Robert Löwe

Research

Publications

• Secondary fans and secondary polyhedra of punctured Riemann surfaces
  with Michael Joswig and Boris Springborn (publication) (arxiv)
  Experimental Mathematics, Volume 29, Issue 4 (2020)
  
  Abstract: A famous construction of Gel'fand, Kapranov and Zelevinsky associates to each finite point configuration \(A \subset \mathbb{R}^d\) a polyhedral fan, which stratifies the space of weight vectors by the combinatorial types of regular subdivisions of \(A\). That fan arises as the normal fan of a convex polytope. In a completely analogous way, we associate to each hyperbolic Riemann surface \(\mathcal{R}\) with punctures a polyhedral fan. Its cones correspond to the ideal cell decompositions of \(\mathcal{R}\) that occur as the horocyclic Delaunay decompositions which arise via the convex hull construction of Epstein and Penner. Similar to the classical case, this secondary fan of \(\mathcal{R}\) turns out to be the normal fan of a convex polyhedron, the secondary polyhedron of \(\mathcal{R}\).
  
  
  
• The Newton polytope of the discriminant of a quaternary cubic form
  with Lars Kastner (publication) (arxiv)
  Le Matematiche, Vol 75 No 2 (2020)
  
  Abstract: We determine the 166104 extremal monomials of the discriminant of a quaternary cubic form. These are in bijection with D-equivalence classes of regular triangulations of the 3-dilated tetrahedron. We describe how to compute these triangulations and their D-equivalence classes in order to arrive at our main result. The computation poses several challenges, such as dealing with the sheer number of triangulations effectively, as well as devising a suitably fast algorithm for computation of a D-equivalence class.
  
  
  
• Minkowski decompositions for generalized associahedra of acyclic type
  with Dennis Jahn and Christian Stump (publication) (arxiv)
  Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 757-775
  
  Abstract: We give an explicit subword complex description of the generators of the type cone of the g-vector fan of a finite type cluster algebra with acyclic initial seed. This yields in particular a description of the Newton polytopes of the F-polynomials in terms of subword complexes as conjectured by S. Brodsky and the third author. We then show that the cluster complex is combinatorially isomorphic to the totally positive part of the tropicalization of the cluster variety as conjectured by D. Speyer and L. Williams.
  
  
  

Preprints

• On Moduli Spaces of Convex Projective Structures on Surfaces: Outitude and Cell-Decomposition in Fock-Goncharov Coordinates
  with Robert Haraway, Dominic Tate and Stephan Tillmann (arxiv)
  
  Abstract Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichmüller space of punctured surfaces. We extend this cell decomposition to the moduli space of decorated strictly convex projective structures of finite volume on punctured surfaces. The proof uses Fock and Goncharov's A-coordinates for doubly decorated structures. In addition, we describe a simple, intrinsic edge-flipping algorithm to determine the canonical cell decomposition associated to a point in moduli space, and show that Penner's centres of Teichmüller cells are also natural centres of the cells in moduli space. We show that in many cases, the associated holonomy groups are semi-arithmetic.
  
  
  

Thesis

Projects

• Discretization in Geometry and Dynamics
• 27 questions on cubic surfaces

Software development

Teaching

• Analysis I (since 2020)
• Lineare Algebra I (since 2020)
• Mathematik & Schach (since 2020)

Former Teaching

• Optimization and Tropical Geometry (2019)
• Analysis I für Ingenieurswissenschaften (2016)
• Topologie (2016)
• Analysis I-III für Mathematiker (2013 - 2015)
• Lineare Algebra I-II für Mathematiker (2011 - 2013)

Events

Upcoming

• DMV-Jahrestagung 2022, Berlin 12-16, 2022

Some past events

• DMV-Jahrestagung 2019, Karlsruhe, September 23-26, 2019
  Talk: Decomposing the moduli space of decorated convex projective structures on non-compact surfaces
• Combinatorial and algebraic aspects of geometric structures, Chiang Mai, July 22-26, 2019
  Talk: Decomposing the moduli space of decorated convex projective structures on non-compact surfaces
• International Conference on Network Games, Tropical Geometry, and Quantum Communication, ZIB, June 3-7, 2019
• Einstein Workshop: Geometric and Topological Combinatorics, Zuse Institute Berlin, October 29-31, 2018
• Conference: Discretization in Geometry and Dynamics, Döllnsee-Schorfheide, October 7-12, 2018
• Annual meeting SFB TRR 195, Tübingen, September 24-28, 2018
  Talk: Secondary polyhedra of punctured Riemann surfaces
• BMS/IMPAN block course on Convex geometry related to Hamiltonian group actions, Bedlewo, March 19-23, 2018
• Einstein Conference on Discrete Geometry and Topology, FU Berlin, March 13-16, 2018
  Poster: Secondary fans and secondary polyhedra of punctured Riemann surfaces
• Jahrestagung der Deutschen Mathematiker-Vereinigung, Paderborn, March 5-9, 2018
  Talk: Secondary fans of punctured Riemann surfaces
• DGD Young Investigators Workshop, Erfurt, March 5-7, 2018
• 9th polymake conference and developer meeting, TU Berlin, February 8-10, 2018
  Talk: Hyperbolic Surfaces
• SFB Workshop "DGD Days 2017", Technische Universität München / Raitenhaslach, October 4-6 , 2017
  Talk: Secondary fans of punctured Riemann surfaces in polymake
• LMS-CMI Research School: Introduction to geometry, dynamics, and moduli in low dimensions, University of Warwick, September 11-15 , 2017
• MSRI Introductory Workshop: Geometric and Topological Combinatorics, Mathematical Sciences Research Institute, Berkeley, September 5-8, 2017