Laureates of the International Banach Prize – edition 2024
MAIN PRIZE
DALIMIL PEŠA
The dissertation title: “Fine properties of certain specific function spaces”
Department of Mathematical Analysis, Charles University, Prague, Czech Republic
supervisor: prof. RNDr. Luboš Pick, CSc., DSc.
The focus of the dissertation of dr. Dalimil Peša are function spaces, a subject that is approached from three distinct angles.
The first part of the thesis is concerned with function spaces in the abstract. It studies the so-called quasi-Banach function spaces, a general class of spaces that covers many examples occurring naturally in many fields such as harmonic analysis and operator theory. A major contribution of the thesis is the introduction of the so-called Wiener–Luxemburg amalgam spaces, an abstract framework for constructing rearrangement-invariant amalgams, that is, spaces where the conditions on the local and global behaviour can be prescribed separately.
The second part provides a comprehensive treatment of Lorentz–Karamata spaces. This is a fairly recent class of function spaces that in the last two decades has found significant applications, especially for treating the limiting cases of interpolation or Sobolev embeddings. The thesis provides a unified approach to said spaces that is also more general than some of the previous attempts and solves several interesting open problems.
The final part of the dissertation is dedicated to applications of the function space theory. The thesis establishes a reduction principle for a class of kernel-type integral operators that is closely related to optimal higher-order Sobolev embeddings, and also several variants of strongly non-linear Gagliardo–Nirenberg inequalities.
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KAROL DUDA
The dissertation title: “Dynamics and computability in Geometric Group Theory”
Faculty of Mathematics and Computer Science, University of Wrocław
supervisors: prof. dr hab. Aleksander Iwanow, dr Damian Osajda
The dissertation by Dr. Duda concerns two areas of geometric group theory.
The first part of the dissertation concerns computable aspects of amenability. Dr. Duda proves a computable version of Tarski’s Alternative Theorem. The proof is based on a new, computable version of Hall’s harem Theorem. There are several computable versions of this theorem in the dissertation. By applying one of these to non-amenable computable coarse spaces Dr. Duda obtains a computable version of the generalization of Whyte results concerning geometric Von Neumann conjecture.
The second part is devoted to the locally elliptic actions of groups on small cancellation complexes. Dr. Duda proves that there are no infinite torsion subgroups in groups defined by C(6), C(4)-T(4), or C(3)-T(6) small cancellation presentations. This result was specifically distinguished by the referees of the dissertation. It is an answer to the problem that has been open for many years. This follows from a more general result of the dissertation on the existence of fixed points for locally elliptic actions of groups on simply connected small cancellation complexes.
The methods used in the proof give other, stronger results in the case of C(3)-T(6) complexes. In particular, simply connected C(3)-T(6) complexes may be equipped with a CAT(0) metric. This implies the Tits Alternative for groups acting on simply connected C(3)-T(6) small cancellation complexes.
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JAKUB SKRZECZKOWSKI
The dissertation title: “Singular limits and rough behavior in evolutionary equations arising in physics and biology”
Institute of Mathematics, Polish Academy of Sciences, Warsaw
supervisor: Prof. dr hab. Piotr Gwiazda
The dissertation addresses three important issues in the theory of partial differential equations inspired by biology and mathematical physics: models of fast chemical reactions responsible for the propagation of the electrical signal in synaptic connections in the brain, the degenerate Cahn-Hilliard equation occurring in materials sciences and models of cell-cell adhesion as well as variational analysis of two-phase functionals used to describe materials composed of two components of different hardness. One of the most important results of the thesis is the proof of the limit of the non-local degenerate Cahn-Hilliard equation to its local counterpart, which completes the program of deriving this equation initiated by Giacomin and Lebowitz in the 1990s. The results presented in the thesis were published in journals such as Communications on Pure and Applied Mathematics, Journal of Functional Analysis and Communications in Mathematical Physics.
Laureates of the International Banach Prize – edition 2023
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BLAS FERNÁNDEZ
The dissertation title: “On certain problems related with Terwilliger algebras and Distance-Balanced graphs”,
supervisor: Prof. Dr. Štefko Miklavič, University of Primorska
The dissertation by Dr. Fernandez is concerned with Terwilliger algebras, which are algebraic objects defined by the graph’s incidence matrix. They are studied mainly with algebraic tools, and one of the goals is their classification. The contribution by Dr. Fernandez relies on an interpretation of certain properties of Terwilliger algebras in terms of combinatorial properties, such as path counting in the graphs describing them. These results are considered very original by Prof. Terwilliger himself. Later in this extensive dissertation, Dr. Fernandez deals in detail with properties of distance-balanced graphs. He managed to establish their partial classification, as well as construct families of examples that provide answers to conjectures previously hypothesized by other mathematicians.
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SZYMON CYGAN
The dissertation title: “Patterns in nonlocal and degenerate models from mathematical biology”,
supervisor: prof. dr hab. Grzegorz Karch, University of Wroclaw
The Dissertation by Dr. Cygan studies mathematical models of biological phenomena. In the first part of the dissertation the model Kondo is considered which, from a mathematical point of view, leads to a differential equation with an additional non-linear term depending on the variable function via an operator with an integral kernel. This quite intricate situation becomes simplified if only stationary solutions are sought. The existence of such solutions is demonstrated by means of the fixed point theorem, stability conditions are given, as well as the construction of solutions of a specific type together with a numerical implementation. This part of the dissertation characterizes with great independence. In the second part of the dissertation, models of diffusion-reaction type are considered. It is shown that stationary solutions near equilibrium are unstable. Further results concern the existence of discontinuous weak solutions and conditions of their stability. The strength of the results obtained is illustrated by their application to a series of classically known models such as the Grey-Scott model, Fitz-Nagumo model, Brusselator and Oregonator.
Laureates of the International Banach Prize – edition 2022
MAIN PRIZE
MARCIN SROKA
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.
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AGNIESZKA HEJNA
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.
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DOMINIK BUREK
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
MAIN PRIZE
MICHAŁ MIŚKIEWICZ
The dissertation title: “Singularities of minimizing harmonic maps into closed manifolds”,
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.
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TOMASZ DĘBIEC
The dissertation title: “Weak convergence methods for equations of 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.
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WOJCIECH GÓRNY
The dissertation title: “Anisotropic least gradient problems”,
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