Laureates of the International Banach Prize – edition 2022
Laureates of the International Banach Prize – edition 2022
The dissertation title: “Monge-Ampère’s equation in hypercomplex geometry”,
supervisor: prof. Sławomir Kołodziej, Jagiellonian University in Cracow
The dissertation of dr. Marcin Sroka concerns solutions of the Monge-Ampère equation on a pseudoconvex domain in Hn (i.e., with n quaternion variables), and on compact manifolds with quaternion structure and equipped with a hyper-Kähler metric with torsion (HKT).
The subject-matter stems from quantum mechanics and became an object of mathematical studies around 20 years ago. Leading names in the theory include Alesker, Verbitsky, Harvey and Lawson.
In the context of Hn, dr. Sroka in his dissertation proved the existence of solutions of the Dirichlet problem with the right hand side in Lp for p > 2. This is an optimal result and significantly improves an earlier one, obtained by Harvey and Lawson, which required continuity.
In the context of HKT, the dissertation provides a new and simpler proof of the so-called a priori C0–estimate, obtained by Alesker and Shelukhin via a long and complicated argument.
The significance for geometric analysis of these results (which improve earlier achievements of leading specialists) is unquestionable. The community’s reaction is close to enthusiasm, which is best manifested in the recommendation letter by Valentino Tosatti from Courant Institute. Finally, an evidence of the authors own contribution is the fact that the results are published as single-authored papers in leading mathematical journals.
The dissertation title: “Harmonic analysis and Hardy spaces in the rational Dunkl setting”,
supervisor: prof. Jacek Dziubański, University of Wrocław
The dissertation concerns harmonic analysis in the Dunkl setting. Generally speaking, Dunkl analysis is a non-standard approach to analysis based on a modified notion of directional derivative involving an additional, non-local factor, proportional to a function invariant under a symmetry group. Such an approach is natural in studying some models in particle physics. In this setting one can define and investigate objects alternative to those lying at the base of classical analysis, such as exponential function, Fourier transform, convolution, Laplace operator, heat semigroup, Schrödinger operator or Hardy spaces. Reconstruction of harmonic analysis in the Dunkl setting is the subject-matter of the extensive dissertation of dr. Hejna. The problem is highly nontrivial even if only for the non-local character of the operators or lack of the Leibniz integral rule. This work requires great originality and courage, as the success of such a research program was a priori far from obvious.
The dissertation title: “Arithmetic properties of finite quotients of Calabi-Yau type manifolds”,
supervisor: prof. Sławomir Cynk, Jagiellonian University in Cracow
The dissertation concerns algebraic geometry, in particular, construction and properties of Calabi-Yau manifolds. These were known earlier in complex dimension 2, where they are hyper-Kähler and have been called by Weil K3 surfaces (partly after the names of three mathematicians involved in their investigation, partly as an allusion to K2 – one of the peaks hardest to conquer). Calabi-Yau manifolds are their higher-dimensional generalizations. They have rich geometric properties but effective constructions are far from obvious. The right away noticeable opening result of the dissertation is exactly such a construction of a rich class of Calabi-Yau manifolds as quotients under actions of finite groups. The construction is valid in any dimension. For the examples so constructed, dr. Burek provides a formula for the Hodge number, and, what is most impressive, an expression for the Weil zeta function describing the distribution of rational points.
Laureates of the International Banach Prize – edition 2021
The dissertation title: “Singularity of minimalization map harmonic into closed varieties”,
supervisor: dr hab. Anna Zatorska-Goldstein, prof. UW, University of Warsaw
Michał Miśkiewicz’s dissertation concerns the set of singularities of transformations minimizing the Dirichlet functional between manifolds. The subject matter goes back to the 80’s, when Schoen and Hlenbeck have shown that such transformations are regular except on a set of codimension 3. Since then, properties of the set of singularities attracted the attention of many outstanding mathematicians, such as Brezis, Eells, Lieb, Lin or Yau. The problem is mathematically interesting because the fact that the set of singularities is nonempty is by itself already unexpected, let alone that it has rich analytic and geometric properties. One can thus say that the subject matter fits in the mainstream of interests of modern geometric analysis.
The dissertation title: “Weak methods of convergence of equations in mathematical physics and biology”,
supervisor: dr hab. Agnieszka Świerczewska-Gwiazda, prof. UW, University of Warsaw
The dissertation of Tomasz Dębiec is concerned with weak solutions in a multitude of important models of dynamics of continua and hydrodynamics. The applications of these models, apart from mathematical physics, also reach biology. The cases under consideration include, among others, the Euler-Korteweg equations, compressible Euler and Navier-Stokes equations, and the cancer growth model with two cell types.
The dissertation title: “Anisotropic methods of the smallest gradients”,
supervisor: prof. dr hab. Piotr Rybka, University of Warsaw
The dissertation of Wojciech Górny concerns the least gradient problem. It offers a modern approach to a number of questions originating from the classic geometric analysis, such as investigating minimal surface, as well as arising from applications to the optimal transportation theory, modeling of electrical conductivity, and more general properties of materials.
Laureates of the International Stefan Banach Prize (editions 2009-2017)
2017 Anna Szymusiak (Poland)
for the dissertation „Minimization of the entropy of measurement for symmetric POVMs and their informational power” under the supervision of dr. Wojciech Słomczyński, Faculty of Mathematics and Computer Science of the Jagiellonian University
2016 Adam Kanigowski (Poland)
Institute of Mathematics, Polish Academy of Sciences in Warsaw for the dissertation „Własności ergodyczne gładkich potoków na powierzchniach” under the supervision of prof. Mariusz Lemańczyk, Nicolaus Copernicus University in Toruń and dr. Joanna Kułaga-Przymus, Polish Academy of Sciences & Nicolaus Copernicus University in Toruń
2015 Joonas Ilmavirta (Finland)
for the dissertation “On the Broken Ray Transform” under the supervision of prof. Mikko Salo, University of Jyvasyla
2014 Dan Petersen (Denmark)
University of Kopenhagen, for the dissertation “Topology of moduli spaces and operads” under the supervision of prof. Carel Faber, KTH Royal Institute of Technology in Stockholm
2013 Marcin Pilipczuk (Poland)
for the dissertation „Nowe techniki stosowane przy rozwiązywaniu wybranych problemów NP-trudnych” under the supervision of dr. Łukasz Kowalik, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw
2012 Andras Mathe (Hungary)
for the dissertation „Izomorficzne problemy miar Hausdorffa i ograniczenia Hoeldera dla funkcji” under the supervision of prof. Miklos Laczkovich, Institute of Mathematics, Eötvös Loránd University, Budapest
2011 Łukasz Pańkowski (Poland)
for the dissertation „Twierdzenia o łącznej uniwersalności a twierdzenie Kroneckera o aproksymacjach diofantycznych” under the supervision of prof. Jerzy Kaczorowski, Adam Mickiewicz University, Poznań
2010 Jakub Gismatullin (Poland)
for the dissertation „G-zwartość i grupy” under the supervision of prof. Ludomir Newelski, University of Wroclaw
2009 Tomasz Elsner (Poland)
for the dissertation „Teoria powierzchni minimalnych w przestrzeniach systolicznych” under the supervision of prof. Tadeusz Januszkiewicz, University of Wroclaw