Abstract: The tropical asymmetric distance is a quasi-metric on the n-dimensional tropical torus and in in particular on the Bergman fan \mathcal{B}(K_N) of the complete graphical matroid on N elements such that n is equal to N choose 2. Using the correspondence between Bergman fan \mathcal{B}(K_N) and the space of equidistance trees with N leaves it gives rise to a consensus method, called tropical median consensus, which unifies multiple phylogenetic trees into one. In this talk, I will present and analyse a clustering algorithm for equidistant phylogenetic trees based on the tropical median consensus.
Abstract : One of the most well studied invariants in algebraic optimization is the Euclidean distance degree. The ED degree provides a measure of complexity of measuring nearest point problems to real algebraic varieties. Here, we introduce the EuclideanDistanceDegree package for Macaulay2 which provides several approaches for computing these degrees. This includes symbolic methods involving multidegrees as well as functions using probabilistic methods. In addition, we provide numerical methods for computing ED degrees, generic ED degrees, and ED degree defects that rely on homotopy continuation from numerical algebraic geometry. We provide a host of test cases that also serve as benchmarks.
Abstract: Matroids are a combinatorial structure that abstracts linear independence. They have a deep connection with Grassmannians, the variety of subspaces of a vector space cut out by the Plücker equations. Grassmannians and matroids are inherently 'type A' objects and have generalizations to other root systems giving rise to what is known as Coxeter matroids. As such, one can generalize (through representation theory) these Grassmannians and Plücker equations to other Coxeter types using miniscule varieties and equations that define the quotient G/P of a simply connected complex Lie Group G by a maximal parabolic subgroup P with a miniscule fundamental representation. In addition, Borovik, Gelfand and White give a description of a strong exchange property for Coxeter matroids. It turns out there is a strong link between the strong exchange property and the tropicalization of the equations that define G/P. In this talk, we describe this link through representation theory, tropical geometry and polytope theory. (This is joint work with Kieran Calvert, Alex Fink and Ben Smith).
Abstract: Viro’s patchworking theorem produces real plane algebraic
curves of degree d from a regular unimodular triangulation of d · ∆2
together with a sign assignment on the vertices of the triangulation.
The topological type of the real scheme of the resulting curve is en-
coded in this combinatorial data alone, but extracting it efficiently is
nontrivial. We describe a fast algorithm for this task. Given the trian-
gulation and sign assignment, the algorithm determines the real scheme
by a purely combinatorial procedure on the reflected signed triangula-
tion: connected components are identified via union-find, regions through
antipodal boundary identifications, and the nesting tree of ovals by a
breadth-first traversal of the region adjacency graph. This avoids cell
complex constructions or homology computations required by earlier
approaches. The algorithm is implemented in C++ and Rust; its effi-
ciency was essenti1al for the recent exhaustive verification that all 121
real schemes of degree seven are realizable as T-curves (real algebraic
curves produced by Viro patchworking), which required classifying bil-
lions of sign distributions per triangulation.
Joint work with:Zoe Geiselmann, Michael Joswig, Lars Kastner, Konrad Mundinger,
Sebastian Pokutta, Christoph Spiegel, Marcel Wack, and Max Zimmer
Abstract: Coherent matching fields, introduced by Sturmfels and Zelevinsky (1993), play
an important role in the study of the Grassmannian and its tropical counterpart. They
can be identified with the vertices of a Minkowski sum of Birkhoff polytopes
associated with the maximal minors of an $n \times r$ matrix.
We present an algorithm for enumerating coherent matching fields up to row and column
permutations. Our approach adapts the reverse search method of Avis and Fukuda (1992)
to enumerate the orbits of the vertices of a polytope under a symmetry group. We
further show how the shadow-vertex rule can be used to perform inexpensive pivots
between coherent matching fields.
We implemented this algorithm in Rust with distributed parallelism via MPI. This
implementation enabled us to enumerate the $n \times r$ coherent matching fields for
$n \leq 10$ when $r = 3$ and for $n \leq 8$ when $r = 4$, up to row and column
permutations. As an application, we show that there exist configurations of nine
points in $\mathbb{R}^2$ in general position whose order type cannot be realized by
configurations of tropical points in general position, highlighting a new discrepancy
between classical and tropical realizability.
Joint work with Xavier Allamigeon.
Abstract: From a statistical standpoint, Markov bases serve as a way to construct a connected Markov chain
for performing conditional tests of a discrete exponential family. However, from an algebraic point of view,
a Markov basis is simply a generating set of a toric ideal. The distance-reduction property for a Markov
basis then ensures that the associated Markov chain can navigate the sample space efficiently. We fix a
convenient metric and, given any two states in the sample space, we ask whether one can always apply a
move from the Markov basis to one of them so as to decrease their distance. If this is the case, then the
Markov basis is called distance-reducing.
In this talk, we will look at distance reduction: firstly, in the context of toric ideals coming from graphs
and, secondly, from the point of view of the bouquet structure of the toric ideals. The link between these
two ideas comes from the fact that, for homogeneous toric ideals, the distance-reduction condition simplifies
to a more combinatorial statement. Toric ideals of graphs are particularly nice, as they are homogeneous,
whilst also providing visual intuition for the elements of the toric ideal. Furthermore, in the case that the
toric ideal is a complete intersection, there is a necessary and sufficient condition depending on the circuits
to check if a minimal Markov basis is distance-reducing. Finally, we will discuss homogeneous toric ideals with
the same bouquet structure and signature and show, by considering the poset of signatures, that
distance-reducing properties are always preserved.
Abstract: The OSCAR (Open Source Computer Algebra Research) System was developed to integrate the capabilities of ANTIC, GAP, Polymake, and Singular. Individually, these software systems can be used to study number theory, group theory, polyhedral geometry, and algebraic geometry as independent fields. Operating within Julia, the OSCAR system creates a unified framework which allows the user to apply the strengths of the cornerstone systems and take advantage of the synergies between these areas of mathematics.
We give an introduction to OSCAR, its basic functionality, and highlight the features which have been relevant in our research. Joint with Oliver Clarke.