Kaj Nyström

I am professor of Mathematical Analysis and I serve as the head of the research program Analysis and Probability at UU. Currently my fields of research are mainly partial differential equations, mathematical analysis for AI, but I also work on control theory and optimization, and risk management.

If you have an interest in my fields of analysis, feel free to contact me.

2024-2025 I am teaching Non-Linear Partial Differential Equations (1MA338) and Foundations of Data Science (1MS048).

Education and Academic degrees

• 1999. Docent in Mathematics, Umeå University.
• 1995-1996. Postdoctoral position at The University of Chicago and MIT (Boston).
• 1994. Doctor of Philosophy in Mathematics, Umeå University. Supervisor: Hans Wallin. PhD thesis: Smoothness Properties of Solutions to Dirichlet Problems in Domains with a Fractal Boundary.
• 1991. Master of Science in Mathematics, Umeå University. Master thesis: Packing problems at SCA-shipping.
• 1989-1990. Military service, Boden. Trained in military communication services, in particular
  cryptology. Leadership and military psychology. Trained to lead forces of ten.
• 1985-1988. Technical Gymnasium, Wargentinskolan, Östersund. Final grade: 5.0/5.0.

Positions

• 2011-xxxx. Professor of Mathematical Analysis, Uppsala University.
• 2010-2011. Professor in Mathematics, Umeå University.
• 2001-2009. Associated Professor in Mathematics, Umeå University.
• 2001-2003. Modeling intelligence and risk analyst, Swedbank, Stockholm. I developed models for all parts of Economic capital i.e., frameworks for the quantification of market, credit and operational risk, capital allocation and models for consistent pricing of financial instruments.
• 1997-2001. Assistent Professor in Mathematics, Umeå University.
• 1994-1995. Lecturer in Mathematics, Uppsala University.
• 1991-1994. PhD student in Mathematics, Umeå University.

Awards

• 2015. Göran Gustafsson prize in Mathematics, Royal Academy of the Sciences.
• 2013. Wallmarkska prize, Royal Academy of the Sciences.
• 2010. Wallenberg prize, Swedish Society of Mathematicians.

Recent research grants

• 2023-2026. Swedish scientific council (VR), 3 800 000 SEK, New Perspectives on Evolutionary Free Boundary Problems.
• 2023-2026. Knut and Alice Wallenberg Foundation, 2 000 000 SEK, Postdoctoral program: New Perspectives on Evolutionary Free Boundary Problems.
• 2021-2024. The Verge Foundation, 2 000 000 SEK, Postdoctoral program: Equations of Kolmogorov type and their Ramifications.
• 2020-2025 WASP Artificial Intelligence program, 2 000 000 SEK, PhD project: Mathematics for AI - Continuous Methods in Deep Learning.
• 2018-2022. Swedish scientific council (VR), 3 600 000 SEK, A Novel Approach to Boundary Value Problems for Parabolic Equations and Systems.
• 2019-2021. Handelsbankens forskningsstiftelser, 1 500 000 SEK, Blockchains and their cryptocurrencies.
• 2016-2019. Royal Academy of the Sciences, 4 000 000 SEK, Göran Gustafsson prize in Mathematics.
• 2014-2018. Swedish scientific council (VR), 4 000 000 SEK, Equations of Kolmogorov type in Analysis, Finance and Physics.
• 2010-2015. Handelsbankens forskningsstiftelser, 2 000 000 SEK, Optimal Switching Problems and their Applications in Economics and Finance.
• 2010-2015. Riksbanken Jubileumsfond, 4 150 000 SEK, Liquidity risk.

A selection of recent responsibilities

• 2015-xxxx. Head of the research program Analysis and Probability at UU.
• 2018-2020. Member (vice-chairman) of the Faculty Advisory Board for Research at UU.
• 2014-2016. Head of the board for CIM - Center for interdisciplinary mathematics - at UU.

Supervision of postgraduate students

• For my record concerning students with completed PhD in mathematics, see Math Genealogy
• Currently with me as 1st supervisor: Alireza Ataei, Hadid Rehman.
• Currently with me as 2nd supervisor: Mingyi Hou, David Lundberg, Emiliano Peña.

Supervision of Postdoctoral students

• Currently: Sunghan Kim.
• Previously: Benny, Avelin, Paolo Baroni, Jens Berg, Alejandro Castro, Chiara Cinti, Prashanta Garain, Elin Götmark, Kai Li, Sidi Mohamed Ould Aly, Anton Savostianov, Martin Strömqvist, Matias Vestberg, Shoyeb Waliullah, Changyong Zhang.

I am working in research theme Analysis and Partial Differential Equation. Follow the link to find seminars, thesis, degree projects, and other activities. Otherwise, feel free to contact me directly if you are interested in discussing. For more data concerning my research and activities I refer to Google Scholar, ResearchGate, and Math Genealogy.

Summary of my research:

Partial differential equations: Parabolic equations and systems, complex coefficients, Kato square problems, equations of Kolmogorov-Fokker-Planck type, quasi-linear PDEs, p-Laplace operator, p -parabolic equation, non-local equations, fractional heat operator, tug-of-war games and degenerate PDEs, subelliptic PDEs, boundary value problems, regularity, boundary behavior, obstacle problems, free boundary problems, harmonic and parabolic measure, p-harmonic measure, geometric measure theory, parabolic uniform rectifiable sets, low-dimensional sets, fractals, layer potentials, harmonic analysis.

Mathematical analysis for AI: Deep learning, stochastic search and optimization algorithms, kinetic theory, Fokker-Planck, nonlinear/nonlocal approximation theory, applications to PDEs, inverse problems for PDEs, data-driven discovery, dimension reduction, diffusion geometry.

Control theory and optimization: Optimal switching problems, stochastic filtering theory, backward stochastic differential equations, the Skorohod problem, Tug-of-wars, option pricing, American options, high frequency trading, Monte Carlo methods, stochastic differential equations, adaptive numerical methods for stochastic differential equations.

Risk management: Market risk, operational risk, credit risk, liquidity risk, structured finance, capital allocation, pricing and hedging of financial derivatives, extreme value theory, econometrics, time-series analysis, Monte Carlo methods.

A selection of publications devoted to Analysis and PDE

• 2024 (with Bortz, S., Hofmann, S., and Martell, J.M.) Solvability of the L^p Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph, Inventiones Mathematicae, to appear.
• 2020. (with Auscher, P. and Egert, M.) L^2 well-posedness of boundary value problems and the Kato square root problem for parabolic systems with measurable coefficients, Journal of the European Mathematical Society, 22(9): 2943-3058.
• 2018. (with Auscher, P. and Egert, M.) The Dirichlet problem for second order parabolic operators in divergence form, Journal de l'Ecole polytechnique Mathematiques, 5, 407-441.
• 2018. (with Lewis, J.) Quasi-linear PDEs and low-dimensional sets, Journal of the European Mathematical Society, 20(7): 1689-1746.
• 2017. (with Azzam, J., Hofmann, S., Martell, J. and Toro, T.) A new characterization of chord-arc domains, Journal of the European Mathematical Society, 19(4): 967-981.
• 2017. (with Hofmann, S., Le, P. and Martell, J.) The weak-A_infty property of harmonic and p-harmonic measures implies uniform rectifiability, Analysis & PDE10(3): 513-558.
• 2013. (with Lewis, J., Vogel, A.) On the Dimension of p-Harmonic Measure in Space, Journal of the European Math Society, 15, 2197-2256.
• 2012 (with Lewis, J.) Regularity and free boundary regularity for the p-Laplace operator in Reifenberg flat and Ahlfors regular domains, Journal of the American Mathematical Society, {25}, 827-862.
• 2010. (with Lewis, J.) Regularity of Lipschitz Free Boundaries in Two-phase Problems for the p-Laplace Operator, Advances in Mathematics 225, 2565-2597.
• 2010. (with Lewis, J.) Boundary Behaviour and the Martin Boundary Problem for p-Harmonic Functions in Lipschitz Domains, Annals of Mathematics 172, 1907-1948.
• 2010. (with Önskog, T.) The Skorohod Oblique Reflection Problem in Time-dependent Domains, Annals of Probability, 38, 2170-2223.

At Publications below you can see an essentially complete record of my publications. My recent preprints can these days usually by found at Math ArXiv. My most recent preprints/papers are

• Higher order interpolative geometries and gradient regularity in evolutionary obstacle problems, Journal des Mathématiques Pures et Appliquées, to appear, with Kim, S.
• Solvability of the L^p Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph, Inventiones Mathematicae, to appear, with Bortz, S., Hofmann, S., and Martell, J.M.
• A Galerkin type method for kinetic Fokker Planck equations based on Hermite expansions, Kinetic and Related Models, to appear, with Avelin, B. and Hou, M.
• A note on fundamental solutions and Gaussian bounds for degenerate parabolic equations with time-dependent coefficients, Potential Analysis, to appear, with Ataei, A.
• The Kato square root problem for parabolic operators with an anti-symmetric part in BMO, with Ataei, A.
• The Kato square root problem for weighted parabolic operators, Analysis & PDE, to appear, with Ataei, A. and Egert, M.
• Square function estimates for the evolutionary p-Laplace equation. Discrete and Continuous Dynamical Systems, to appear.
• On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients, Math. Eng. 5 (2023), no. 2, Paper No. 043, 37 pp, with Garain, P.
• Deep learning, stochastic gradient descent and diffusion maps, Journal of Computational Mathematics and Data Science, 4 (2022), with Fjellström, C.
• Tug-of-war with Kolmogorov, J. Diff. Eqs, to appear, with Fjellström, C. and Vestberg, M.
• Asymptotic mean value formulas, nonlocal space-time parabolic operators and anomalous tug-of-war games, J. Diff. Eqs, 342 (2023), 150–178, with Fjellström, C. and Wang, Y.