Research Interests
- Positive polynomials
- Convex algebraic geometry
- Classical algebraic geometry
- Algebraic foundations of optimization
DFG-Project Geometry of Hyperbolic Polynomials
(joint with Rainer Sinn, FU Berlin)"At the heart of this project are hyperbolic polynomials and their hyperbolicity cones. These are real polynomials in several variables bounding a convex cone, with a particular reality condition on the roots. They originate in complex analysis and the theory of partial differential equations, but have more recently arisen in optimization, combinatorics, and probability theory. [..]"
Advances in Geometry
I am a member of the editorial board of Advances in Geometry. The journal covers a wide range of topics in geometry. (Please send submissions to the managing editor, following instructions on the journal's homepage.)Publications
Liste auf arXiv.org-
Computing totally real hyperplane sections and linear series on algebraic curves
with Huu Phuoc Le and Dimitri Manevich. Le Matematiche Vol. 77(1), 119-141 (2022) -
Families of faces and the normal cycle of a convex semi-algebraic set
with Rainer Sinn and Jannik Lennart Wesner. Beiträge zur Algebra und Geometrie (2022) -
Kippenhahn's Theorem for joint numerical ranges and quantum states
with Rainer Sinn and Stephan Weis. SIAM Journal on Applied Algebra and Geometry, 5, No. 1, 86-113 (2021) -
Testing hyperbolicity of real polynomials
with Papri Dey. Mathematics in Computer Science. 14, 111-121 (2020) -
Spectrahedral representations of plane hyperbolic curves
with Mario Kummer and Simone Naldi. Pacific Journal of Mathematics Vol. 303, No. 1, 243–263 (2019) -
Sixty-Four Curves of
Degree Six
with Nidhi Kaihnsa, Mario Kummer, Mahsa Sayyary Namin, and Bernd Sturmfels. Experimental Mathematics 28(2), 132-150 (2019) -
Symbolic computation in
hyperbolic programming
with Simone Naldi. Journal of Algebra and its Applications 17(10)(2018) -
Gram spectrahedra
with Lynn Chua, Rainer Sinn, and Cynthia Vinzant. Contemporary Mathematics 697 (2017) -
Low-rank
sum-of-squares representations on varieties of minimal degree
with Grigoriy Blekherman, Rainer Sinn, and Cynthia Vinzant. International Mathematics Research Notices 2019(1), 33-54 (2017) -
Toric completions and
bounded functions on real algebraic varieties
with Claus Scheiderer. Journal of the London Mathematical Society (2) 94, 598-616 (2016) -
Computing hermitian determinantal
representations of hyperbolic curves
with Rainer Sinn, David E Speyer, and Cynthia Vinzant. International Journal of Algebra and Computation 25, No. 8, 1327–1336 (2015) -
A relative Grace theorem
for complex polynomials
with Mihai Putinar. Mathematical Proceedings of the Cambridge Philosophical Society, 161, No. 1, 17-30 (2016) -
Hyperbolic polynomials, interlacers, and sums of squares
with Mario Kummer and Cynthia Vinzant. Mathematical Programming 153, no.1, Ser. B, 223-245 (2015). -
Determinantal representations of hyperbolic curves via polynomial
homotopy continuation
with Anton Leykin. Mathematics of Computation 86(308), 2877-2888 (2017). (Supplementary material) -
Determinantal
representations of hyperbolic plane curves: An elementary
approach
with Cynthia Vinzant. Journal of Symbolic Computation 57, No. 1, 48-60 (2013) -
Determinantal
Representations and the Hermite Matrix
with Tim Netzer and Andreas Thom. Michigan Mathematical Journal 62, 407-420 (2013) -
The ring of bounded polynomials on a semi-algebraic set
with Claus Scheiderer. Transactions of the American Mathematical Society 364, No. 9, 4663-4682 (2012). -
Positivity of continuous piecewise polynomials
Bulletin of the London Mathematical Society, 44, No. 4, 749-757 (2012). -
Computing linear matrix
representations of Helton-Vinnikov curves
with Bernd Sturmfels and Cynthia Vinzant. Volume in honor of Bill Helton, Mathematical Methods in Systems, Optimization and Control, (eds. Harry Dym, Mauricio de Oliveira, Mihai Putinar), Operator Theory: Advances and Applications, Birkhäuser, Basel, 2011. (Supplementary material) -
Quartic curves and their bitangents
with Bernd Sturmfels and Cynthia Vinzant. Journal of Symbolic Computation 46, No. 6, 712-733 (2011). (Supplementary material) -
Exposed faces of semidefinitely representable sets
with Tim Netzer and Markus Schweighofer. SIAM Journal on Optimization 20, No. 4, 1944-1955 (2010) -
Sums
of squares on reducible real curves
Mathematische Zeitschrift 265, No. 4, 777-797 (2009) -
Bounded polynomials, sums of squares and the moment
problem
PhD thesis (Universität Konstanz). Konstanzer Online-Publikationssystem (2008)
Text books
- Einführung in die Algebraische Geometrie (Springer-Spektrum, 2020)
- Algebra (Springer-Spektrum, 2023)
- Geometry of Linear Matrix Inequalities (Birkhäuser, 2023)
Lecture Notes
- Real Algebraic Geometry (Short Tutorial at IHP, Paris, 12 October 2023)