Research Interests

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

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

Publications

You can find my articles on the ArXiv.

Research Articles

  1. "Lopsided Approximation of Amoebas"
    with J. Forsgård, L. Matusevich, and N. Mehlhop; to appear in Mathematics of Computation; see ArXiv 1608.08663.
  2. "A Positivstellensatz for Sums of Nonnegative Circuit Polynomials"
    with M. Dressler and S. Iliman; to appear in the SIAM Journal on Applied Algebra and Geometry; see ArXiv.
    Was accepted for a talk at "MEGA 2017".
  3. "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.
  4. "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".
  5. "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".
  6. "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".
  7. "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".
  8. "Amoebas of Genus at Most One",
    with T. Theobald; Advances in Mathematics 239 (2013), 190-213; see ArXiv.

Preprints

  1. "An Approach to Constrained Polynomial Optimization via Nonnegative Circuit Polynomials and Geometric Programming"
    with M. Dressler and S. Iliman, see ArXiv 1602.06180.
  2. "Imaginary Projections of Polynomials"
    with T. Jörgens and T. Theobald, see ArXiv 1602.02008.
    Was accepted for a talk at "MEGA 2017".
  3. "Intersections of Amoebas",
    with M. Juhnke-Kubitzke, see ArXiv 1510.08416.
    Was accepted for a poster presentation at "FPSAC '16".
  4. "A Sharp Upper Bound for the Complexity of Labeled Oriented Trees",
    with M. Christmann, see ArXiv 1412.7257.
  5. "The Boundary of Amoebas",
    with F. Schroeter, see ArXiv 1310.7363.
  6. "Low Dimensional Test Sets for Nonnegativity of Even Symmetric Forms",
    with S. Iliman, see ArXiv 1303.4241.
  7. "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. "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.
  2. "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.
  3. "Viro.sage".
    A SAGE class for working with Viro's Patchworking
    with C. O'Neill and E. Owusu Kwaakwah (alumna).
    Latest Version: (0.3a). Latest Update: 05/31/16. Development: Spring 2015 - present.