Research Interests: Geometry (algebraic, convex, discrete, real)
- Positive polynomials and sums of squares
- Convex, semi-algebraic sets
- Positive Geometry
PhD students
- Joris Köfler (IMPRS, MPI MiS)
- Julian Weigert (SPP Combinatorial Synergies, starting 10/2024)
- Jannik Wesner (jointly supervised with Daniel Plaumann)
Postdocs
- Lorenzo Baldi (Humboldt Postdoctoral Fellowship, June 2024 to May 2026)
Graduated students
- Marie-Charlotte Brandenburg
- Chiara Meroni (jointly supervised with Bernd Sturmfels)
- Laith Rastanawi (jointly supervised with Günter Rote)
- Hannah Sjöberg (jointly supervised with Günter Ziegler) at Freie Universität Berlin
Affiliations
Scientific Member of the International Max Planck Research School Mathematics in the Sciences (IMPRS MiS)Grants
- Co-Spokesperson of MaRDI for TA6 (Data Culture and Community Integration), funding period 2021 - 2026
Mathematik in der Nationalen Forschungsdateninfrastruktur, with Christiane Görgen, Mitteilungen der DMV, 29(3), 2021 - DFG project Positive Geometry as part of the priority program SPP 2458 Combinatorial Synergies, funding period 2024 - 2027
- DFG project Geometry of Hyperbolic Polynomials (with Daniel Plaumann), funding period 2019 - 2022
- SAXAG: Saxonian Algebraic Geometry (incubator grant of the Saxonian Ministry of Science and Art), funding period 2022
SIAM Journal on Applied Algbera and Geometry
I am associate editor for the SIAM Journal of Applied Algebra and Geometry (first term from January 1, 2024).Publications
Book
Panorama der Mathematik with Andreas Loos and Günter M. Ziegler, Springer Berlin, 2022.see also Was macht Mathematik effektiv? in Physik in unserer Zeit, 54, No. 4, 176-183 (2023)
Articles
Here are links to my publication record on arXiv.org, MathSciNet, and zbMATH. In the following list, original research articles (accepted for publication) are counted as [Z..] and survey articles as [S..].- Spurious local minima in nonconvex sum-of-squares optimization
with Grigoriy Blekherman, Mauricio Velasco & Shixuan Zhang
Preprint (2024) - Pythagoras Numbers for Ternary Forms
with Grigoriy Blekherman & Alex Dunbar
Preprint (2024) - Nonnegativity certificates on real algebraic surfaces
with Grigoriy Blekherman, Greg Smith & Mauricio Velasco
Preprint (2024) - Plane quartics and heptagons
with Daniele Agostini, Daniel Plaumann & Jannik Lennart Wesner
Preprint (2024) - Nonnegative Polynomials and Moment Problems on Algebraic Curves
with Lorenzo Baldi & Grigoriy Blekherman
Preprint (2024) -
Weak and Strong Extremal Biquadratics
with Grigoriy Blekherman, Bogdan Raiță & Isabelle Shankar
Preprint (2022) -
Moments, Sums of Squares, and Tropicalization
with Grigoriy Blekherman, Felipe Rincon, Cynthia Vinzant & Josephine Yu
Preprint (2022) -
Do Alcoved Lattice Polytopes Have Unimodal h*-Vector?
with Hannah Sjöberg
Preprint (2021) -
[Z26] Adjoints and Canonical Forms of Polypols
with Kathlén Kohn, Ragni Piene, Kristian Ranestad, Felix Rydell, Boris Shapiro, Miruna-Stefana Sorea & Simon Telen
to appear in Documenta Mathematica - [Z25] Adjoints and Canonical Forms of Tree Amplituhedra with Kristian Ranestad & Simon Telen
Mathematica Scandinavica, 130(3), 433-466 (2024) -
[S2] Landau on Chess Tournaments and Google's PageRank
with Günter M. Ziegler
to appear in Edmund Landau: Collected Works, Volume 10, edited by A. Hüntelmann, A. Kreuzer, H. Wefelscheid and U. Wefelscheid -
[Z24] Convex hulls of surfaces in fourspace
with Chiara Meroni & Kristian Ranestad
Collectanea Mathematica (2024) -
[Z23] Tropical Positivity and Determinantal Varieties
with Marie Brandenburg & Georg Loho
Algebraic Combinatorics, 6(4), 999-1040 (2023) -
[Z22] Families of Faces and the Normal Cycle of a Convex Semi-algebraic Set
with Daniel Plaumann & Jannik Lennart Wesner
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 64, 851-875 (2023) -
[Z21] The Chiral Domain of a Camera Arrangement
with Sameer Agarwal, Andrew Pryhuber & Rekha Thomas
Journal of Mathematical Imaging and Vision, 64, 948-967 (2022) -
[Z20] On the Existence of Two View Chiral Reconstructions
with Andrew Pryhuber & Rekha Thomas
SIAM Journal on Applied Algebra and Geometry, 6, No. 1, 41-76 (2022) -
[Z19] Hyperbolic Secant Varieties of M-Curves
with Mario Kummer
Journal für die Reine und Angewandte Mathematik, 787, 125-162 (2022) -
[Z18] Sums of Squares and Quadratic Persistence on Real Projective Varieties
with Grigoriy Blekherman, Gregory G. Smith & Mauricio Velasco
Journal of the EMS, 24, No. 3, 925-965 (2022) -
[Z17] Maximum Likelihood Estimation for Nets of Conics
with Stefan Dye, Kathlén Kohn & Felix Rydell
Le Matematiche, 76, No. 2, 399-414 (2021) -
[S1] Sums of Squares: A real projective story
with Grigoriy Blekherman, Gregory G. Smith & Mauricio Velasco
Notices of the AMS, 68, No. 5, 734-747 (2021) -
[Z16] On the Dimensions of the Realization Spaces of Polytopes
with Laith Rastanawi & Günter M. Ziegler
Mathematika, 67, No. 2, 342-365 (2021) -
[Z15] Conic Programming: Infeasibility Certificates and Projective Geometry
with Simone Naldi
Journal of Pure and Applied Algebra, 225, No. 7, 22 pages (2021) -
[Z14] Kippenhahn's Theorem for Joint Numerical Ranges and Quantum States
with Daniel Plaumann & Stephan Weis
SIAM Journal on Applied Algebra and Geometry, 5, No. 1, 86-113 (2021) -
[Z13] Combinatorial inscribability obstructions for higher-dimensional polytopes
with Joseph Doolittle, Jean-Philippe Labbé, Carsten Lange, Jonathan Spreer & Günter M. Ziegler
Mathematika, 66, No. 4, 927-953 (2020) -
[Z12] Typical and Generic Ranks in Matrix Completion
with Daniel Bernstein & Grigoriy Blekherman
Linear Algebra and Applications, 585, 71-104 (2020) -
[Z11] Positive Semidefinite Univariate Matrix Polynomials
with Christoph Hanselka
Mathematische Zeitschrift, 292, No. 1-2, 83-101 (2019)
Read-only link -
[Z10] Maximum Likelihood Threshold and Generic Completion Rank of Graphs
with Grigoriy Blekherman
Discrete and Computational Geometry, 61, No. 2, 303-324 (2019) -
[Z9] Gram Spectrahedra
with Lynn Chua, Daniel Plaumann & Cynthia Vinzant
in Ordered Algebraic Structures and Related Topics, vol. 697 of Contemporary Mathematics, 81-105. American Mathematical Society, Providence, RI (2017) -
[Z8] Low-Rank Sum-of-Squares Representations on Varieties of Minimal Degree
with Grigoriy Blekherman, Daniel Plaumann & Cynthia Vinzant
International Mathematical Research Notices, 2019, No. 1, 33-54 (2019) -
[Z7] Do Sums of Squares Dream of Free Resolutions?
with Grigoriy Blekherman & Mauricio Velasco
SIAM Journal on Applied Algebra and Geometry, 1, No. 1, 175-199 (2017) -
[Z6] Extreme Rays of the Hankel Spectrahedra for Ternary Forms
with Grigoriy Blekherman
Journal of Symbolic Computation, 79, Part 1, 23-42 (2017) -
[Z5] Real Rank with Respect to Varieties
with Grigoriy Blekherman
Linear Algebra and its Applications, 505, 344-360 (2016) -
[Z4] Computing hermitian determinantal representations of hyperbolic curves
with Daniel Plaumann, David E Speyer & Cynthia Vinzant
Our implementation of the algorithm in Mathematica is available here.
International Journal on Algebra and Computation, 25, No. 8, 1327-1336 (2015) -
[Z3] Generic Spectrahedral Shadows
with Bernd Sturmfels
SIAM Journal on Optimization, 25, No. 2, 1209-1220 (2015) -
[Z2] Algebraic Boundaries of Convex Semi-algebraic Sets
Research in the Mathematical Sciences, 2, No. 1 (2015)
-
[T] Algebraic Boundaries of Convex Semi-algebraic Sets
PhD thesis (Universität Konstanz, 2014) -
[Z1] Algebraic Boundaries of SO(2)-Orbitopes
Discrete and Computational Geometry, 50, No. 1, 219-235 (2013) -
[A1] A Note on the Convex Hull of Finitely Many Projections of Spectrahedra
Tim Netzer
Appendix of [Z15]