arXiv subjects
Publications and source records attributed to Omid Amini.
Given a divisor on a tropical curve, we associate to each point of the curve a Weierstrass gap sequence. We investigate structural properties of these gap sequences and explore their relationship with the Weierstrass gap sequences of line bundles on algebraic curves via the tropicalization process.
A Lefschetz module is a module over a graded algebra $A$ that satisfies analogues of Poincar\'{e} duality, the Hard Lefschetz property, and the Hodge--Riemann relations with respect to an open convex cone $\mathscr{K}$ in the degree one part of $A$. We analyze its decomposition into indecomposable modules over subrings of $A$ that are generated by elements in the closure of $\mathscr{K}$, establishing structural results that parallel the decomposition theorem for morphisms of complex projective varieties. We use our theorems to recover key statements in combinatorial Hodge theory and illuminate the Hodge-theoretic aspects of the decomposition theorem in algebraic geometry.
To a compact tropical variety of arbitrary dimension, we associate a collection of intermediate Jacobians defined in terms of tropical homology and tropical monodromy. We then develop an Abel-Jacobi theory in the tropical setting by defining functorial Abel-Jacobi maps. We introduce, in particular, tropical Albanese varieties and formulate obstructions to algebraic equivalence of tropical cycles. In dimension 1, we show that this recovers the existing Abel-Jacobi theory for tropical curves. As an application, we consider the Ceresa class of a tropical curve which is defined as the image of the Ceresa cycle in an appropriate intermediate Jacobian under the Abel-Jacobi map. We give an explicit formula for this class entirely in terms of the combinatorics of the tropical curve.
The tropical rank of a semimodule of rational functions on a metric graph mirrors the concept of rank in linear algebra. Defined in terms of the maximal number of tropically independent elements within the semimodule, this quantity has remained elusive due to the challenges of computing it in practice. We establish that the tropical rank is, in fact, precisely equal to the topological dimension of the semimodule, one more than the dimension of the associated linear system of divisors. This implies that the equality of divisorial and tropical ranks in the definition of tropical linear series is equivalent to the pure dimensionality of the corresponding linear system. We then address the question of computing the tropical rank. In particular, we show that checking whether a given family of tropical rational functions is tropically independent is equivalent to solving a turn-based stochastic mean-payoff game, whereas calculating the tropical rank of a finitely generated semimodule of tropical rational functions is NP-hard. We conclude with several complementary results and questions regarding combinatorial and topological properties of the tropical rank.
Given an algebraic variety defined over a discrete valuation field and a skeleton of its Berkovich analytification, the tropicalization process transforms function field of the variety to a semifield of tropical functions on the skeleton. Our main result offers a purely polyhedral characterization of this semifield: we show that a tropical function is in the image of the tropicalization map if and only if it takes the same slope near infinity along parallel half-lines of the skeleton. This extends a result of Baker and Rabinoff in dimension one to arbitrary dimensions. We use this characterization to establish that this semifield is finitely generated over the semifield of tropical rational numbers, providing a new proof of a recent result by Ducros, Hrushovski, Loeser and Ye in the discrete valued field case. As a second application, we present a new proof of the faithful tropicalization theorem by Gubler, Rabinoff and Werner in the discrete valuation field case. The proof is constructive and provides explicit coordinate functions for the embedding of the skeleton, extending the existing results in dimension one to arbitrary dimensions.
We describe the limits of canonical series along families of curves degenerating to a nodal curve which is general for its topology, in the weak sense that the branches over nodes on each of its components are in general position. We define a fan structure on the space of edge lengths on the dual graph of the limit curve, and construct a projective variety parametrizing the limits, organized in strata associated to the cones of this fan. This extends to all topologies the works by Eisenbud-Harris (Invent. Math. 87: 496-515, 1987) on curves of compact type and Esteves-Medeiros (Invent. Math. 149: 267-338, 2002) on two-component curves.
A level graph is the data of a pair $(G,\pi)$ consisting of a finite graph $G$ and an ordered partition $\pi$ on the set of vertices of $G$. To each level graph on $n$ vertices we associate a polytope in $\mathbb R^n$ called its residue polytope. We show that residue polytopes are compatible with each other in the sense that if $\pi'$ is a coarsening of $\pi$, then the polytope associated to $(G,\pi)$ is a face of the one associated to $(G,\pi')$. Moreover, they form all the faces of the residue polytope of $G$, defined as the polytope associated to the level graph with the trivial ordered partition. The results are used in a companion work to describe limits of spaces of Abelian differentials on families of Riemann surfaces approaching a stable Riemann surface on the boundary of the moduli space.
We say that a tropical fan is homologically smooth if each of its open subsets verify tropical Poincare duality. A tropical homology manifold is a tropical variety that is locally modelled by open subsets of homologically smooth tropical fans. We show that homological smoothness is a T-stable property in the category of tropical fans. This implies in particular that quasilinear fans are homologically smooth, and tropical varieties locally modelled by them are tropical homology manifolds. Previously, this was known only for locally matroidal tropical varieties. In order to show the above results, we prove a tropical analogue of the Deligne weight spectral sequence for homologically smooth tropical fans. This allows to describe the cohomology of tropical modifications, and will be of importance in our companion work which develops a Hodge theory in the tropical setting.
We prove that the Chow ring of any simplicial fan is isomorphic to the middle degree part of the tropical cohomology ring of its canonical compactification. Using this result, we prove a tropical analogue of Kleiman's criterion of ampleness for fans. In the case of tropical fans that are homology manifolds, we obtain an isomorphism between the Chow ring of the fan and the entire tropical cohomology of the canonical compactification. When applied to matroids, this provides a new representation of the Chow ring of a matroid as the cohomology ring of a projective tropical manifold.
We show that tropicalization of linear series on curves gives rise to two-parameter families of tilings by polymatroids, with one parameter arising from the theory of divisors on tropical curves and the other from the reduction of linear series of rational functions in non-Archimedean geometry. In order to do this, we introduce a general framework that produces tilings of vector spaces and their subsets by polymatroids. We furthermore show that these tilings are regular and relate them to work by Kapranov and Lafforgue on Chow quotients of Grassmannians.
We introduce combinatorial objects named matricubes that provide a generalization of the theory of matroids. As matroids provide a combinatorial axiomatization of hyperplane arrangements, matricubes provide a combinatorial axiomatization of arrangements of initial flags in a vector space. We give cryptomorphic axiomatic systems in terms of rank function, flats, circuits, and independent sets, and formulate a duality concept. We also provide precise links between matricubes, permutation arrays and matroids, and raise several open questions.
The RoboCup competition was started in 1997, and is known as the oldest RoboCup league. The RoboCup 2D Soccer Simulation League is a stochastic, partially observable soccer environment in which 24 autonomous agents play on two opposing teams. In this paper, we detail the main strategies and functionalities of CYRUS, the RoboCup 2021 2D Soccer Simulation League champions. The new functionalities presented and discussed in this work are (i) Multi Action Dribble, (ii) Pass Prediction and (iii) Marking Decision. The Multi Action Dribbling strategy enabled CYRUS to succeed more often and to be safer when dribbling actions were performed during a game. The Pass Prediction enhanced our gameplay by predicting our teammate's passing behavior, anticipating and making our agents collaborate better towards scoring goals. Finally, the Marking Decision addressed the multi-agent matching problem to improve CYRUS defensive strategy by finding an optimal solution to mark opponents' players.
Soccer Simulation 2D (SS2D) is a simulation of a real soccer game in two dimensions. In soccer, passing behavior is an essential action for keeping the ball in possession of our team and creating goal opportunities. Similarly, for SS2D, predicting the passing behaviors of both opponents and our teammates helps manage resources and score more goals. Therefore, in this research, we have tried to address the modeling of passing behavior of soccer 2D players using Deep Neural Networks (DNN) and Random Forest (RF). We propose an embedded data extraction module that can record the decision-making of agents in an online format. Afterward, we apply four data sorting techniques for training data preparation. After, we evaluate the trained models' performance playing against 6 top teams of RoboCup 2019 that have distinctive playing strategies. Finally, we examine the importance of different feature groups on the prediction of a passing strategy. All results in each step of this work prove our suggested methodology's effectiveness and improve the performance of the pass prediction in Soccer Simulation 2D games ranging from 5\% (e.g., playing against the same team) to 10\% (e.g., playing against Robocup top teams).
This paper is the first in a series devoted to the development of a Hodge theory for tropical varieties. We introduce a notion of T-stability for tropical fans and prove that various geometric properties of tropical fans are T-stable. As a consequence, we establish K\"ahler properties for the Chow ring in a large class of tropical fans, going beyond the case of matroids and their Bergman fans. As a by-product, we obtain a new proof of the K\"ahler package for combinatorial geometries. The approach makes it possible to deal with tropical fans with general weights.
We introduce higher rank inner products on real and complex vector spaces and study their corresponding Voronoi tilings. We use the framework to describe metric degenerations of polarized tori and Hausdorff limits of Voronoi tilings of discrete sets.
Soccer, also known as football in some parts of the world, involves two teams of eleven players whose objective is to score more goals than the opposing team. To simulate this game and attract scientists from all over the world to conduct research and participate in an annual computer-based soccer world cup, Soccer Simulation 2D (SS2D) was one of the leagues initiated in the RoboCup competition. In every SS2D game, two teams of 11 players and one coach connect to the RoboCup Soccer Simulation Server and compete against each other. Over the past few years, several C++ base codes have been employed to control agents' behavior and their communication with the server. Although C++ base codes have laid the foundation for the SS2D, developing them requires an advanced level of C++ programming. C++ language complexity is a limiting disadvantage of C++ base codes for all users, especially for beginners. To conquer the challenges of C++ base codes and provide a powerful baseline for developing machine learning concepts, we introduce Pyrus, the first Python base code for SS2D. Pyrus is developed to encourage researchers to efficiently develop their ideas and integrate machine learning algorithms into their teams. Pyrus base is open-source code, and it is publicly available under MIT License on GitHub
Given the tropicalization of a complex subvariety of the torus, we define a morphism between the tropical cohomology and the rational cohomology of their respective tropical compactifications. We say that the subvariety of the torus is cohomologically tropical if this map is an isomorphism for all closed strata of the tropical compactification. We prove that a sch\"on subvariety of the torus is cohomologically tropical if and only if it is wundersch\"on and its tropicalization is a tropical homology manifold. The former property means that the open strata in the boundary of a tropical compactification are all connected and the mixed Hodge structures on their cohomology are pure of maximum possible weight; the latter property requires that, locally, the tropicalization verifies tropical Poincar\'e duality. We study other properties of cohomologically tropical and wundersch\"on varieties, and show that in a semistable degeneration to an arrangement of cohomologically tropical varieties, the Hodge numbers of the smooth fibers are captured in the tropical cohomology of the tropicalization. This extends the results of Itenberg, Katzarkov, Mikhalkin, and Zharkov.
The RoboCup competitions hold various leagues, and the Soccer Simulation 2D League is a major one among them. Soccer Simulation 2D (SS2D) match involves two teams, including 11 players and a coach, competing against each other. The players can only communicate with the Soccer Simulation Server during the game. This paper presents the latest research of the CYRUS soccer simulation 2D team, the champion of RoboCup 2021. We will explain our denoising idea powered by long short-term memory networks (LSTM) and deep neural networks (DNN). The CYRUS team uses the CYRUS2D base code that was developed based on the Helios and Gliders bases.