Prof. Dr. Karl-Theodor Sturm

Coordinator HCM

E-mail: sturm(at)
Phone: +49 228 73 4874
Fax: +49 228 73 6716
Room: 3.030
Location: Mathematics Center
Institute: Institute for Applied Mathematics
Research Areas: Research Area B1
Research Area B3
Research Area A3

Academic Career


Dr. rer. nat., University of Erlangen-Nürnberg

1989 - 1997

Postdoc, Universities of Zürich (Switzerland), Erlangen-Nürnberg, Bonn; Max Planck Institute, Leipzig


Habilitation, University of Erlangen-Nürnberg

Since 1997

Professor (C3/W2/W3), University of Bonn

Research Profile

My research addresses a broad variety of problems from analysis, geometry and probability. Particular foci in recent years had been questions of optimal transport and effects of curvature in various contexts, including Riemannian manifolds, Finsler spaces, and infinite dimensional spaces (like path spaces, loop spaces, configuration spaces, or spaces of measures). With foundational publications I contributed to establish and to promote a new hot area in mathematics: metric measure spaces with synthetic Ricci bounds. Other research topics are matching problems, random measures, interacting particle systems and their scaling limits, gradient flows, geometric and functional inequalities, stochastic calculus on manifolds.

Major research directions for the future include: precise geometric structure of metric measures spaces with curvature-dimension condition (singular/regular points, boundaries) and their relation to Cheeger-Colding limits and Alexandrov spaces, analysis on time-dependent mm-spaces and their evolutions (e.g. under generalizations of Ricci flow), extension of ‘curvature’ concepts to spaces which so far had been out of reach (including branching or non-convex boundary conditions), stochastic calculus on mm-spaces, higher dimensional random geometries.

Research Projects and Activities

DFG Cluster of Excellence “Hausdorff Center for Mathematics”
Member of the Board of Directors, since 2009, and coordinator, since 2012

DFG Collaborative Research Center SFB 1060 “The Mathematics of Emergent Effects” (projects B3, C1, C5)
Principal investigator and member of the steering committee, since 2013

DFG Collaborative Research Center SFB 611 “Singular Phenomena and Scaling in Mathematical Models” (projects A1, A2, A5, A9)
PI, vice director and member of the steering committee, 2002 - 2012

DFG Priority Program “Interacting stochastic systems of high complexity”
Principal investigator, 1997 - 2003

ERC Advanced Investigator Grant “Metric measure spaces and Ricci curvature - analytic, geometric, and probabilistic challenges”

Organizer of more than 15 International Conferences and Workshops within the last decade, among them e.g.

Trimester Program “Complex Stochastic Systems: Discrete vs. Continuous” at the Hausdorff Research Institute HIM (Bonn) with 5 embedded workshops and 9 lecture series, 2007 - 2008

Workshops at IPAM (Los Angeles), CIRM (Luminy/Marseille), RIMS (Kyoto), Lebesgue Center (Rennes), CRM (Pisa), 2008 – 2013

Ongoing series of Oberwolfach Workshops on “Heat Kernels, Stochastic Processes and Functional Inequalities”, corresponding organizer, 2005, 2013, 2016

International Conference on Stochastic Analysis and Applications (SFB 611, Bonn)
– more than 250 registered participants, 2011

International Conference “New Trends in Optimal Transport” (SFB 1060, Bonn, 2015)

Conference “Panorama of Mathematics” (Bonn),
Organizer, 2015

Intense activity period on “Metric measure spaces and Ricci bounds”, 4 weeks of lecture series and invited/contributed talks at MPIM Bonn, 2017

Contribution to Research Areas

Research Area G
One of the central ongoing topics of my research is the study of (deterministic or random) evolutions in environments of complex geometric structure. The evolutions under consideration might describe functions (e.g. states, particle densities), interacting particle systems, maps, or spaces. The complexity of the environment is due to irregularity, infinite dimensionality, or randomness. A major focus of my research in recent years is on optimal transport and its many applications, in particular, the interpretation of heat flows and other dissipative evolutions as gradient flows of entropy-like functionals on the Wasserstein space. Convexity properties of these functionals yield rates for equilibration and concentration, in many cases with explicit formulas for modifying these rates under tensorization, limits, conformal transformations, or time changes. Coupling of stochastic processes and optimal transport, leads to new deep insights in the study of time-dependent spaces e.g. evolving under (super-) Ricci flow. Important breakthroughs have been achieved in constructing optimal couplings between random measures, e.g. between point processes on <br>mathbb R^n.
Research Area A
The topics of my research which in recent years attracted most international attention and publicity are the synthetic Ricci bounds for metric measure spaces and the far reaching geometry and geometric analysis developed on these spaces.
With two publications [1] and [2], - together with Lott and Villani (2009) who independently presented a similar concept - I laid the foundations to this innovative, flourishing field with many fascinating applications and stimulating interactions. In subsequent years, many further insights and results had been added, among them Ricci bounds for constructions (cones, suspensions), rough curvature bounds for discrete spaces, local-to-global property, essential non-branching, and existence of optimal maps.
As a landmark result, [3] established the equivalence of the `entropic' curvature-dimension condition - defined in terms of optimal transport - with the celebrated `energetic' curvature-dimension condition (or generalized `Bochner's inequality’) introduced already 30 years ago by Bakry-Emery in terms of the so-called <br>Gamma calculus for diffusion operators. Besides this complete equivalence of the Eulerian and the Lagrangian approach to heat flow and regularity issues on mm-spaces, various new estimates (space-time gradient estimate, Wasserstein control, N-log Sobolev inequality) had been derived in [3]. And the paper opened the door for many new developments and initiated various innovative research directions.

Selected Publications

[1] Karl-Theodor Sturm
On the geometry of metric measure spaces. I
Acta Math. , 196: (1): 65--131
DOI: 10.1007/s11511-006-0002-8
[2] Karl-Theodor Sturm
On the geometry of metric measure spaces. II
Acta Math. , 196: (1): 133--177
DOI: 10.1007/s11511-006-0003-7
[3] Matthias Erbar, Kazumasa Kuwada, Karl-Theodor Sturm
On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces
Invent. Math. , 201: (3): 993--1071
DOI: 10.1007/s00222-014-0563-7
[4] Karl-Theodor Sturm
Gradient Flows for Semiconvex Functions on Metric Measure Spaces - Existence, Uniqueness and Lipschitz Continuity
eprint arXiv:1410.3966, to appear in Proc. AMS
[5] Janna Lierl, Karl-Theodor Sturm
Neumann heat flow and gradient flow for the entropy on non-convex domains
eprint arXiv:1704.04164, to appear in Calc.Var.PDE
[6] Eva Kopfer, Karl-Theodor Sturm
Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows
eprint arXiv:1611.02570, accepted for Comm. Pure Appl. Math
[7] Max-K. von Renesse, Karl-Theodor Sturm
Transport inequalities, gradient estimates, entropy, and Ricci curvature
Comm. Pure Appl. Math.
58: (7): 923--940
DOI: 10.1002/cpa.20060
[8] Max-K. von Renesse, Karl-Theodor Sturm
Entropic measure and Wasserstein diffusion
Ann. Probab. , 37: (3): 1114--1191
DOI: 10.1214/08-AOP430
[9] Martin Huesmann, Karl-Theodor Sturm
Optimal transport from Lebesgue to Poisson
Ann. Probab. , 41: (4): 2426--2478
DOI: 10.1214/12-AOP814
[10] Shin-Ichi Ohta, Karl-Theodor Sturm
Bochner-Weitzenböck formula and Li-Yau estimates on Finsler manifolds
Adv. Math. , 252: : 429--448
DOI: 10.1016/j.aim.2013.10.018

Publication List

MathSciNet Publication List (external link)

ArXiv Preprint List (external link)



Habilitationsstipendium (DFG)




Professor Invité, Toulouse III, France


Professor Invité, Paris VI, France


ERC Advanced Grant “Metric measure spaces and Ricci curvature – analytic, geometric, and probabilistic challenges”


“Hirzebruch Professor”, Max Planck Institute for Mathematics, Bonn

Selected Invited Lectures


Plenary speaker, Annual Meeting of the German Mathematical Society (DMV)


Plenary speaker, 34th Conference on Stochastic Processes and Their Applications, Osaka, Japan


Within one year, 6 lecture series (Imperial, SISSA Trieste, SMS Montreal, Top-Math Munich, Midwest Probability Evanston, UK Easter Warwick)


International conference dedicated to the centenary of L. V. Kantorovich, St. Petersburg, Russia


Fields Medal Symposium in honour of C. Villani, Toronto, ON, Canada


International conference dedicated to the centenary of K. Ito, Kyoto, Japan



Chair Professor, Imperial College, UK


Distinguished Professor, Kansas University, USA


Full Professor, PennState University, USA


Full Professor, Northwestern University, USA

Supervised Theses

  • Diplom theses: 35
  • PhD theses: 13, currently 3
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