## Research Interests

My research is focused on **real and computational algebraic geometry**. My key interests are:

- Nonnegativity of real polynomials and sums of squares (real algebraic geometry).
- Amoeba theory (dealing with complex algebraic varieties under norm constraints).
- Applications of algebraic geometry (and combinatorics) in science and engineering.

My work is interdisciplinary since these topics have natural connections to other fields in math. Specifically, I use methods from:

- Semidefinite and Polynomial Optimization,
- Combinatorics, particularly Discrete Geometry,
- Tropical Geometry.

## Publications

You can find my articles on the ArXiv.

### Research Articles

**"Nondegenerate Multistationarity in Small Reaction Networks"**

with A. Shiu; to appear in Discrete and Continuous Dynamical Systems, Series B; see ArXiv 1802.00306.**"An Approach to Constrained Polynomial Optimization via Nonnegative Circuit Polynomials and Geometric Programming"**

with M. Dressler and S. Iliman; to appear in Journal of Symbolic Computation (special issue for "MEGA 2017"); see ArXiv 1602.06180.**"Imaginary Projections of Polynomials"**

with T. Jörgens and T. Theobald; to appear in Journal of Symbolic Computation (special issue for "MEGA 2017"); see ArXiv 1602.02008.

Was accepted for a talk at "MEGA 2017".**"Lopsided Approximation of Amoebas"**

with J. Forsgård, L. Matusevich, and N. Mehlhop; to appear in Mathematics of Computation; see ArXiv 1608.08663.**"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".**"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.**"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".**"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".**"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".**"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".**"Amoebas of Genus at Most One"**,

with T. Theobald; Advances in Mathematics**239**(2013), 190-213; see ArXiv.

### Preprints

**"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.**"Optimization over the Boolean Hypercube via Sums of Nonnegative Circuit Polynomials"**

with M. Dressler and A. Kurpisz, see ArXiv 1802.10004.

Is accepted for a talk at "MFCS 2018" and at "FLoC '18".**"The Lattice of Amoebas"**

with J. Forsgård, see ArXiv 1711.02705.**"Intersections of Amoebas"**,

with M. Juhnke-Kubitzke, see ArXiv 1510.08416.

Was accepted for a poster presentation at "FPSAC '16".**"A Sharp Upper Bound for the Complexity of Labeled Oriented Trees"**,

with M. Christmann, see ArXiv 1412.7257.**"The Boundary of Amoebas"**,

with F. Schroeter, see ArXiv 1310.7363.**"Low Dimensional Test Sets for Nonnegativity of Even Symmetric Forms"**,

with S. Iliman, see ArXiv 1303.4241.**"Polytopes with Special Simplices"**,

see ArXiv 1009.6158.

### Surveys

**"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".**"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

**"On the Geometry, Topology and Approximation of Amoebas"**,

PhD thesis for mathematics (Dissertation), 2013, download as PDF.**"Goodmans "New Riddle of Induction" - Eine Analyse auf mathematischer Grundlage"**,

Master thesis for philosophy (Magisterarbeit), 2010.**"Polytope mit speziellen Simplizes"**,

Master thesis for mathematics (Diplomarbeit), 2008, download as PDF (German).

### Software

**""**A Python based software for the computation of SONC and SOS certificates in polynomial optimization.

with H. Seidler.

Latest Version: (0.1.1.0(a)). Prototype; for test purposes only. Latest Update: 04/09/18. Development: July 2017 - present.**"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.**"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.**"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.