Convex and Discrete Geometry

Univ.-Prof. Dr. Monika Ludwig

Short Curriculum Vitae

• 1990: Master degree (Dipl.-Ing.), Technische Universität Wien
  
• 1990 - 1994: Vertragsassistentin, Technische Universität Wien
• 1994: Doctor of Technical Sciences sub auspiciis praesidentis, Technische Universität Wien
• 1994 - 2000: Assistant Professor, Technische Universität Wien
• 1998: Hlawka-Prize of the Austrian Academy of Sciences
• 1999 - 2000: Visiting Scholar, University College London
• 2000: Habilitation, Technische Universität Wien
• 2000 - 2001: Visiting Scholar, Polytechnic University New York
• 2000 - 2007: Associate Professor, Technische Universität Wien
• 2/2002 - 6/2002: Visiting Professor, University of Bern
  
• 2004: Förderungspreis of the Austrian Mathematical Society
• 2007 - 2010: Professor of Mathematics, Polytechnic Institute of New York University
  
• since 2010: Professor of Mathematics, Technische Universität Wien
  
• 2011 - 2013: Corresponding Member of the Austrian Academy of Sciences
  
• 2012: Fellow of the American Mathematical Society, Inaugural Class
  
• 2013 - 2015: Department Head, Institute for Discrete Mathematics and Geometry
  
• since 2013: Full Member of the Austrian Academy of Sciences
  
• 8/2017 - 12/2017: Research Professor, MSRI, Berkeley.

Projects and Fellowships

• 2024 - 2028: FWF Project Anisotropic fractional perimeters and Riesz energies (Award number P 37030)
• 2021 - 2025: FWF Project Valuations on convex functions (Award number P 34446)
• 2016 - 2021: FWF Project Valuations on lattice polytopes (Award number I 3027; joint with Károly J. Böröczky)
• 2013 - 2017: FWF Project Valuations on functions spaces (Award number P25515)
• 2012 - 2014: FWF Project Linearly intertwining maps on convex bodies (Award number P23639; joint with Christoph Haberl)
• 2010: NSF Special Meeting Award: Asymptotic geometric analysis (Award number 0963819)
• 2008 - 2011: NSF Project Affine geometric analysis (Award number 0805623)
• 2008: NSF Award The State of Geometry and Functional Analysis; Tel Aviv 2009 (Award number 0901911)
• 2005 - 2007: FWF Project Valuations on convex bodies (Award number P18308; joint by Matthias Reitzner)
• 2005 - 2007: Scientist in charge of the Austrian node of the EU Research and Training Network Phenomena in High Dimensions
  
• 2003 - 2007: FWF Project Affinely associated bodies (Award number P16547)
• 2000 - 2001: Erwin Schrödinger Fellowship of the FWF (Award number J1961)
• 1999 - 2000: Erwin Schrödinger Fellowship of the FWF (Award number J1769)

Selected Talks 

• Geometric Valuation Theory,
  Plenary address at the 8TH EUROPEAN CONGRESS OF MATHEMATICS, Portorož, Slovenia, 2021
• On the Geometric Classification of Functions,
  Fields Medal Symposium, Fields Institute, Toronto, 2014
• Additive functions on convex bodies
  Invited address at the joint meeting of the Austrian and German Mathematical Societies, Graz, 2009
• SL(n) invariant notions of surface area
  Invited address at the AMS Sectional Meeting, Wesleyan University, 2008  

PhD Students

• 2017 - 2021: Andreas Kreuml
  Thesis: "Fractional perimeters and symmetrization"
• 2015 - 2017: Fabian Mussnig
  Thesis: "Valuations on Convex and Log-concave Functions"
• 2014 - 2017: Laura Silverstein
  Thesis: "Tensor Valuations on Lattice Polytopes"
• 2013 - 2015: Michael Ober
  Thesis: "L_p-Minkowski Valuations on Function Spaces"
• 2012 - 2015: Dan Ma
  Thesis: "Analysis of Sobolev Spaces in the Context of Convex Geometry and Investigations of the Busemann-Petty Problem"
• 2010 - 2013: Tuo Wang
  Thesis: "Affine Sobolev Inequalities"
• 2008 - 2010: Andy Tsang
  Thesis: "Valuations on L_p-Spaces"
• 2005 - 2007: Christoph Haberl
  Thesis: "Valuations and the Dual L_p Brunn-Minkowski Theory"
• 2003 - 2005: Franz Schuster
  Thesis: "Convolutions and Multiplier Transformations of Convex Bodies"

Editorial Work

• since 2024: Member of the Editorial Board of the Canadian Journal of Mathematics
• since 2024: Member of the Editorial Board of the Canadian Mathematical Bulletin
• since 2022: Member of the Editorial Board of Discrete & Computational Geometry
• since 2021: Member of the Editorial Board of the Monatshefte für Mathematik
• since 2007: Member of the Editorial Board of the Advances in Applied Mathematics
• 2023: Convex Geometry. Cetraro, Italy 2021
  Lecture Notes in Mathematics 2332, CIME Foundation Subseries, Spinger, Cham.
  Editor, joint with Andrea Colesanti.
• 2013: Asymptotic Geometric Analysis. Proceedings of the Fall 2010 Fields Institute Thematic Program.
  Fields Institute Communications, Vol. 68, London, Springer.
  Editor, joint with Vitali Milman, Vladimir Pestov and Nicole Tomczak-Jaegermann.

Conference Organization

• 9/2023: Convex Geometry and Geometric Probability at the Paris Lodron University Salzburg
  
• 6/2023: Convex Geometry - Analytic Aspects at Cortona
  
• 6/2023: Geometric Valuation Theory - from convex sets to functions at CIEM Castro Urdiales
  
• 4/2022: Szeged Workshop on Convexity at the University of Szeged
  
• 12/2021: Convex Geometry and its Application in Oberwolfach
  
• 8/2021: Summer School: Convex Geometry in Cetraro
  
• 9/2019: Convex and Integral Geometry, Special Session at DGA2019 in Hradec Králové
  
• 5/2019: Analytic-Geometric Inequalities and Related Topics at the Institut Mittag-Leffler in Stockholm
  
• 4/2019: Szeged Workshop on Convexity at the University of Szeged
  
• 12/2018: Convex Geometry and its Application in Oberwolfach
  
• 8/2017: Introductory Workshop: Phenomena in High Dimensions at MSRI, Berkeley
  
• 7/2016: Conference on Convex and Discrete Geometry in Vienna
• 12/2015: Convex Geometry and its Application in Oberwolfach
  
• 11/2015: Szeged Workshop on Convexity in Szeged
  
• 9/2015: Affine Geometric Analysis in Oaxaca, Mexico
  
• 12/2012: Convex Geometry and its Application in Oberwolfach
  
• 4/2012: Convex Geometry and Asymptotic Geometric Analysis at CRM, Montreal
  
• 10/2011: Colloquium on the Occasion of the 70th Birthday of Peter M. Gruber in Vienna
  
• 9/2011: Conference on Convex and Integral Geometry at the Goethe University Frankfurt
  
• 9/2010: Workshop on Asymptotic Geometric Analysis and Convexity at the Fields Institute, Toronto
  
• 11/2009: Convex Geometry and its Application in Oberwolfach
  
• 7/2009: Conference on Convex and Discrete Geometry in Vienna
• 6/2009: The State of Geometry and Functional Analysis in Tel Aviv
  
• 1/2009: Affine Convex Geometric Analysis in Banff
• 10/2008: Special Session on Convex and Integral Geometry in Middletown, CT
  
• 7/2005: Convex Geometry and High Dimensional Phenomena in Vienna
  

Preprints

• Unimodular valuations beyond Ehrhart
  Joint with Ansgar Freyer and Martin Rubey.
• Affine Hardy-Littlewood-Sobolev inequalities
  Joint with Julián Haddad.

Publications

• Affine fractional Sobolev and isoperimetric inequalities
  JOURNAL OF DIFFERENTIAL GEOMETRY, in press.
  Joint with Julián Haddad.
• The Hadwiger theorem on convex functions. II
  AMERICAN JOURNAL OF MATHEMATICS, in press.
  Joint with Andrea Colesanti and Fabian Mussnig.
• The Hadwiger theorem on convex functions. I
  GEOMETRIC AND FUNCTIONAL ANALYSIS 34, 1839-1898 (2024).
  Joint with Andrea Colesanti and Fabian Mussnig.
• Affine Lp fractional Sobolev inequalities
  MATHEMATISCHE ANNALEN 388, 1091-1115 (2024).
  Joint with Julián Haddad.
• Valuations on convex bodies and functions
  In: Convex Geometry, Lecture Notes in Mathematics 2332,
  CIME Foundation Subseries (pp. 19-78), Springer, Cham (2023).
  Joint with Fabian Mussnig.
• Geometric valuation theory
  In: A. Hujdurovic, K. Kutnar, D. Marusic, S. Miklavic, T. Pisanski, & P. Sparl (Eds.),
  EUROPEAN CONGRESS OF MATHEMATICS (pp. 93-123). EMS Press (2023).
• The Hadwiger theorem on convex functions, IV: The Klain approach
  ADVANCES IN MATHEMATICS 413, 108832 (2023).
  Joint with Andrea Colesanti and Fabian Mussnig.
• The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Ampère measures
  CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS 61, Art. 181 (2022).
  Joint with Andrea Colesanti and Fabian Mussnig.
• A homogeneous decomposition theorem for valuations on convex functions
  JOURNAL OF FUNCTIONAL ANALYSIS 279, Art. 108573 (2020).
  Joint with Andrea Colesanti and Fabian Mussnig.
• Hessian valuations
  INDIANA UNIVERSITY MATHEMATICS JOURNAL 69, 1275-1315 (2020).
  Joint with Andrea Colesanti and Fabian Mussnig.
• Minkowski valuations on lattice polytopes
  JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY (JEMS) 21, 163-197 (2019).
  Joint with Károly J. Böröczky.
• Valuations on convex functions
  INTERNATIONAL MATHEMATICS RESEARCH NOTICES NOTICES 2019, No. 8, 2384-2410 (2019).
  Joint with Andrea Colesanti and Fabian Mussnig.
• Weighted floating bodies and polytopal approximation
  TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 370, 7129-7148 (2018).
  Joint with Florian Besau and Elisabeth Werner.
• Minkowski valuations on convex functions
  CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS 56, Art. 162 (2017).
  Joint with Andrea Colesanti and Fabian Mussnig.
• Tensor valuations on lattice polytopes
  ADVANCES IN MATHEMATICS 319, 76-110 (2017).
  Joint with Laura Silverstein.
• Valuations on lattice polytopes
  In: Tensor Valuations and their Applications in Stochastic Geometry and Imaging (M. Kiderlen and E. Vedel Jensen, eds.),
  SPRINGER LECTURE NOTES IN MATHEMATICS 2177, 213-234 (2017).
  Joint with Károly J. Böröczky.
• SL(n) invariant valuations on polytopes
  DISCRETE AND COMPUTATIONAL GEOMETRY 57, 571 - 581 (2017).
  Joint with Matthias Reitzner.
• Anisotropic fractional perimeters
  JOURNAL OF DIFFERENTIAL GEOMETRY 96, 77 - 93 (2014).
• Anisotropic fractional Sobolev norms
  ADVANCES IN MATHEMATICS 252, 150 - 157 (2014).
• Covariance matrices and valuations
  ADVANCES IN APPLIED MATHEMATICS 51, 359-366 (2013).
• Valuations on Sobolev spaces
  AMERICAN JOURNAL OF MATHEMATICS 134, 824 - 842 (2012).
• Fisher information and matrix-valued valuations
  ADVANCES IN MATHEMATICS 226, 2700 - 2711 (2011).
• Sharp convex Lorentz-Sobolev inequalities
  MATHEMATISCHE ANNALEN 350, 169 - 197 (2011).
  Joint with Jie Xiao and Gaoyong Zhang.
• Valuations on function spaces
  ADVANCES IN GEOMETRY 11, 745 - 756 (2011).
• A classification of SL(n) invariant valuations
  ANNALS OF MATHEMATICS 172, 1219 - 1271 (2010).
  Joint with Matthias Reitzner.
• Minkowski areas and valuations
  JOURNAL OF DIFFERENTIAL GEOMETRY 86, 133 - 161 (2010).
• General affine surface areas
  ADVANCES IN MATHEMATICS 224, 2346 - 2360 (2010).
• Intersection bodies and valuations
  AMERICAN JOURNAL OF MATHEMATICS 128, 1409 - 1428 (2006).
• A characterization of Lp intersection bodies
  INTERNATIONAL MATHEMATICS RESEARCH NOTICES, Art ID 10548, 1 - 29 (2006).
  Joint with Christoph Haberl.
• Elementary moves on triangulations
  DISCRETE & COMPUTATIONAL GEOMETRY 35, 527 - 536 (2006).
  Joint with Matthias Reitzner.
• Valuations in the affine geometry of convex bodies
  PROCEEDINGS OF THE CONFERENCE ''INTEGRAL GEOMETRY AND CONVEXITY'',
  Wuhan 2004, World Scientific, Singapore, 49 - 65 (2006).
• Approximation of the Euclidean ball by polytopes
  STUDIA MATHEMATICA 173, 1 - 18 (2006).
  Joint with Carsten Schütt and Elisabeth Werner.
• Minkowski valuations
  TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 357, 4191 - 4213 (2005).
• Ellipsoids and matrix valued valuations
  DUKE MATHEMATICAL JOURNAL 119, 159 - 188 (2003).
• Projection bodies and valuations
  ADVANCES IN MATHEMATICS 172, 158 - 168 (2002).
• Moment vectors of polytopes
  RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO (2) Suppl. 70, part II, 123 - 138 (2002).
• Valuations on polytopes containing the origin in their interiors
  ADVANCES IN MATHEMATICS 170, 239 - 256 (2002).
• On the semicontinuity of curvature integrals
  MATHEMATISCHE NACHRICHTEN 227, 99 - 108 (2001).
• Upper semicontinuous valuations on the space of convex discs
  GEOMETRIAE DEDICATA 80, 263 - 279 (2000).
• A characterization of affine surface area
  ADVANCES IN MATHEMATICS 147, 138 - 172 (1999).
  Joint with Matthias Reitzner.
• A characterization of affine length and asymptotic approximation of convex discs
  ABHANDLUNGEN AUS DEM MATHEMATISCHEN SEMINAR DER UNIVERSITÄT HAMBURG 69, 75 - 88 (1999).
• Asymptotic approximation of smooth convex bodies by general polytopes
  MATHEMATIKA 46, 103 - 125 (1999).
• Approximation of convex bodies and a momentum lemma for power diagrams
  MONATSHEFTE DER MATHEMATIK 127, 101 - 110 (1999).
  Joint with Károly Böröczky, jr.
• Asymptotic approximation by quadratic spline curves
  ANNALES UNIVERSITATIS SCIENTIARUM BUDAPESTINENSIS. SECTIO MATHEMATICA 42, 133 - 139 (1999).
• Asymptotic approximation of convex curves; the Hausdorff metric case
  ARCHIV DER MATHEMATIK 70, 331 - 336 (1998).
• A Helmholtz-Lie type characterization of ellipsoids, II
  DISCRETE & COMPUTATIONAL GEOMETRY 16, 55 - 67 (1996).
  Joint with Peter M. Gruber.
• Asymptotic approximation of convex curves
  ARCHIV DER MATHEMATIK 63, 377 - 384 (1994).