICREA Research Professor. My current research in mathematics focuses in Fourier analysis, geometric measure theory, and geometric function theory.

My main scientific achievements are the proof of the semiadditivity of analytic capacity and contribution to the Painlevé problem (2003), and the solution of the David Semmes problem in codimension 1, with F. Nazarov and A. Volberg (2014).

Current and previous positions

Assistant professor, Universitat de Barcelona – (1994-1999)

Post-doctoral Research. Chalmers University of Technology (1999 – 2000)

Post-doctoral Research. Université Paris-Sud 11 (2000 – 2001)

Ramón y Cajal, Universitat Autònoma de Barcelona – (2001-2003)

ICREA Research Professor, Universitat Autònoma de Barcelona – Matemàtiques (2003 – present)

Prizes

Salem Prize (2002)

Prize of the European Mathematical Society (2004)

ERC Advanced Grant (2013-2018), to develop the project ”Geometric Analysis in the Euclidean Space”

Ferran Sunyer i Balaguer Prize 2013, for the monograph “Analytic capacity, the Cauchy transform, and non-homogneous Carderón-Zigmund theory” (2013)

Member of the Editorial Board of the “Journal of the European Mathematical Society”. (2014)

**Joan Verdera**

Professor at UAB since 1987.

Visiting professor at University of California at Los Angeles during 1983-1984 and 2001-2002.

I contributed to the understanding of analytic capacity in a series of papers during the nineties and the beginning of 2000. I also worked on the maximal singular integral in connection with existence of principal values. More recently, I became interested in Fluid Dynamics, and I am working presently in the subject using methods from Classical Analysis.