## 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)