Tropical Geometry Research Articles

Floor diagrams and enumerative invariants of line bundles over an elliptic curve

We use the tropical geometry approach to compute absolute and relative enumerative invariants of complex surfaces which are$\mathbb {C} P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block–Göttsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to compute these invariants along with the associated interpretation as operators acting on some Fock space. The floor diagram algorithm allows one to prove the piecewise polynomiality of the relative invariants, and the quasi-modularity of their generating series.

On moduli spaces of roots in algebraic and tropical geometry

In this paper we construct a tropical moduli space parametrizing roots of divisors on tropical curves. We study the relation between this space and the skeleton of Jarvis moduli space of nets of limit roots on stable curves. We show that the combinatorics of the moduli space of tropical roots is governed by the poset of flows, a poset parametrizing certain flows on graphs.

Exponential sums equations and tropical geometry

Zilber’s Exponential-Algebraic Closedness Conjecture states that algebraic varieties in {{mathbb {C}}}^n times ({{mathbb {C}}}^times )^n intersect the graph of complex exponentiation, unless that contradicts the algebraic and transcendence properties of {textrm{exp}}. We establish a case of the conjecture, showing that it holds for varieties which split as the product of a linear subspace of the additive group and an algebraic subvariety of the multiplicative group. This amounts to solving certain systems of exponential sums equations, and it generalizes old results of Zilber, which required the linear subspace to either be defined over a generic subfield of the real numbers, or it to be any subspace defined over the reals assuming unproved conjectures from Diophantine geometry and transcendence theory. The proofs use the theory of amoebas and tropical geometry.

A tropical geometric approach to exceptional points

Non-Hermitian systems have been widely explored in platforms ranging from photonics to electric circuits. A defining feature of non-Hermitian systems is exceptional points (EPs), where both eigenvalues and eigenvectors coalesce. Tropical geometry is an emerging field of mathematics at the interface between algebraic geometry and polyhedral geometry, with diverse applications to science. Here, we introduce and develop a unified tropical geometric framework to characterize different facets of non-Hermitian systems. We illustrate the versatility of our approach using several examples and demonstrate that it can be used to select from a spectrum of higher-order EPs in gain and loss models, predict the skin effect in the non-Hermitian Su-Schrieffer-Heeger model, and extract universal properties in the presence of disorder in the Hatano-Nelson model. Our work puts forth a framework for studying non-Hermitian physics and unveils a connection of tropical geometry to this field.

Proto-exact categories of modules over semirings and hyperrings

Proto-exact categories, introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a non-additive setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor.In this paper, we show that the categories of modules over semirings and hyperrings - algebraic structures which have gained prominence in tropical geometry - carry proto-exact structures.In the first part, we prove that the category of modules over a semiring is equipped with a proto-exact structure; modules over an idempotent semiring have a strong connection to matroids. We also prove that the category of algebraic lattices L has a proto-exact structure, and furthermore that the subcategory of L consisting of finite lattices is equivalent to the category of finite B-modules as proto-exact categories, where B is the Boolean semifield. We also discuss some relations between L and geometric lattices (simple matroids) from this perspective.In the second part, we prove that the category of modules over a hyperring has a proto-exact structure. In the case of finite modules over the Krasner hyperfieldK, a well-known relation between finite K-modules and finite incidence geometries yields a combinatorial interpretation of exact sequences.

Bitangents to plane quartics via tropical geometry: rationality, $$\mathbb {A}^1$$-enumeration, and real signed count

We explore extensions of tropical methods to arithmetic enumerative problems such as $$\mathbb {A}^1$$ -enumeration with values in the Grothendieck–Witt ring and rationality over Henselian valued fields, using bitangents to plane quartics as a test case. We consider quartic curves over valued fields whose tropicalizations are smooth and satisfy a mild genericity condition. We then express obstructions to rationality of bitangents and their points of tangency in terms of twisting of edges of the tropicalization; the latter depends only on the tropicalization and the initial coefficients of the defining equation modulo squares. We also show that the $${{\,\textrm{GW}\,}}$$ -multiplicity of a tropical bitangent, i.e. the multiplicity with which its lifts contribute to the $$\mathbb {A}^1$$ -enumeration of bitangents as defined by Larson and Vogt (Res Math Sci 8(26):1–21, 2021), can be computed from the tropicalization of the quartic together with the initial coefficients of the defining equation. As an application, we show that the four lifts of most tropical bitangent classes contribute $$2\mathbb {H}$$ , twice the class of the hyperbolic plane, to the $$\mathbb {A}^1$$ -enumeration. These results rely on a degeneration theorem relating the Grothendieck–Witt ring of a Henselian valued field to the Grothendieck–Witt ring of its residue field, in residue characteristic not equal to two.

On the Decision Boundaries of Neural Networks: A Tropical Geometry Perspective.

This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to characterize the decision boundaries of a simple network of the form (Affine, ReLU, Affine). Our main finding is that the decision boundaries are a subset of a tropical hypersurface, which is intimately related to a polytope formed by the convex hull of two zonotopes. The generators of these zonotopes are functions of the network parameters. This geometric characterization provides new perspectives to three tasks. (i) We propose a new tropical perspective to the lottery ticket hypothesis, where we view the effect of different initializations on the tropical geometric representation of a network's decision boundaries. (ii) Moreover, we propose new tropical based optimization reformulations that directly influence the decision boundaries of the network for the task of network pruning. (iii) At last, we discuss the reformulation of the generation of adversarial attacks in a tropical sense. We demonstrate that one can construct adversaries in a new tropical setting by perturbing a specific set of decision boundaries by perturbing a set of parameters in the network.

Convergent Hahn series and tropical geometry of higher rank

AbstractWe propose to study the tropical geometry specifically arising from convergent Hahn series in multiple indeterminates. One application is a new view on stable intersections of tropical hypersurfaces. Another one is perturbations of rank one tropical polytopes, which is beneficial for algorithmic purposes.

Dynamical Tropicalisation

We analyse the dynamics of the pullback of the map z longmapsto z^m on the complex tori and toric varieties. We will observe that tropical objects naturally appear in the limit, and review several theorems in tropical geometry.

Tropical Monte Carlo quadrature for Feynman integrals

We introduce a new method to evaluate algebraic integrals over the simplex numerically. This new approach employs techniques from tropical geometry and exceeds the capabilities of existing numerical methods by an order of magnitude. The method can be improved further by exploiting the geometric structure of the underlying integrand. As an illustration of this, we give a specialized integration algorithm for a class of integrands that exhibit the form of a generalized permutahedron. This class includes integrands for scattering amplitudes and parametric Feynman integrals with tame kinematics. A proof-of-concept implementation is provided with which Feynman integrals up to loop order 17 can be evaluated.

A Novel Tropical Geometry-Based Interpretable Machine Learning Method: Pilot Application to Delivery of Advanced Heart Failure Therapies.

A model's interpretability is essential to many practical applications such as clinical decision support systems. In this article, a novel interpretable machine learning method is presented, which can model the relationship between input variables and responses in humanly understandable rules. The method is built by applying tropical geometry to fuzzy inference systems, wherein variable encoding functions and salient rules can be discovered by supervised learning. Experiments using synthetic datasets were conducted to demonstrate the performance and capacity of the proposed algorithm in classification and rule discovery. Furthermore, we present a pilot application in identifying heart failure patients that are eligible for advanced therapies as proof of principle. From our results on this particular application, the proposed network achieves the highest F1 score. The network is capable of learning rules that can be interpreted and used by clinical providers. In addition, existing fuzzy domain knowledge can be easily transferred into the network and facilitate model training. In our application, with the existing knowledge, the F1 score was improved by over 5%. The characteristics of the proposed network make it promising in applications requiring model reliability and justification.

Identifying potential candidates for advanced heart failure therapies using an interpretable machine learning algorithm.

Systems level barriers to heart failure (HF) care limit access to HF advanced therapies (heart transplantation, left ventricular assist devices). There is a need for automated systems that can help clinicians ensure patients with HF are evaluated for HF advanced therapies at the appropriate time to optimize outcomes. We performed a retrospective study using the REVIVAL (Registry Evaluation of Vital Information for VADs in Ambulatory Life) and INTERMACS (Interagency Registry for Mechanically Assisted Circulatory Support) registries. We developed a novel machine learning model based on principles of tropical geometry and fuzzy logic that can accommodate clinician knowledge and provide recommendations regarding need for advanced therapies evaluations that are accessible to end-users. The model was trained and validated using data from 4,694 HF patients. When initiated with clinical knowledge from HF and transplant cardiologists, the model achieved an F1 score of 43.8%, recall of 51.1%, and precision of 46.9%. The model achieved comparable performance compared with other commonly used machine learning models. Importantly, our model was 1 of only 3 models providing transparent and parsimonious clinical rules, significantly outperforming the other 2 models. Eleven clinical rules were extracted from the model which can be leveraged in clinical practice. A machine learning model capable of accepting clinical knowledge and making accessible recommendations was trained to identify patients with advanced HF. While this model was developed for HF care, the methodology has multiple potential uses in other important clinical applications.

Complex and tropical counts via positive characteristic

We study two classical families of enumerative problems: inflection lines of plane curves and theta-hyperplanes of canonical curves. In these problems the complex counts and the tropical counts disagree. Each problem suggests a prime with special behavior. On the one hand, we analyze the reduction modulo these special primes, and we prove that the complex solutions coalesce in uniform clusters. On the other hand, we observe that the counts in special characteristic and in tropical geometry match.

On the tropicalization of D-ideals

Following the Gröbner basis theoretic study of tropical geometry due to Maclagan and Sturmfels, we construct a notion of tropicalization for ideals in various rings of differential operators. We first treat the classical case, tropicalizing left ideals in the ring of differential operators with analytic or formal coefficients. Our construction is benefited from the recent development of Gröbner basis theory in this setting. Most of our elaboration will focus on the tropicalization of left ideals in the Weyl algebra over a characteristic 0 field with valuation. In order to achieve this, the relevant Gröbner basis theory in this generality is developed. In both cases, we will prove the existence of a tropical basis and relate the tropicalization of a D-ideal with the usual tropicalization of its characteristic varieties.

Hypergraph characterization of split matroids

We provide a combinatorial study of split matroids, a class that was motivated by the study of matroid polytopes from a tropical geometry point of view. A nice feature of split matroids is that they generalize paving matroids, while being closed under duality and taking minors. Furthermore, these matroids proved to be useful in giving exact asymptotic bounds for the dimension of the Dressian, and also implied new results on the rays of the tropical Grassmannians.In the present paper, we introduce the notion of elementary split matroids, a subclass of split matroids that contains all connected split matroids. We give a hypergraph characterization of elementary split matroids in terms of independent sets, and show that the proposed class is closed not only under duality and taking minors but also truncation. We further show that, in contrast to split matroids, the proposed class can be characterized by a single forbidden minor. As an application, we provide a complete list of binary split matroids.

Cross-ratio degrees and perfect matchings

Cross-ratio degrees count configurations of points z 1 , … , z n ∈ P 1 z_1,\ldots ,z_n\in \mathbb {P}^1 satisfying n − 3 n-3 cross-ratio constraints, up to isomorphism. These numbers arise in multiple contexts in algebraic and tropical geometry, and may be viewed as combinatorial invariants of certain hypergraphs. We prove an upper bound on cross-ratio degrees in terms of the theory of perfect matchings on bipartite graphs. We also discuss several of the many perspectives on cross-ratio degrees—including a connection to Gromov-Witten theory—and give many example computations.

Invariant Factors as Limit of Singular Values of a Matrix

The paper concerns a result in linear algebra motivated by ideas from tropical geometry. Let $A(t)$ be an $n \times n$ matrix whose entries are Laurent series in $t$. We show that, as $t \to 0$, logarithms of singular values of $A(t)$ approach the invariant factors of $A(t)$. This leads us to suggest logarithms of singular values of an $n \times n$ complex matrix as an analogue of the logarithm map on $(\mathbb{C}^*)^n$ for the matrix group $GL(n, \mathbb{C})$.

The Algebraic Boundary of the Sonc-Cone

In this article, we explore the connections between nonnegativity, the theory of $A$-discriminants, and tropical geometry. For an integral support set $A \subset \mathbb{Z}^n$, we cover the boundary of the sonc-cone by semialgebraic sets that are parametrized by families of tropical hypersurfaces. As an application, we give sufficient conditions for equality between the sonc-cone and the sparse nonnegativity cone for generic support sets, and we describe a semialgebraic stratification of the boundary of the sonc-cone in the univariate case.

Mathematical Progress — On Maddy and Beyond

AbstractA key question of the ‘maverick’ tradition of the philosophy of mathematical practice is addressed, namely what is mathematical progress. The investigation is based on an article by Penelope Maddy devoted to this topic in which she considers only contributions ‘of some mathematical importance’ as progress. With the help of a case study from contemporary mathematics, more precisely from tropical geometry, a few issues with her proposal are identified. Taking these issues into consideration, an alternative account of ‘mathematical importance’, broadly within the framework of progress Maddy offers, is developed with a special focus on mathematicians’ peer-review practice.

An interpretable neural network for outcome prediction in traumatic brain injury

BackgroundTraumatic Brain Injury (TBI) is a common condition with potentially severe long-term complications, the prediction of which remains challenging. Machine learning (ML) methods have been used previously to help physicians predict long-term outcomes of TBI so that appropriate treatment plans can be adopted. However, many ML techniques are “black box”: it is difficult for humans to understand the decisions made by the model, with post-hoc explanations only identifying isolated relevant factors rather than combinations of factors. Moreover, such models often rely on many variables, some of which might not be available at the time of hospitalization.MethodsIn this study, we apply an interpretable neural network model based on tropical geometry to predict unfavorable outcomes at six months from hospitalization in TBI patients, based on information available at the time of admission.ResultsThe proposed method is compared to established machine learning methods—XGBoost, Random Forest, and SVM—achieving comparable performance in terms of area under the receiver operating characteristic curve (AUC)—0.799 for the proposed method vs. 0.810 for the best black box model. Moreover, the proposed method allows for the extraction of simple, human-understandable rules that explain the model’s predictions and can be used as general guidelines by clinicians to inform treatment decisions.ConclusionsThe classification results for the proposed model are comparable with those of traditional ML methods. However, our model is interpretable, and it allows the extraction of intelligible rules. These rules can be used to determine relevant factors in assessing TBI outcomes and can be used in situations when not all necessary factors are known to inform the full model’s decision.