Mathias Braun
Mathias Braun
Bernoulli Instructor @ EPFL
Institute of Mathematics, EPFL, 1015 Lausanne, Switzerland
German, English, French
Workshop “MDPDE 2026”

Welcome to my homepage! I am a mathematician working in probability, nonlinear PDEs, and mathematical relativity. My tools come from optimal transport, gradient flows, and stochastic, geometric, and numerical analysis. Highlights of my work include the discovery of ellipticity of a nonlinear \(p\)-d'Alembert operator together with elliptic proofs of the Lorentzian splitting theorems, the development of a second-order vector calculus on Dirichlet spaces, and sharp pathwise estimates for classical or nonlinear Brownian motion. I am also interested in AI4Math.

Currently, I am a Bernoulli Instructor @ EPFL. My mentor is Professor Martin Hairer. Before, I held a Postdoctoral Fellowship in the group of Professor Robert McCann @ Department of Mathematics, University of Toronto. In between, I was a Fields Postdoctoral Fellow @ Fields Institute for Research in Mathematical Sciences, advised by Professors Robert Haslhofer, Vitali Kapovitch, Robert McCann, and Adrian Nachman. I was associated with the Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry. I completed my doctorate advised by Professor Theo Sturm @ Institute for Applied Mathematics, University of Bonn.

If you have any questions, do not hesitate to get in touch with me!

News

Visit my LinkedIn profile to read more posts and thoughts!

1 October 2026

Why should we continue learning cognitive skills that AI can or will do better than humans?

Let me give you three reasons.

1️⃣ To use AI well, you need expertise to understand the subject. The best prompts come from people who know what they want. In my research, a prompt guided by intuition and a specific idea clearly outperforms a generic "solve this problem". The output has much more impact on my process of understanding. On the other hand, even when AI produces a valid solution, without relevant background it takes considerable time before you can make use of it.

2️⃣ Learning changes you, not just your knowledge. AI can produce a result, but it often doesn‘t expose the experience of the path that leads there: the dead ends, the rough terrain, the moments where something suddenly makes sense. That process of realizing what works and what doesn‘t builds intuition, judgment, and knowledge.

3️⃣ Even where AI is better than humans, the distance remains smallest in what I believe will continue to matter for human influence: judgment, contextualization, and unexpected connections. From this perspective, learning means getting good where it counts. And maybe you'll end up contributing to the breathtaking growth of AI itself.

🌱 Don't let insecurity about the rapid growth of AI influence your decisions for or against an education path, a degree, a new skill, etc. The best investment you can make is in yourself. No one has ever advanced by standing still.

#AI #Education #Learning #PhDLife #Research

15 September 2026

〰️ The radial part of Brownian motion has profound connections to the geometry of the space it lives in. And its nonlinear cousin?

My new preprint "The radial part of p-Brownian motion" is out. It initiates a geometric theory of p-Brownian motion, the nonlinear Markov process associated with the p-Laplacian, introduced by Barbu–Rehmeier–Röckner. The centerpiece is an analysis of its radial process: Tanaka–Meyer semimartingale formulas, a characterization of its local time at the center (with notable differences to Brownian motion), and sharp exit time and scaling estimates.

One thing I enjoy about this project: it sits at the intersection of stochastic analysis, PDEs, and geometry. Letting ideas from different fields interact is one of the most rewarding aspects of mathematics.

The timing is also convenient. The paper will be on the table at the upcoming workshop "Nonlinear Markov processes, Dirichlet forms, and PDEs" I coorganize with Giovanni Brigati, PhD, Lorenzo Dello Schiavo, and Marco Rehmeier this October. Link in the comments. Looking forward to discussing it!

#Mathematics #Research #StochasticAnalysis #PDEs #Geometry

26 August 2026

I'm excited to share that my work "Exact d'Alembertian for Lorentz distance functions" has been accepted for publication in Calculus of Variations and Partial Differential Equations! I would like to thank the reviewers for their thorough assessment and constructive suggestions. One of their comments led to a generalization of one of the main results, a volume incompleteness theorem stated in Theorem 6.31. I'm grateful for the careful feedback and happy to see this work moving towards publication.

#Mathematics #LorentzianGeometry #GeometricAnalysis #CalculusOfVariations

Publications

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Refereeing. I am a reviewer in the AMS Mathematical Reviews® community.

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Instructorships @ EPFL

Instructorships @ University of Toronto

Assistantships @ University of Bonn

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Education