# Kaj Nyström

Professor at Department of Mathematics; Analysis and Partial Differential Equations

- Telephone:
- +46 18 471 32 85
- E-mail:
- kaj.nystrom@math.uu.se
- Visiting address:
- Ångströmlaboratoriet, Lägerhyddsvägen 1
- Postal address:
- Box 480

751 06 UPPSALA

More information is available to staff who log in.

## Short presentation

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.

2023-2024 I am teaching Non-Linear Partial Differential Equations (1MA338) and Mathematical Topics in Data Science (1MS046).

## Keywords

- control theory
- data science
- deep learning
- financial risk management
- harmonic analysis
- mathematical analysis
- mathematical analysis for ai
- partial differential equations
- risk management
- stochastic optimization

## Biography

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 (interdisciplinary) experiences

- 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.
- Currently with me as 2nd supervisor: Mingyi Hou, David Lundberg.

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.

## Research

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

- 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, 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, 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.

## Publications

### Selection of publications

- Potential theory for a class of strongly degenerate parabolic operators of Kolmogorov type with rough coefficients (2022)
- On the fine properties of parabolic measures associated to strongly degenerate parabolic operators of Kolmogorov type (2021)
- L2 well-posedness of boundary value problems and the Kato square root problem for parabolic systems with measurable coefficients (2020)
- The Dirichlet problem for second order parabolic operators in divergence form (2018)
- Quasi-linear PDEs and low-dimensional sets (2018)
- A new characterization of chord-arc domains (2017)
- The A∞-Property of the Kolmogorov Measure (2017)
- Square function estimates and the Kato problem for second order parabolic operators in Rn+1 (2016)
- Kolmogorov-Fokker-Planck Equations (2016)
- The Hadamard variational formula and the Minkowski problem for p-Capacity (2015)
- Sharp regularity for evolutionary obstacle problems, interpolative geometries and removable sets (2014)
- On the dimension of p-harmonic measure in space (2013)
- Regularity and free boundary regularity for the
*p*-Laplace operator in Reifenberg flat and Ahlfors regular domains (2012) - Regularity of Lipschitz free boundaries in two-phase problems for the $p$-Laplace operator (2010)
- Boundary behavior and the Martin boundary problem for $p$ harmonic functions in Lipschitz domains (2010)
- The Skorohod oblique reflection problem in time-dependent domains (2010)
- Boundary behaviour for $p$ harmonic functions in Lipschitz and starlike Lipschitz ring domains (2007)
- Caloric measure in parabolic flat domains (2004)

### Recent publications

- Higher order interpolative geometries and gradient regularity in evolutionary obstacle problems (2024)
- The Kato square root problem for weighted parabolic operators (2023)
- A note on fundamental solutions and Gaussian bounds for degenerate parabolic equations with time-dependent coefficients (2023)
- A Galerkin type method for kinetic Fokker-Planck equations based on Hermite expansions (2023)
- Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets (2023)

### All publications

#### Articles

- Higher order interpolative geometries and gradient regularity in evolutionary obstacle problems (2024)
- The Kato square root problem for weighted parabolic operators (2023)
- A note on fundamental solutions and Gaussian bounds for degenerate parabolic equations with time-dependent coefficients (2023)
- A Galerkin type method for kinetic Fokker-Planck equations based on Hermite expansions (2023)
- Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets (2023)
- Corona Decompositions for Parabolic Uniformly Rectifiable Sets (2023)
- Solvability of the L^p Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in thecase of a parabolic Lipschitz graph (2023)
- Tug-of-war with Kolmogorov (2023)
- Asymptotic mean value formulas, nonlocal space-time parabolic operators and anomalous tug-of-war games (2023)
- On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients (2023)
- A structure theorem for elliptic and parabolic operators with applications to homogenization of operators of Kolmogorov type (2023)
- On local regularity estimates for fractional powers of parabolic operators with time-dependent measurable coefficients (2023)
- Square function estimates for the evolutionary p-Laplace equation (2023)
*Solving the Dirichlet problem for the Monge-Ampère equation using neural networks*(2023)- The Kato square root problem for parabolic operators with an anti-symmetric part in BMO (2022)
- On big pieces approximations of parabolic hypersurfaces (2022)
- CORONIZATIONS AND BIG PIECES IN METRIC SPACES (2022)
- Deep learning, stochastic gradient descent and diffusion maps (2022)
- Potential theory for a class of strongly degenerate parabolic operators of Kolmogorov type with rough coefficients (2022)
- Hawkes-Based Models for High Frequency Financial Data (2022)
- Neural ODEs as the deep limit of ResNets with constant weights (2021)
- Neural networks as smooth priors for inverse problems for PDEs (2021)
- On the fine properties of parabolic measures associated to strongly degenerate parabolic operators of Kolmogorov type (2021)
- The Dirichlet problem for Kolmogorov-Fokker-Planck type equations with rough coefficients (2021)
- L2 well-posedness of boundary value problems and the Kato square root problem for parabolic systems with measurable coefficients (2020)
- Boundary behavior of solutions to the parabolic p-Laplace equation (2019)
- Data-driven discovery of PDEs in complex datasets (2019)
- The Dirichlet problem for second order parabolic operators in divergence form (2018)
- A unified deep artificial neural network approach to partial differential equations in complex geometries (2018)
- Quasi-linear PDEs and low-dimensional sets (2018)
- A new characterization of chord-arc domains (2017)
*L*^{2}Solvability of boundary value problems for divergence form parabolic equations with complex coefficients (2017)- The A∞-Property of the Kolmogorov Measure (2017)
- Tug-of-war, market manipulation, and option pricing (2017)
- Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients (2016)
- Square function estimates and the Kato problem for second order parabolic operators in Rn+1 (2016)
- Reflected BSDE of Wiener-Poisson type in time-dependent domains (2016)
- Kolmogorov-Fokker-Planck Equations (2016)
- Extension properties and boundary estimates for a fractional heat operator (2016)
- The Hadamard variational formula and the Minkowski problem for p-Capacity (2015)
- Optimal switching problems under partial information (2015)
- Boundary estimates for non-negative solutions to non-linear parabolic equations (2015)
- Boundary estimates for solutions to linear degenerate parabolic equations (2015)
- Remarks on the Skorohod Problem and Reflected Levy Driven SDEs in Time-dependent Domains (2015)
- Harnack estimates for degenerate parabolic equations modeled on the subelliptic $p-$Laplacian (2014)
- Estimates for Solutions to Equations of
*p*-Laplace type in Ahlfors regular NTA-domains (2014) - Sharp regularity for evolutionary obstacle problems, interpolative geometries and removable sets (2014)
- A Boundary Harnack Inequality for Singular Equations of p-Parabolic Type (2014)
- Systems of variational inequalities for non-local operators related to optimal switching problems (2014)
- Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type (2014)
- A backward in time Harnack inequality for non-negative solutions to fully non-linear parabolic equations (2014)
- Market Making and Portfolio Liquidation under Uncertainty (2014)
- Cauchy integrals for the $p$-Laplace equation in planar Lipschitz domains (2014)
- Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures (2013)
- A Carleson-type estimate in Lipschitz type domains for non-negative solutions to Kolmogorov equations (2013)
- Boundary behaviour of non-negative solutions to degenerate sub-elliptic equations (2013)
- On the dimension of p-harmonic measure in space (2013)
- p-Harmonic Functions in the Heisenberg Group (2013)
- A framework for the Modeling of Order Book Dynamics based on Event Sizes (2013)
- A boundary estimate for non-negative solutions to Kolmogorov operators in non-divergence form (2012)
- Non-divergence form parabolic equations associated with non-commuting vector fields (2012)
- The Obstacle Problem for Parabolic Non-divergence Form Operators of Hörmander type (2012)
- Regularity and free boundary regularity for the
*p*-Laplace operator in Reifenberg flat and Ahlfors regular domains (2012) - Regularity of flat free boundaries in two-phase problems for the
*p*-Laplace operator (2012) - On an Inverse Type Problem for the Heat Equation in Parabolic Regular Graph Domains (2012)
- Boundary estimates for solutions to operators of $p$-Laplace type with lower order terms (2011)
- Optimal doubling, Reifenberg flatness and operators of p-Laplace type (2011)
- New results for $p$ harmonic functions (2011)
- p-Harmonic Measure in Simply Connected Domains (2011)
- The boundary Harnack inequality for solutions to equations of Aronsson type in the plane (2011)
- A note on Harnack inequalities and propagation sets for a class of hypoelliptic operators (2010)
- Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type (2010)
- Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options (2010)
- Regularity of Lipschitz free boundaries in two-phase problems for the $p$-Laplace operator (2010)
- Boundary behavior and the Martin boundary problem for $p$ harmonic functions in Lipschitz domains (2010)
- Regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolic operators (2010)
- The Skorohod oblique reflection problem in time-dependent domains (2010)
- Pricing and hedging of financial derivatives using a posteriori error estimates and adaptive methods for stochastic differential equations (2010)
- Weak approximation of obliquely reflected diffusions in time-dependent domains (2010)
- On a two-phase free boundary condition for $p$-harmonic measures (2009)
- On Monte Carlo algorithms applied to Dirichlet problems for parabolic operators in the setting of time-dependent domains (2009)
- Boundary Harnack inequalities for operators of p-Laplace type in Reifenberg flat domains (2008)
- Boundary behaviour of $p$-harmonic functions in domains beyond Lipschitz domains (2008)
- The boundary Harnack inequality for infinity harmonic functions in the plane (2008)
- Regularity and free boundary regularity for the $p$ Laplacian in Lipschitz and $C^1$ domains (2008)
- On area integral estimates for solutions to parabolic systems in time-varying and non-smooth cylinders (2008)
- On the behaviour near expiry for multi-dimensional American options (2008)
- On deposit volumes and the valuation of non-maturing liabilities (2008)
- Harmonic analysis, quadratic forms and asymptotic expansions of risk measures (2008)
- Boundary behaviour for $p$ harmonic functions in Lipschitz and starlike Lipschitz ring domains (2007)
- On a parabolic symmetry problem (2007)
- On the Rating and Pricing of Mortgage Portfolios through Structured Finance (2007)
- Free boundary regularity for multi-dimensional American options through blow-ups and global solutions (2007)
- Regularity below the continuous threshold in a two-phase parabolic free boundary problem (2006)
- Boundary value problems and duality between $L^p$ Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains (2006)
- On blow-ups and the classification of global solutions to parabolic free boundary problems (2006)
- Caloric measure and Reifenberg flatness (2006)
- A Credit Risk Model for Large Dimensional Portfolios with Application to Economic Capital (2006)
- Theory and Practice in Risk Based Capital Assessment Methodology (2005)
- Efficient Filtering of Financial Time Series and Extreme Value Theory (2005)
- Caloric measure in parabolic flat domains (2004)
- Towards Operational Risk Management (2004)
- Existence of big pieces of graphs for parabolic problems (2003)
- Dirichlet problems for a nonstationary linearized system of Navier-Stokes equations in non-cylindrical domains (2002)
- Boundary value problems for parabolic Lamé systems in time-varying domains (1999)
- Unique continuation on the boundary for Dini domains (1998)
- An estimate of a polynomial capacity (1998)
- The Dirichlet problem for second order parabolic operators (1997)
- Integrability of Green potentials in fractal domains (1996)