Tomasz Kania
Functional Analysis · Banach Algebras · Operator Theory

Tomasz Kania

Research Interests

My research in functional analysis develops along two complementary directions. The first concerns the geometry of Banach spaces—structural and quantitative phenomena such as bases, projection properties, injectivity, and rigidity. This geometric perspective naturally extends to operators and their behaviour.

The second strand is more algebraic, focusing on Banach algebras and especially algebras of operators. I am interested in how algebraic structure interacts with operator theory and the geometry of the underlying space.

Across both areas, I draw on logic and set theory: combinatorial and foundational methods that illuminate structural questions in infinite-dimensional spaces.

More recently, I have turned to numerical methods, stochastic analysis, and applied probability—particularly in risk management, where abstract analytic structures meet quantitative modelling.

Publications arXiv kania, tomasz GitHub ResearchGate MathOverflow Math.SE Map of Banach Space Theorists Erdős number: 3 linktr.ee Instagram Wikipedia

Publications

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Open Problems

Questions I'm thinking about

Is convolution in $\ell^1(\mathbb{Z})$ an open map?

I would like to know whether convolution is open (maps open sets to open sets) when viewed as a map $\ell_1(\mathbb{Z}) \times \ell_1(\mathbb{Z}) \to \ell_1(\mathbb{Z})$; equivalently whether multiplication in the Wiener algebra is open.

→ View on MathOverflow

Toroidal Elton–Odell and the rephasing trap in the Ramsey reduction

I want to circumvent combinatorial obstacles for establishing a toroidal version of the Elton–Odell separation theorem for infinite-dimensional complex normed spaces.

→ View on MathOverflow

Selected Talks & Events

Jan 2026

53rd Winter School in Abstract Analysis

Hotel Pavla, Vlachovice-Sykovec, Czech Republic — There is no universal separable Banach algebra
Nov 2025

Functional Analysis in the Pacific Northwest

University of Oregon, Eugene, US — The π-property along a filter
Sep 2025

XXXIX International Summer Conference on Real Functions Theory

Rowy, Poland — Openness of multiplication in Banach functional algebras — Invited talk
Jan 2025

52nd Winter School in Abstract Analysis

Hotel Pavla, Vlachovice-Sykovec, Czech Republic — Discrete subgroups of normed spaces are free Abelian
Sep 2024

Analysis Seminar Innsbruck

Innsbruck, Austria — Lattice points in Banach spaces
Aug 2024

IWOTA 2024 — 35th International Workshop on Operator Theory and its Applications

University of Kent, Canterbury, UK — Polish spaces of Banach lattices
Jul 2024

Geometry of Banach and Metric Spaces and Fixed Point Theory

9th European Congress of Mathematics (9ECM), Sevilla, Spain — Uniform property (S) — Invited minisymposium talk
May 2024

Lattice Structures in Analysis and Applications

ICMAT, Madrid, Spain — Projectively universal objects and Polish spaces of spaces
Jan 2024

51st Winter School in Abstract Analysis

Hotel Pavla, Vlachovice-Sykovec, Czech Republic
Jul 2023

SUMIRFAS — Summer Informal Regional Functional Analysis Seminar

Texas A&M University, College Station, US — Invited talk
Jul 2023

Concentration Week on Ideals and Algebras of Operators on Banach Spaces

Texas A&M University, College Station, US — Invited talk
Jan 2023

WorC(K)shop

University of Extremadura, Badajoz, Spain — Complex C(K)-spaces of topological stable rank 1 — Invited talk
Jan 2023

50th Winter School in Abstract Analysis

Sněžné-Milovy, Czech Republic
Jun 2022

Mikrokonferencja z procesów stochastycznych

University of Silesia, Katowice, Poland — Wzajemna ciągłość rozkładów procesów Ornsteina-Uhlenbecka — Invited talk
May 2022

Workshop on Banach spaces and Banach lattices

ICMAT, Madrid, Spain — Invited talk
Jan 2022

49th Winter School in Abstract Analysis

Sněžné, Czech Republic — $(λ+)$-injective Banach spaces
Jan 2020

48th Winter School in Abstract Analysis

Svratka, Czech Republic — Isomorphic Kottman constant of a Banach space
Nov 2019

Analysis Seminar Innsbruck

Innsbruck, Austria — Tsirelson and Schreier spaces, and their operators
Sep 2019

Workshop on Banach spaces and Banach lattices

ICMAT, Madrid — Classification of ideals of algebras of operators on Banach spaces
Sep 2019

Jubilee Congress for the 100th anniversary of the Polish Mathematical Society

Kraków, Poland — When are algebras of operators Grothendieck spaces? (in Polish)
Apr 2019

Preserver Weekend in Szeged

Bolyai Institute, Szeged, Hungary — Maps respecting the compatibility ordering
Jul 2018

Function Spaces XII

Kraków, Poland — On the problem of openness of convolutions on discrete groups
Jun 2018

Geometrical Aspects of Banach Spaces

Birmingham, UK — $(1+)$-separation in the unit sphere of a Banach space
May 2017

Geometric Topology and Geometry of Banach Spaces

Eilat, Israel — Separated subsets of the unit sphere of a Banach space
Jan 2017

45th Winter School in Abstract Analysis

Svratka, Czech Republic — Recovering a compact space from its continuous functions
Oct 2016

Virginia Operator Theory and Complex Analysis Meeting (VOTCAM)

University of Richmond, USA — Recovering a compact space from C(X) — Invited talk
Jul 2016

Young Mathematicians in C*-Algebras

Münster, Germany — Compatibility morphisms between commutative C*-algebras
Jul 2016

Transfinite methods in Banach spaces and algebras of operators

Będlewo, Poland — Closed ideals of operators on Banach spaces of functions with countable support
Jul 2016

The Stone–Čech Compactification: Theory and Applications

University of Cambridge, UK
Jun 2016

The mathematical legacy of Uffe Haagerup

University of Copenhagen, Denmark
Jan 2016

44th Winter School in Abstract Analysis

Svratka, Czech Republic — Ideal structure of the algebra of bounded operators
Aug 2015

Young Mathematicians in C*-Algebras

University of Copenhagen, Denmark — On C*-algebras that do not decompose into tensor products
Apr 2015

QOP Meeting: Quantum Probability, Groups and Geometry

IMPAN, Warsaw, Poland
Sep 2014

Heidelberg Laureate Forum

Heidelberg, Germany — Workshop organised: Independence results in Mathematics
Jul 2014

Workshop on Applications of model theory to C*-algebras

Münster, Germany
Jun 2014

Workshop on Ordered Banach Algebras

Lorentz Centre, Leiden, The Netherlands
Jun 2014

Integration, Vector Measures and Related Topics

Będlewo, Poland — Which algebras of operators are Grothendieck?
Apr 2014

Young Functional Analysts' Workshop 2014

Lancaster University, UK — Universality problems in operator algebras via model theory
Sep 2013

Heidelberg Laureate Forum

Heidelberg, Germany
Sep 2013

Classifying Structures for Operator Algebras and Dynamical Systems

Aberystwyth, UK
Sep 2013

School and workshop: Topics in Operator Algebras and Applications

Madrid, Spain
Aug 2013

Banach algebras and applications

Gothenburg, Sweden — Maximal left ideals of operators acting on a Banach space
Jul 2013

QOP Meeting: C*-algebras and Banach algebras

Warsaw, Poland
Apr 2013

QOP Meeting: Banach and operator spaces

Lancaster, UK — Operators from C(K)-spaces annihilating chains of functions — Invited talk
Mar 2013

Young Functional Analysts' Workshop 2013

University of Sheffield, UK — Grothendieck algebras of operators on Banach spaces
Sep 2012

Thematic Program on Forcing: Workshop on Applications to Operator Algebras

Fields Institute, Toronto, Canada
Sep 2012

Appalachian Set Theory Workshop

Fields Institute, Toronto, Canada
Aug 2012

Geometry of Banach spaces

CIRM, Luminy, Marseille, France
Jul 2012

Trends in set theory

Warsaw, Poland — Operator ideal of weakly compactly generated operators
Jun 2012

Banach Spaces Workshop

University of Birmingham, UK
Apr 2012

Stochastic Analysis and Stochastic PDEs

University of Warwick, UK
Mar 2012

EPSRC Symposium Workshop – 2012 UK Easter Probability Meeting

University of Warwick, UK
Mar 2012

Young Functional Analysts' Workshop 2012

University of Oxford, UK — Closed ideals of bounded operators on C[0,ω₁]
Oct 2011

North British Functional Analysis Seminar

University of Edinburgh, UK
Sep 2011

Analytical and Computer Assisted Methods in Mathematical Models

Bydgoszcz, Poland
Aug 2011

Banach Algebras 2011

Waterloo, Ontario, Canada — Uniqueness of the maximal ideal on C([0,ω₁])
Jul 2011

14th International Workshop for Young Mathematicians "Algebra"

Kraków, Poland — Radical-theoretic approach to ring theory
Apr 2011

Operator Theory and its Applications in honour of Victor Shulman

Gothenburg, Sweden
Apr 2011

LMS–EPSRC Short Course 2011

Mathematical Institute, University of Oxford, UK
Feb 2011

Set-theoretic Techniques in Functional Analysis — Banach Space Meeting 2011

Castro Urdiales, Cantabria, Spain
Feb 2010

6th International Students' Conference on Analysis

Síkfőkút, Noszvaj, Hungary
Feb 2008

4th International Students' Conference on Analysis

Zamárdi, Hungary — A Remark on homomorphisms on C(X)

Selected Seminar Talks

Feb 2026

Hot Topics Seminar

Technical University of Łódź — A no-limits approach to Probability and Statistics: an Introduction to Non-Standard Analysis and its applications to risk estimation
Dec 2025

Seminar

University of Warsaw, MIM UW — Polish spaces of categories enriched over separable Banach spaces
Apr 2025

Seminar

University of Silesia, Katowice — Dyskretne podprzestrzenie przestrzeni unormowanych
Mar 2022

Noncommutative Probability Seminar

University College Cork, Ireland
Feb 2022

Functional Analysis Seminar

Adam Mickiewicz University, Poznań — Stopping-time spaces and other spaces related to the Haar system from the operator-ideal viewpoint
Dec 2021

Functional Equations Seminar

University of Silesia, Katowice
Jan 2021

Functional Analysis Seminar

Adam Mickiewicz University, Poznań — Large cardinal numbers vs continuity of coordinate functionals
Nov 2020

Real Analysis Seminar

Technical University of Łódź — Geometry of the sphere of a normed space
Jun 2020

Seminar on Geometry of Banach spaces

Jagiellonian University, Kraków — Filter bases in Banach spaces and large cardinals
May 2020

Analysis Seminar

University of Portsmouth — Estimating Kalton's constant
Apr 2020

Banach Spaces Webinar

University of North Texas — Quantifying Kottman's constant
Nov 2019

Seminar on Geometry of Banach spaces

Jagiellonian University, Kraków
Oct 2019

Výjezdní zasedání Matematického ústavu AV ČR

Čejkovice, Czech Republic — Separation in the sphere: state of the art
May 2019

Colloquium

Jagiellonian University, Kraków — Surjective images of operator algebras
Dec 2018

Seminar on Geometry of Banach spaces

Jagiellonian University, Kraków — On separated subsets of the sphere of a Banach space
Oct 2018

Abstract Analysis Seminar

Czech Academy of Sciences, Prague — Openness of multiplication in Banach algebras
Jun 2018

Seminar

University of Extremadura, Badajoz, Spain — Separated sets and Auerbach systems in Banach spaces
Apr 2018

Functional Analysis Seminar

Adam Mickiewicz University, Poznań — Some new results on separated subsets of the unit sphere
Mar 2018

Mathematics Seminar

University of Kielce, Poland — When is multiplication in a Banach algebra open?
Feb 2018

Real Analysis Seminar

Technical University of Łódź
Jan 2018

Operator Algebras Seminar

University of Münster — When is multiplication in a Banach algebra open?
Mar 2017

Real Analysis Seminar

Technical University of Łódź
Jan 2017

Analysis Seminar

University of Portsmouth
Dec 2016

Seminar Talks

Sardar Patel University & University of Delhi, India — C(X) and compact spaces
Oct 2016

Seminar Talk

Washington and Lee University, Virginia — Steinhaus' lattice-point problem
Oct 2016

Analysis Seminar

University of Virginia — Separated subsets in Banach spaces
Mar 2014

Set-theory Seminar

University of Toronto — A chain condition for operators from C(K)-spaces
Jan 2014

Descriptive Set Theory Seminar

University of Copenhagen — Maximal left ideal structure of operator algebras
Nov 2013

Analysis Seminar

University of Oxford — Maximal left ideals of operators
Jul 2013

Analysis Seminar

University of Glasgow — Grothendieck algebras of operators
Jan 2012

Real Analysis Seminar

University of Silesia, Katowice — Maximal left ideals in Banach algebras
Jan 2012

Working Group in Applications of Set Theory

IMPAN, Warsaw — On classical operator ideals and the structure of the operator algebra on C([0,ω₁])
Jan 2012

Functional Analysis Seminar

IMPAN, Warsaw — Operators and vector measures with range in a WCG Banach space
Apr 2011

Real Analysis Seminar

Technical University of Łódź — Closed ideals in algebras of bounded operators on C(K)-spaces
May 2015

Lancaster Pure Maths Postgraduate Forum

Lancaster University — How many norm-one continuous functions with distance greater than 1 are there?
Jan 2015

Lancaster Pure Maths Postgraduate Forum

Lancaster University — Linear algebra from analyst's point of view
Nov 2013

Lancaster Pure Maths Postgraduate Forum

Lancaster University — The Johnson–Lindenstrauss space
Oct 2013

Lancaster Pure Maths Postgraduate Forum

Lancaster University — Killing uncountable chains of continuous functions
Feb 2013

Lancaster Pure Maths Postgraduate Forum

Lancaster University — Around Pełczyński's decomposition method
Mar 2011

Lancaster Pure Maths Postgraduate Forum

Lancaster University — Beyond ZFC: Beauty of Jensen's Diamond Principle
Jan 2011

Lancaster Pure Maths Postgraduate Forum

Lancaster University — Some Remarks on the Stone–Čech Compactification

Teaching

All materials →

Python in Finance, Finance in Python

Jagiellonian University · Graduate level · 6 ECTS · English

Python is one of the most widely used object-oriented programming languages, whose popularity continues to grow thanks to Data Science libraries such as numpy, pandas, and scikit-learn. This course presents selected topics in Finance including interest rate models, option pricing (Black–Scholes, Cox–Ross–Rubinstein, and other models), derivative pricing, and simulations using binomial trees/lattices and Monte Carlo methods—all implemented in Python.

Topics include algorithmic trading, Value-at-Risk estimation, portfolio liquidity, and Big Data techniques. While emphasis is placed on programming (mainly in Jupyter notebooks), mathematical content such as properties of solutions to stochastic differential equations is treated rigorously.

Python NumPy / Pandas Black–Scholes Model Monte Carlo Methods Value-at-Risk Algorithmic Trading SDEs

View Full Syllabus (PDF)

I have also taught courses in Functional Analysis, Banach Spaces, and related topics. For lecture notes and additional materials, visit my teaching page.

Education

2020
Habilitation in Mathematics
Jagiellonian University, Kraków, Poland
Series of papers: Surjective images and ideals of Banach algebras with an emphasis on algebras of operators on Banach spaces
2021
MSc in Computer Science
University of Economics in Katowice, Poland
Thesis: Methods of Topological Data Analysis in Decision Theory
Supervisor: Tomasz Wachowicz
2013
PhD in Mathematics
Lancaster University, England
Thesis: Closed ideals of operators on Banach spaces of continuous functions
Supervisor: Niels J. Laustsen
2010
Integrated MSc in Pure Mathematics
University of Silesia, Katowice, Poland
Thesis: On the applications of Martin's Axiom
Supervisor: Aleksander Błaszczyk
2005
Matura
I Liceum Ogólnokształcące im. Waleriana Łukasińskiego, Dąbrowa Górnicza, Poland
Form Tutor: Adam Emeryk

Past Academic Positions

2016–2017
Research Associate
Mathematics Institute, University of Warwick, UK
Participant of the ERC Advanced Grant project "Local Structure of Sets, Measures and Currents"
PI: Professor David Preiss, FRS
2014–2015
Temporary Lecturer
School of Mathematical Sciences, UCC Cork, Ireland
2014–2015
Temporary Lecturer
Department of Mathematics and Statistics, Lancaster University, UK
2014
Research Fellow
Fields Institute, Toronto, Canada
Participant of the Thematic Program on Abstract Harmonic Analysis, Banach and Operator Algebras
2014
Assistant Professor
Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
2010–2013
Undergraduate Tutor
Lancaster University, England

Professional Service

Editorial Board

Banach Journal of Mathematical Analysis Springer Concrete Operators De Gruyter

Workshop and Conference Organisation

Sep 2022
IWOTA 2022 Conference: Conference co-organisation and organisation of the panel Operator ideals and operators on Banach spaces together with K. Beanland and N. Laustsen, Kraków
Sep 2021
IWOTA 2021 Conference: Organisation of the panel Operator ideals and operators on Banach spaces together with K. Beanland and N. Laustsen, Lancaster, UK
Jul 2020
Functions and operators, 10 years after: Co-organiser, Kraków, Poland
Jul 2017
Geometric Measure Theory: Local co-organiser, University of Warwick, UK

Grant Referee

  • Natural Sciences and Engineering Research Council of Canada (NSERC)
  • Polish National Agency for Academic Exchange (NAWA)
  • Estonian Research Council (6×)

Member of the Mathematics and Computer Science panel of the Estonian Research Council (2022–2023)

Habilitation Referee

Jan Rozendaal IMPAN

PhD Thesis Referee

Agnieszka Gergont UMCS Lublin Krzysztof Zakrzewski MIMUW Karol Aleksandrowicz UMCS Lublin Bartłomiej Zawalski IMPAN

PhD Supervision

Natalia Maślany 2026
Combinatorial Banach spaces and related topics: algebraic structure and self-isometries
Mariusz Niwiński exp. 2028
Descriptive set theory of Banach lattices

MSc Supervision

Jakub Kamiński 2025
On the Convergence of Deep Q-Learning in the Guess 2/3 of the Average Game
Konstantin Belozerov 2023
Topologiczna analiza danych w zastosowaniach finansowych
Marlena Kostrzewa 2021
Pricing convertible bonds
Aleksandra Bogusz 2021
Topological summary statistics and statistical inference in Topological Data Analysis
Emily Billau 2015
Brouwer's fixed point theorem
Ross Tattersall 2015
The Ax–Grothendieck Theorem

Grants and Awards

Awards

2021 Otto Wichterle Award

Awarded by the Czech Academy of Sciences in Prague, given to researchers under 35 years of age. Named after Otto Wichterle, one of the first members of the Czechoslovak Academy of Sciences, chemist and inventor of contact lenses. Recognised in mathematics for a series of publications on the geometry of Banach spaces, specifically qualitative and quantitative descriptions of separated sets on spheres of Banach spaces.

2013 & 2014 Heidelberg Laureate Forum

Heidelberg Laureate Forum: Abel-, Fields- and Turing-Laureates Meet the Next Generation — alumnus.

2011, 2012 & 2013 Dean's Award for Excellence

Lancaster University Faculty of Science and Technology Dean's Award for Excellence in PhD Studies.

Grants

2020–2024 Principal Investigator

NCN SONATA 15

Geometria lipschitzowska przestrzeni metrycznych oraz stowarzyszona z nią teoria operatorów i analiza harmoniczna

Lipschitz geometry of metric spaces and associated operator theory and harmonic analysis

Narodowe Centrum Nauki (National Science Centre, Poland)

2019–2021 Principal Investigator

GAČR Project 19-07129Y

Metody lineární analýzy v operátorových algebrách a naopak

Linear-analysis techniques in operator algebras and vice versa

Postdoctoral Researchers
Szymon Draga 2019 Bence Horváth 2019–2021 Jarosław Swaczyna 2020–2022
Mar–Apr 2018 Participant

Doktorská škola pro vzdělávání v oblasti matematických metod a nástrojů v HPC

Doctoral School for Education in Mathematical Methods and Tools in HPC

Visiting the University of Cambridge for the purpose of transferring know-how concerning the organisation of PhD studies in High-Performance Computing. The final report serves the Czech government in the ongoing process of modernisation of programmes in the area of HPC.

Project details →