Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
|
|
|
Session Overview | |
|
Location: Unitobler, F013 53 seats, 74m^2 |
| Date: Tuesday, 09/Jul/2019 | |
| 10:00am - 12:00pm | MS122: Tropical and combinatorial methods in economics |
| Unitobler, F013 | |
|
|
10:00am - 12:00pm
Tropical and combinatorial methods in economics Over the past ten years, combinatorial auctions and mechanism designs have posed interesting challenges at the intersection of tropical geometry, matroid theory, discrete convex analysis and integer programming. This minisymposium features experts who work at this intersection discussing the latest developments and potential approaches to major conjectures concerning valuated matroids (also known as gross substitutes or $M^natural$-concave functions). (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) On the Construction of Substitutes Gross substitutability is a central concept in Economics and is connected to important notions in Discrete Convex Analysis, Number Theory and the analysis of Greedy algorithms in Computer Science. Many different characterizations are known for this class, but providing a constructive description remains a major open problem. The construction problem asks how to construct all gross substitutes from a class of simpler functions using a set of operations. Since gross substitutes are a natural generalization of matroids to real-valued functions, matroid rank functions form a desirable such class of simpler functions. Shioura proved that a rich class of gross substitutes can be expressed as sums of matroid rank functions, but it is open whether all gross substitutes can be constructed this way. Our main result is a negative answer showing that some gross substitutes cannot be expressed as positive linear combinations of matroid rank functions. En route, we provide necessary and sufficient conditions for the sum to preserve substitutability, uncover a new operation preserving substitutability and fully describe all substitutes with at most 4 items. Joint work with Renato Paes Leme. Connection Between Discrete Convex Analysis and Auction Theory Discrete convex analysis is a theory of discrete convexity in combinatorial optimization. In this talk we explain the connection between discrete convex analysis and auctions with multiple differentiated items. In particular, we show that computation of an equilibrium price vector can be reduced to the minimization of an L-convex function, and the existing iterative auction algorithms can be regarded as specialized implementation of the algorithms for L-convex function minimization. Unimodular schemes In 2003 we with V Danilov proved that an ample class of discrete convexity is a pure systems such that its one dimensional generators form a totally unimodular system. I explain how to glue such pure systems into a unimodular scheme and to build a class of discrete convex functions. Transversal valuated matroids As every first course in linear algebra tells us, there are a great number of ways to describe a linear subspace of K^n when K is a field. When K is the tropical semiring, many of these cease to agree, but tropical geometers have a consensus as to which of them give the correct notion of tropical linear subspace (one that does is Plücker vectors). My subject will be one of the "wrong" descriptions, namely row spaces of matrices, which give a subset of the tropical linear spaces. We obtain tropical analogues of results from the '70s on presentations of transversal matroids. Some of this work is joint with Felipe Rincón; the rest is joint with Jorge Alberto Olarte. |
| 3:00pm - 5:00pm | MS138: Computational aspects of tropical geometry |
| Unitobler, F013 | |
|
|
3:00pm - 5:00pm
Computational aspects of tropical geometry The aim of this session is to demonstrate the effictive use of tropical geometry to tackle problems from optimization and various applications. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Condition numbers of stochastic mean payoff games and what they say about nonarchimedean semidefinite programming Semidefinite programming can be considered over any real closed field, including fields of Puiseux series equipped with their nonarchimedean valuation. Nonarchimedean semidefinite programs encode parametric families of classical semidefinite programs, for sufficiently large values of the parameter. Recently, a correspondence has been established between nonarchimedean semidefinite programs and stochastic mean payoff games with perfect information. This correspondence relies on tropical geometry. It allows one to solve generic nonarchimedean semidefinite feasibility problems, of large scale, by means of stochastic game algorithms. Computing tropical hypersurface intersections A tropical hypersurface is a balanced polyhedral complex of codimension 1 while a tropical prevarity is a finite intersection of such hypersurfaces. The dynamic enumeration strategy for prevarity computation proposed by Mizutani, Takeda and Kojima (2006) appears in the context of polyhedral homotopy continuation in polynomial system solving. It produces the prevariety for zero-dimensional intersections in general position. In contrast, the non-generic situation is more involved as the intersection may be non-pure and numerical exactness becomes critical. Extending joint work with Sommars and Verschelde we propose a dynamic decomposition strategy for reducing the number of vertices in the enumeration tree. We report on experimental results with our implementation. Algebraic systems and exterior semi-algebras In this talk, we describe negation maps and ``systems,'' and their application to Grassmann (exterior) algebra in a rather general framework that includes tropical algebra, hyperfields, and fuzzy rings. Tropical volume by tropical Ehrhart polynomials We are motivated by the problem of intrinsically defining a volume concept for tropical polytopes. In Euclidean space, the number of integral points contained in a large dilate of a given polytope approximates its volume. Even more, by Ehrhart's theorem, the function that counts integral points in dilates of an integral polytope is a polynomial in the dilation factor, whose leading coefficient equals the volume of the polytope. After introducing a suitable and natural concept of a tropical lattice, we aim to establish an analogous result for tropical lattice polytopes based on their covector decomposition. |
| Date: Wednesday, 10/Jul/2019 | |
| 10:00am - 12:00pm | MS156: Tropical geometry in statistics |
| Unitobler, F013 | |
|
|
10:00am - 12:00pm
Tropical geometry in statistics Classically, statistics is the branch of mathematics that deals with data. The challenges of modern data demand the development of new statistical methods to handle them. Modern data collection technology brings not only “big data” that are extremely high dimensional, but additionally, they are made up of complex structures, which can be prohibitive to the Euclidean setting of classical statistics. Tropical geometry defines and studies piecewise linear structures in an algebraic framework that, if interpreted appro- priately, is amenable to modern data structures and challenges. This session focuses on leveraging the potential of tropical geometry to reinterpret classical statistics and enhance the utility of statistical methodology in the face of modern data challenges. Specifically, we seek to adapt the linearizing properties of the tropical semiring to statistical settings that rely on principles of linear algebra and optimization. These encompass fundamental descriptive and inferential statistics, such as the computation of Fréchet means, principal component analysis, linear regression, and hypothesis testing. This is a very new direction of research with potential for wide-reaching applications from biology to economics, and it is our hope to bring together researchers to develop and advance the interaction between tropical geometry and statistics. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Tropical principal component analysis We introduce a notion of principal component analysis in the setting of tropical geometry. We also describe some results on the containment of a Stiefel linear space within a larger tropical linear space and apply them to our setting of tropical principal component analysis. Tropical Foundations for Probability and Statistics on Phylogenetic Tree Spaces A geometric approach to phylogenetic tree space was first introduced by Billera, Holmes, and Vogtmann. We reinterpret the tree space via tropical geometry and introduce a novel framework for the statistical analysis of phylogenetic trees: the palm tree space, which represents phylogenetic trees as points in a space endowed with the tropical metric. We show that the palm tree space possesses a variety of properties that allow for the definition of probability measures, and thus expectations, variances, and other fundamental statistical quantities. In addition, they lead to increased computational efficiency. Our approach provides a new, tropical basis for a statistical treatment of evolutionary biological processes represented by phylogenetic trees. This is a joint work with Anthea Monod (Columbia University, USA) and Ruriko Yoshida (Naval Postgraduate School, USA). Tropical Gaussians There is a growing need for a systematic study of probability distributions in tropical settings. Over the classical algebra, the Gaussian measure is arguably the most important distribution to both theoretical probability and applied statistics. In this work, we review the existing analogues of the Gaussian measure in the tropical semiring and outline various research directions. Tropical hardware for data intensive applications: DNA sequence alignment to machine learning New encodings are being explored to allay the energy efficiency concerns, that fundamentally limit performance of data intensive, modern computing systems. One such brain inspired encoding, known as Race Logic, encodes information in the arrival time of signals. With such an encoding, conventional-computing gates such as OR, AND and delay gates perform MIN, MAX and addition-by-constant operations respectively. Hence with we end up with elegant hardware implementations for the fundamental operations of tropical algebra. This allows tropical operations to be easily expressed with temporally-coded hardware, which allows data intensive problems to be solved with low latency, low energy computer architectures. One such architecture is a DNA sequence alignment engine which calculates the edit distance between two input sequences. Our architecture physically implements the dynamic programming nature of tropical graph traversal methods, on a programmable edit-graph. The other architecture describes a programmable methodology of mapping various decision tree based forests to MIN/MAX gates and is used for high throughput in-sensor image classification. The main message of this talk is to stress on the symbiosis between tropical algebra and computing hardware communities, which we believe can lead to development of compact, energy efficient, computing hardware for new classes of complex optimization problems.
|
| 3:00pm - 5:00pm | MS167, part 1: Computational tropical geometry |
| Unitobler, F013 | |
|
|
3:00pm - 5:00pm
Computational tropical geometry This session will highlight recent advances in tropical geometry, algebra, and combinatorics, focusing on computational aspects and applications. The area enjoys close interactions with max-plus algebra, polyhedral geometry, combinatorics, Groebner theory, and numerical algebraic geometry. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) The tropical geometry of shortest paths We study parameterized versions of classical algorithms for computing shortest paths. This is most easily expressed in terms of tropical geometry. Applications include the enumeration of polytropes, i.e., ordinary convex polytopes which are also tropically convex, as well as shortest paths in traffic networks with variable link travel times. Tropicalization of semialgebraic sets arising in convex optimization Linear programming (LP) is the simplest and most studied class of conic optimization problems. It consists in minimizing a linear function over a convex cone that is polyhedral. Important generalizations of LP include semidefine and hyperbolic programming, in which we allow the underlying cone to have a more complicated structure. In the case of semidefinite programming, the cone is defined by linear matrix inequalities, while a hyperbolicity cone is defined by imposing a positivity condition on the eigenvalues of a hyperbolic polynomial. In all three cases, the underlying cones are semialgebraic, which implies that we can study them over arbitrary real closed fields, such as the nonarchimedean field of real Puiseux series. The tropicalization of such cone is then defined as its image under the nonarchimedean valuation. In this talk, we discuss the structure of these tropicalizations and the related computational problems. In particular, we study how the structure of tropical spectrahedral cones is more restrictive in comparison to the structure of arbitrary tropical convex cones. To obtain these results, we study the structure of arbitrary tropical semialgebraic sets. We also show how tropical convex cones encode stochastic mean payoff games and how this can be used, in the case of generic tropical spectrahedra, to solve the associated feasibility problems. Linear algebra and convexity over symmetrized semirings, hyperfields and systems Rowen introduced a notion of algebraic structure, called systems, which unifies symmetrized tropical semirings, supertropical semirings, and hyperfields. We study linear algebra and convexity over systems. We identify cases in which the row rank, column rank, and submatrix rank of a matrix are equal. We also discuss Helly and Carathéodory numbers. Linear algebra and convexity over symmetrized semirings, hyperfields and systems. Priority mechanisms are a key element of the management of emergency call centers. These mechanisms can be modeled by dynamical systems with a piecewise affine transition mapping, determined by a rational map in the tropical semifield. Performance indicators can be inferred from stationary regimes. The latter are determined by solving structured tropical polynomial systems: these are analogous to the non-linear eigenproblems associated to Markov decision processes, but priority rules lead to negative "probabilities". |
| Date: Thursday, 11/Jul/2019 | |
| 10:00am - 12:00pm | MS144: Tropical geometry in machine learning |
| Unitobler, F013 | |
|
|
10:00am - 12:00pm
Tropical geometry in machine learning A connection between tropical polynomials and neural networks has been recently established. This connection remains to be explored in full. Currently, most basic notions from tropical geometry are used to quantify the number of linear regions in a neural network. Purpose of this session is to present what is currently know about the relationship between tropical polynomials and neural networks and promote further exploration of tropical algebra in the context of machine learning at neural networks. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Tropical geometry of deep neural networks Abstract: We exlore connections between feedforward neural networks with ReLU activation and tropical geometry — we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden layer can be characterized by zonotopes, which serve as building blocks for deeper networks; we relate decision boundaries of such neural networks to tropical hypersurfaces, a major object of study in tropical geometry; and we prove that linear regions of such neural networks correspond to vertices of polytopes associated with tropical rational functions. An insight from our tropical formulation is that a deeper network is exponentially more expressive than a shallow network. Tropical geometry and weighted lattices We present advances on extending the max-plus or min-plus algebraic structure of tropical geometry by using weighted lattices and a max-* algebra with an arbitrary binary operation * that distributes over max or min. The envisioned application areas include geometry, image analysis, optimization and learning. Further, we generalize some tropical geometrical objects using weighted lattices. For example, we outline the optimal solution of max-* equations using weighted lattice adjunctions, and apply it to optimal regression for fitting max-* tropical curves on arbitrary data. A Tropical Approach to Neural Networks with Piecewise Linear Activations This talk revisits the problem of counting the regions of linearity of piecewise linear neural networks. We treat layers of neural networks with piecewise linear activations as tropical signomials, which generalize polynomials in the so-called (max, +) or "tropical" algebra to the case of real-valued exponents. Motivated by the discussion in (Montufar et. al, 2014), this approach enables us to recover tight bounds on linear regions of layers with ReLU / leaky ReLU activations, as well as bounds for layers with arbitrary convex, piecewise linear activations. Our approach crucially relies on exploiting a correspondence between regions of linearity and vertices of Newton polytopes, which also enables us to design a randomized method for counting linear regions in practice. This algorithm relies on sampling vertices and places no restrictions on the range of inputs of the neural network, avoiding the overhead of existing exact approaches which rely on solving a large number of linear or mixed-integer programs. Moreover, it extends beyond rectifier networks. The results presented in the talk are joint work with Petros Maragos. |
| 3:00pm - 5:00pm | MS167, part 2: Computational tropical geometry |
| Unitobler, F013 | |
|
|
3:00pm - 5:00pm
Computational tropical geometry This session will highlight recent advances in tropical geometry, algebra, and combinatorics, focusing on computational aspects and applications. The area enjoys close interactions with max-plus algebra, polyhedral geometry, combinatorics, Groebner theory, and numerical algebraic geometry. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Connectivity of tropical varieties The standard algorithm to compute tropical varieties makes crucial use of the fact that the tropicalization of an irreducible variety is connected. I will discuss joint work with Josephine Yu showing that the tropicalization of a d-dimensional irreducible variety satisfies a stronger d-connectedness property. Tropical convex hull of polytopes Tropical convexity has been mostly focused on tropical convex hull of finitely many points, i.e., tropical polytopes. Moreover there has been some work on polytropes which are convex tropical polytopes. In this talk I will consider the tropical convex hull of polytopes and polyhedra. I will show that these are convex sets and that in some cases tconv(conv(S))=conv(tconv(S)) and tconv(pos (S))=pos(tconv(0,S)) for a finite set S. This will lead the way to compute the tropical convex hull of a tropical variety. Algorithmic questions around tropical Carathéodory Since Imre Bárány found the colourful version of Carathéodory's theorem in 1982, many combinatorial generalizations and algorithmic variations have been considered. This ranges from variations of the colour classes to different notions of convexity. We take a closer look at the tropical convexity version of this theorem. We provide new insights on colourful linear programming and matroid generalizations from a tropical point of view, by considering additional sign informations. We focus on explicit constructions for 'colourful simplices'. The difficulty of the arising algorithmic questions ranges from greedily solvable to NP-hard. Convergent Puiseux series and tropical geometry of higher rank Tropical hypersurfaces arising from polynomials over the Puiseux series are well studied and well understood objects. The picture becomes less clear when considering Puiseux series in multiple indeterminates. Unlike their rank one counterparts, these higher rank tropical hypersurfaces are not ordinary polyhedral complexes, but we shall see they still have a large amount of structure. Moreover, by restricting to convergent Puiseux series we show how one can describe them via the rank one tropical hypersurfaces arising from substitution of indeterminates. We will also consider a couple of applications of this framework, including a new viewpoint for stable intersection in the vein of symbolic perturbation. |
| Date: Friday, 12/Jul/2019 | |
| 10:00am - 12:00pm | MS141, part 1: Chip-firing and tropical curves |
| Unitobler, F013 | |
|
|
10:00am - 12:00pm
Chip-firing and tropical curves The chip-firing game on metric graphs is a simple combinatorial model that serves as a tropical analogue of divisor theory on algebraic curves, and it has been an active and fruitful research direction over the last decade. The behaviors of chip-firing resemble, but not always completely match, the classical situation in algebraic geometry. So on one hand, chip-firing can often be used to prove results (old and new) in algebraic geometry; while on the other hand, the combinatorics of chip-firing is interesting and surprising in its own right. We will focus on three main topics: (I) Tropical analogues (or failure thereof) of classical results of algebraic curves, (II) applications of chip-firing in algebraic geometry and other subjects, and (III) complexity issues of computational problems related to chip-firing. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Introduction to chip firing This session is intended as a preamble, giving a quick outline on chip-firing and introducing concepts likely to appear in several of the talks to follow. These include: graphs, metric graphs, divisor theory (divisors, linear equivalence, rank), Jacobian group, reduced divisors, divisorial gonality, and related concepts. We illustrate with graphs of low genus. Computing divisorial gonality is hard The (divisorial) gonality of a graph G is the smallest degree of a divisor of positive rank in the sense of Baker-Norine. In terms of the classical chip-firing game of Björner-Lovász-Shor it relates to chip configurations that result in a finite game, even after adding a chip at an arbitrary position. We show that computing the gonality of a graph is NP-hard. In fact, it cannot be approximated to within an arbitrary factor in polynomial time (unless P=NP). Recognizing hyperelliptic graphs Based on analogies between algebraic curves and graphs, a new multigraph parameter was defined. This parameter is called divisorial gonality and can be defined using a chip-firing game. In this talk we consider so-called hyperelliptic graphs, which are graphs with divisorial gonality 2. We will see that we can decide in polynomial time whether a graph is hyperelliptic or not. Graphs of gonality three In 2013, Chan classified all metric graphs of gonality two, proving that divisorial gonality and geometric gonality are equivalent in the hyperelliptic case. We show that such a classification extends to combinatorial graphs of divisorial gonality three, under certain edge-connectivity assumptions. We also give a construction for graphs of divisorial gonality three, and provide conditions for determining when a graph is not of divisorial gonality three. |
| 3:00pm - 5:00pm | MS167, part 3: Computational tropical geometry |
| Unitobler, F013 | |
|
|
3:00pm - 5:00pm
Computational tropical geometry This session will highlight recent advances in tropical geometry, algebra, and combinatorics, focusing on computational aspects and applications. The area enjoys close interactions with max-plus algebra, polyhedral geometry, combinatorics, Groebner theory, and numerical algebraic geometry. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Tropicalized quartics and curves of genus 3 Brodsky, Joswig, Morrison and Sturmfels showed that not all abstract tropical curves of genus 3 can be realized as a tropicalization of a quartic in the euclidean space. We focus on the interior of the maximal cones in the moduli space and study all curves which can be realized as a faithful tropicalization in a tropical plane. Reflecting the algebro-geometric world, these are exactly those which are not realizably hyperelliptic. Our approach is constructive: For any not realizably hyperelliptic curve, we explicitly construct a realizable model of the tropical plane and a faithfully tropicalized quartic in it. These constructions rely on modifications resp. tropical refinements. Conversely, we prove that any realizably hyperelliptic curve cannot be embedded in such a fashion. For that, we rely on the theory of tropical divisors and embeddings from linear systems, and recent advances in the realizability of sections of the tropical canonical divisor. Tropical Jucys Covers and refined quasimodularity Hurwitz numbers count genus $g$, degree $d$ covers of the complex projective line with fixed branched locus and fixed ramification data. An equivalent description is given by factorisations in the symmetric group. Simple double Hurwitz numbers are a class of Hurwitz-type counts of specific interest. In recent years a related counting problem in the context of random matrix theory was introduced as so-called monotone Hurwitz numbers. These can be viewed as a desymmetrised version of the Hurwitz-problem. Moreover, the notion of strictly monotone Hurwitz numbers has risen in interest as it is equivalent to a certain Grothendieck dessins d'enfant count. We study monotone and strictly monotone Hurwitz numbers from a bosonic Fock space perspective. This yields a new interpretation in terms of tropical covers involving local multiplicities given by Gromov-Witten invariants. We further discuss applications of this new interpretation with regards to quasimodularity results and wall-crossing formulae. Tropical lines on tropical surfaces In 1849, Arthur Cayley and George Salmon proved that every smooth cubic surface in P3 contains exactly 27 lines. Since the early development of tropical geometry, two natural problems were to understand whether the same statement holds for smooth tropical cubic surfacss and to classify combinatorial positions of their tropical lines. The answer to the first turned out to be false, as smooth tropical surfaces might contain families of tropical lines. Moreover, classifying positions of tropical lines reveals some computational challenges due to the massive number of combinatorial types of smooth tropical cubic surfaces. In this talk we will tell this tropical story. We will introduce motifs of tropical lines on tropical surfaces and study their configurations. Throughout the talk we will highlight the computational aspects. Polyhedral tropical geometry of higher rank In a recent paper, Jell, Scheiderer, and Yu define a notion of real tropicalization for semialgebraic sets. In this talk I will discuss what happens when the semialgebraic set is a linear subspace. In this setting, the real tropicalization is a polyhedral fan that is best understood using the theory of oriented matroids. The main focus of this talk will be on understanding the topological and combinatorial properties of this fan. |
| Date: Saturday, 13/Jul/2019 | |
| 10:00am - 12:00pm | MS141, part 2: Chip-firing and tropical curves |
| Unitobler, F013 | |
|
|
10:00am - 12:00pm
Chip-firing and tropical curves The chip-firing game on metric graphs is a simple combinatorial model that serves as a tropical analogue of divisor theory on algebraic curves, and it has been an active and fruitful research direction over the last decade. The behaviors of chip-firing resemble, but not always completely match, the classical situation in algebraic geometry. So on one hand, chip-firing can often be used to prove results (old and new) in algebraic geometry; while on the other hand, the combinatorics of chip-firing is interesting and surprising in its own right. We will focus on three main topics: (I) Tropical analogues (or failure thereof) of classical results of algebraic curves, (II) applications of chip-firing in algebraic geometry and other subjects, and (III) complexity issues of computational problems related to chip-firing. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Chip-firing and the tropical inverse problem The chip-firing game on graphs gives strong combinatorial constraints on the types of meromorphic functions that can be defined on a Riemann surface. The framework provides useful combinatorial bounds for invariants of the Riemann surface: for instance, it can prove the generic non-existence of linear systems with prescribed numerical invariants. In general, the method is not set up to go the other way around: to prove the existence of geometric objects lifting the combinatorial ones. However, when even simple instances of these lifting statements can be proved, they have substantial applications. I will give a tour of the ideas surrounding this ``tropical inverse problem’’, and mention some applications along the way. Tropical Prym Varieties My talk revolves around combinatorial aspects of Prym varieties with applications to Brill-Noether theory. Prym varieties are a class of Abelian varieties that appear in the presence of covers between Riemann surfaces, and have deep connections with 2-torsion points of Jacobians, bi-tangent lines, and spin structures on curves. In my talk, I will describe the tropical version of Prym varieties in terms of chip-ring, and discuss the relation with their algebraic counterpart. As a consequence of the tropical construction we obtain new results in the geometry of special algebraic curves. This is joint work with Martin Ulirsch. Equidistribution of tropical Weierstrass points The set of (higher) Weierstrass points on a curve of genus g > 1 is an analogue of the set of N-torsion points on an elliptic curve. As N grows, the torsion points "distribute evenly" over a complex elliptic curve. This makes it natural to ask how Weierstrass points distribute, as the degree of the corresponding divisor grows. We will explore how Weierstrass points behave on abstract tropical curves, and explain how their distribution can be described in terms of electrical networks. Submodular functions in tropical geometry: the existence of semibreak divisors In the talk, I would like to show a situation where a statement in tropical geometry is proved using the submodular technique of combinatorial optimization. Break divisors are a very useful concept in tropical geometry that were introduced by Mikhalkin and Zharkov. They give a system of representatives of divisor classes of degree g (where g is the genus). We introduce semibreak divisors, generalizing break divisors for degree less than the genus. We show that every effective divisor class of degree between 0 and g contains a semibreak representative. Semibreak divisors can be used to give elementary proofs for some properties of effective loci in tropical curves. Formerly, the only known proofs for these properties used the counterparts of these properties for algebraic curves. To prove the existence of semibreak divisors in effective divisor classes of degree between 0 and g, we give a characterization of break divisors using a submodular function. Though the submodular function in our case is defined on an infinite set system (on those closed subsets of the metric graph that have finitely many path connected components), the problem has a quasi-discrete nature, which enables us to obtain a proof for the existence of semibreak divisors that very much resembles discrete arguments. We also obtain an algorithm that computes a semibreak representative for a given effective divisor, using a submodular minimization algorithm as subroutine. Joint work with Andreas Gross and Farbod Shokrieh. |
| 3:00pm - 5:00pm | MS167, part 4: Computational tropical geometry |
| Unitobler, F013 | |
|
|
3:00pm - 5:00pm
Computational tropical geometry This session will highlight recent advances in tropical geometry, algebra, and combinatorics, focusing on computational aspects and applications. The area enjoys close interactions with max-plus algebra, polyhedral geometry, combinatorics, Groebner theory, and numerical algebraic geometry. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Massively parallel methods with applications in tropical geometry In this talk, I will discuss the use of massively parallel methods in the context of tropical geometry. I will first address the technical framework, which combines the computer algebra system Singular with the workflow management system GPI-Space. I will then focus on the computation of tropicalizations, and algorithms to determine generating series for Gromov-Witten invariants via Feynman integrals. Tropical Grassmannians Gr_p(3,8) and the Dressian Dr(3,8) The pointwise valuation of an algebraic variety is a polyhedral complex, the tropical variety, which carries information about the algebraic set. A class of prominent examples are tropical Grassmannians Gr_p(d,n) over fields of characteristic p which are set theoretically included in tropical prevarieties, called Dressians. I will introduce these fans with their natural fan structure, inherit from Gröbner bases and regular subdivisions. Moreover, I report about new theoretical results additional to Janko Böhm's presentation of massiv parallelized computations. My focus will be on the relationship between Gr_0(3,8) and Dr(3,8). Computing unit groups of curves The group of units modulo constants of an affine variety over an algebraically closed field is free abelian of finite rank. Computing this group is difficult but of fundamental importance in tropical geometry, where it is desirable to realize intrinsic tropicalizations. We present practical algorithms for computing unit groups of smooth curves of low genus. Our approach is rooted in divisor theory, based on interpolation in the case of rational curves and on methods from algebraic number theory in the case of elliptic curves. A numerical algorithm for tropical membership In 2012, Hauenstein and Sottile proposed a numerical oracle for the Newton polytopes of a hypersurface. Drawing from ideas of Hept and Theobald, we describe how this algorithm may be used to numerically verify membership in tropical hypersurfaces. |
