Research Interests

I work on applied algebra. My research covers a broad range of topics; my key interests are:

• Nonnegativity of real polynomials (real algebraic geometry).
• Polynomial optimization.
• Algebra related applications in science and engineering.
• Computational and algorithmic algebraic geometry, e.g., polynomial system solving.
• Mathematical software development.

There is a range of further topics I work on, am interested in, or have worked on in the past. For example:

• Amoeba theory (dealing with complex algebraic varieties under norm constraints).
• Fewnomial theory / sparsity of polynomial sysem.
• Combinatorial algebraic geometry (e.g., A-discriminants, toric, and tropical geometry).
• Various topics in optimization.

For me, applied Algebra means applied algebra: My group and I develop software, we work with real world data, and we cooperate with researchers from other sciences. For example, we are involved in projects on

• chemical reaction networks,
• material design,
• life science,
• theoretical computer science.

Publications

You can find my articles on the ArXiv.

Research Articles

1. "Re-entrant tensegrity: A three-periodic, chiral, tensegrity structure that is auxetic"
   with M. Oster, M. Dias, and M. Evans; Science Advances, 7 (50), 2021; doi: 10.1126/sci-adv.abj6737; see also ArXiv 2105.04601.
2. "Initial Steps in the Classification of Maximal Mediated Sets"
   with J. Hartzer, O. Röhrig, and O. Yürük; Journal of Symbolic Computation (special issue for ``MEGA 2019''); 109 (2022), 404-425; doi: 10.1016/j.jsc.2020.07.013; see also ArXiv 1910.00502.
3. "Evaluation of Pool-based Testing Approaches to Enable Population-wide Screening for COVID-19"
   with D.~Pflüger, M.~Rehme, J.~Heuer, and M.-I.~Bittner; PLoS ONE 15 (12): e0243692. https://doi.org/10.1371/journal.pone.0243692;
   This project has accompanying code and an accompanying interactive website.
   News about the project at Die Junge Akademie, TU Braunschweig (in German), and Universität Stuttgart (in German); see also ArXiv 2004.11851
4. "Nondegenerate Multistationarity in Small Reaction Networks"
   with A. Shiu; Discrete and Continuous Dynamical Systems, Series B, 24 (6) (June 2019), 2683-2700; see ArXiv 1802.00306.
5. "An Approach to Constrained Polynomial Optimization via Nonnegative Circuit Polynomials and Geometric Programming"
   with M. Dressler and S. Iliman; Journal of Symbolic Computation 91, (2019), 149-172 (special issue for "MEGA 2017"); see ArXiv 1602.06180.
6. "Imaginary Projections of Polynomials"
   with T. Jörgens and T. Theobald, 91, (2019), 181-199 Journal of Symbolic Computation (special issue for "MEGA 2017"); see ArXiv 1602.02008.
   Was accepted for a talk at "MEGA 2017".
7. "Lopsided Approximation of Amoebas"
   with J. Forsgård, L. Matusevich, and N. Mehlhop, 8 (315) (2019), 485-500; see Mathematics of Computation; see ArXiv 1608.08663.
8. "A Positivstellensatz for Sums of Nonnegative Circuit Polynomials"
   with M. Dressler and S. Iliman; SIAM Journal on Applied Algebra and Geometry 1 (1) (2017), 536-555; see ArXiv.
   Was accepted for a talk at "MEGA 2017".
9. "Lower Bounds for Polynomials with Simplex Newton Polytopes Based on Geometric Programming",
   with S. Iliman; SIAM Journal on Optimization 26 (2) (2016), 1128-1146; see ArXiv.
10. "Amoebas, Nonnegative Polynomials and Sums of Squares Supported on Circuits",
    with S. Iliman;, Research in the Mathematical Sciences 3(1) (2016), 1-35; see ArXiv.
    Was accepted for a talk at "MEGA 2015".
11. "Norms of Roots of Trinomials",
    with T. Theobald; Mathematische Annalen, 366 (1) (2016), 219-247; ; see ArXiv.
    Was accepted for a talk at "MEGA 2015".
12. "Separating Inequalities for Nonnegative Polynomials that Are not Sums of Squares",
    with S. Iliman; Journal of Symbolic Computation 68 (2015), part 2, 181-194 (special issue for "MEGA 2013"); see ArXiv.
    Was accepted for a talk at "MEGA 2013".
13. "Approximating Amoebas and Coamoebas by Sums of Squares",
    with T. Theobald; Mathematics of Computation 84 (2015), 455-473; see ArXiv.
    Was accepted for a poster presentation at "MEGA 2011".
14. "Amoebas of Genus at Most One",
    with T. Theobald; Advances in Mathematics 239 (2013), 190-213; see ArXiv.

Conference Publications

15. "Computing the Real Isolated Points of an Algebraic Hypersurface"
    with P.L. Huu, and M. Safey El Din; Proceedings of ISSAC 2020, Kalamata, Greece, July 20-23, 2020 (virtual conference due to COVID-19) (I. Z. Emiris, and L. Zhi (eds.)), ACM, (2020), 297 - 304; see ArXiv 2008.10331.
16. "Global Optimization via the Dual SONC Cone and Linear Programming"
    with M. Dressler, H. Naumann, and J. Heuer; Proceedings of ISSAC 2020, Kalamata, Greece, July 20-23, 2020 (virtual conference due to COVID-19) (I. Z. Emiris, and L. Zhi (eds.)), ACM, (2020), 138 - 145; see ArXiv 2002.09368.
17. "New Dependencies of Hierarchies in Polynomial Optimization"
    with A. Kurpisz; Proceedings of ISSAC 2019, Beijing, China, July 15-18, 2019 (J. H. Davenport, D. Wang, M. Kauers, and R. J. Bradford (eds.)), ACM, (2019), 251-258; see ArXiv 1903.04996.
18. "Exact Optimization via Sums of Nonnegative Circuits and Sums of AM/GM Exponentials"
    with V. Magron and H. Seidler; Proceedings of ISSAC 2019, Beijing, China, July 15-18, 2019 (J. H. Davenport, D. Wang, M. Kauers, and R. J. Bradford (eds.)), ACM, (2019), 291-298; see ArXiv 1902.02123.
19. "A New Method for Computing Elimination Ideals of Likelihood Equations"
    with X. Tang and R. Zhao; Proceedings of ISSAC 2019, Beijing, China, July 15-18, 2019 (J. H. Davenport, D. Wang, M. Kauers, and R. J. Bradford (eds.)), ACM, (2019), 339-346; see ArXiv 1810.05620.
    Suplimentary Material for the computations in the article are available via our website accompanying the article.

Articles Accepted for Publication

20. "The Kinetic Space of Multistationarity in Dual Phosphorylation"
    with E. Feliu, N. Kaihnsa, and O. Yürük; published electronically on Sep. 04, 2020; doi: 10.1007/s10884-020-09889-6; see ArXiv 2001.08285.
21. "Optimization over the Boolean Hypercube via Sums of Nonnegative Circuit Polynomials"
    with M. Dressler and A. Kurpisz; to appear in Foundations of Computational Mathematics; published electronically on April 06, 2021; doi: 10.1007/s10208-021-09496-x;
    Extended abstract was accepted at 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, Liverpool, UK, August 27-31, 2018, (I. Potapov, P. G. Spirakis, and J. Worrell, eds.), LIPIcs, 117, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, (2018), 82:1-82:17; see ArXiv 1802.10004.
    

Preprints

22. "The Duality of SONC: Advances in Circuit-based Certificates"
    with J. Heuer, see ArXiv 2204.03918.
23. "A Polyhedral Homotopy Algorithm For Real Zeros"
    with A. A. Ergür, see ArXiv 1910.01957.
24. "The Algebraic Boundary of the SONC Cone"
    with J. Forsgård, see ArXiv 1905.04776.
25. "An Experimental Comparison of SONC and SOS Certificates for Unconstrained Optimization"
    with H. Seidler, see ArXiv 1808.08431.
    NOTE: The polynomial in Example 4.3 is not displayed correctly in the current Arxiv version. The correct polynomial is available via our website accompanying the article.
    Was accepted for a talk at "MEGA 2019".
26. "The Lattice of Amoebas"
    with J. Forsgård, see ArXiv 1711.02705.
27. "Intersections of Amoebas",
    with M. Juhnke-Kubitzke, see ArXiv 1510.08416.
    Was accepted for a poster presentation at "FPSAC '16".
28. "A Sharp Upper Bound for the Complexity of Labeled Oriented Trees",
    with M. Christmann, see ArXiv 1412.7257.
29. "The Boundary of Amoebas",
    with F. Schroeter, see ArXiv 1310.7363.
30. "Low Dimensional Test Sets for Nonnegativity of Even Symmetric Forms",
    with S. Iliman, see ArXiv 1303.4241.
31. "Polytopes with Special Simplices",
    see ArXiv 1009.6158.

Surveys

1. "Amoebas, Nonnegative Polynomials and Sums of Squares Supported on Circuits",
   an extended abstract for the correspondent article;
   Oberwolfach Report, no. 23, 2015, 1308-1311; for the workshop "Tropical Aspects in Geometry, Topology and Physics".
2. "Amoebas and their Tropicalizations - a Survey", to appear in "Analysis Meets Geometry: The Mikael Passare Memorial Volume", M. Andersson, J. Boman, C. Kiselman, P. Kurasov, R. Sigurdsson (Eds.), Series: "Trends in Mathematics", Birkäuser Mathematics, 2017, 157-190.

Theses

1. "On the Geometry, Topology and Approximation of Amoebas",
   PhD thesis for mathematics (Dissertation), 2013, download as PDF.
2. "Goodmans "New Riddle of Induction" - Eine Analyse auf mathematischer Grundlage",
   Master thesis for philosophy (Magisterarbeit), 2010.
3. "Polytope mit speziellen Simplizes",
   Master thesis for mathematics (Diplomarbeit), 2008, download as PDF (German).

Software

1. "POEM" A Python based software for the computation of SONC and SOS certificates in polynomial optimization.
   with H. Seidler.
   Latest Version: (0.2.1.0(a)). Prototype; for test purposes only. Latest Update: 07/04/19. Development: July 2017 - present.
2. "Maximal_Mediated_Sets.sage ".
   A SAGE class for the computation of maximal mediated sets.
   with J. Hartzer.
   Latest Version: (0.1.0.1(a)). Latest Update: 04/03/17. Development: Fall 2016 - present.
3. "Lopsided_Amoeba_Approximation".
   A SINGULAR/SAGE software package for the approximation of amoebas via lopsided amoebas and cyclic resultants.
   Latest Version: (0.1.0.0(a)). Latest Update: 08/30/16. Development: Spring 2016 - present.
4. "Viro.sage".
   A SAGE class for working with Viro's Patchworking
   with C. O'Neill and E. Owusu Kwaakwah (alumna).
   Latest Version: (0.4b). Latest Update: 05/09/18. Development: Spring 2015 - present.