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We introduce a notion of tropical vector bundle on a tropical toric variety which is a tropical analogue of a torus equivariant vector bundle on a toric variety. Alternatively it can be called a toric matroid bundle. We define equivariant $K$-theory and characteristic classes of these bundles. As a particular case, we show that any matroid comes with tautological tropical toric vector bundles over the permutahedral toric variety and the corresponding equivariant $K$-classes and Chern classes recover the tautological classes of matroids constructed in the recent work of Berger-Eur-Spink-Tseng. In analogy with toric vector bundles, we define sheaf of sections and Euler characteristic as well as positivity notions such as global generation, ampleness and nefness for tropical toric vector bundles. Moreover, we prove a vanishing of higher cohomologies result. Finally, we study the splitting of our tropical toric vector bundles and, in particular, an analogue of Grothendieck's theorem on splitting of vector bundles on projective line.
We study anisotropic scaling limits of topological field theories using tropical geometry. The resulting topological field theories are characterized by foliated geometries and are invariant under foliation-preserving gauge transformations. We demonstrate the tropicalization for the 2D BF theory and generalize the prescription to topological Yang-Mills and Chern-Simons theories. We call the tropical limit of the BF theory, the \textit{TBF} theory, which is an anisotropic generalization of the BF theory with an additional adjoint-valued field $T$ that enforces a projectability condition onto the leaves of the foliation. The TBF theory localizes onto the moduli space of tropicalized flat connections $\mathcal{M}(Σ_g,G)$ on a foliated Riemann surface $Σ_g$ of genus $g$. The tropical connections exhibit anisotropic behavior; their holonomy is sensitive only to the leaves of the foliation. We analyze this moduli space two distinct ways, Firstly, they are classified by leaf-wise holonomy whose dimension can be explicitly calculated for the case of tropical projective space $\mathbb{TP}^1$ by the moduli space isomorphism $\mathcal{M}\left(\mathbb{TP} ^1, G\right) \cong \operatorname{Hom}(
Precision medicine, tailored to individual patients based on their genetics, environment, and lifestyle, shows promise in managing complex diseases like infections. Integrating artificial intelligence (AI) into precision medicine can revolutionize disease management. This paper introduces a novel approach using AI to advance precision medicine in infectious diseases and beyond. It integrates diverse fields, analyzing patients' profiles using genomics, proteomics, microbiomics, and clinical data. AI algorithms process vast data, providing insights for precise diagnosis, treatment, and prognosis. AI-driven predictive modeling empowers healthcare providers to make personalized and effective interventions. Collaboration among experts from different domains refines AI models and ensures ethical and robust applications. Beyond infections, this AI-driven approach can benefit other complex diseases. Precision medicine powered by AI has the potential to transform healthcare into a proactive, patient-centric model. Research is needed to address privacy, regulations, and AI integration into clinical workflows. Collaboration among researchers, healthcare institutions, and policymakers is cruci
During the COVID-19 pandemic, there were over three million infections in Los Angeles County (LAC). To facilitate distribution when vaccines first became available, LAC set up six mega-sites for dispensing a large number of vaccines to the public. To understand if another choice of mega-site location would have improved accessibility and health outcomes, and to provide insight into future vaccine allocation problems, we propose a multi-objective mixed integer linear programming model that balances travel convenience, infection reduction, and equitable distribution. We provide a tractable objective formulation that effectively proxies real-world public health goals of reducing infections while considering travel inconvenience and equitable distribution of resources. Compared with the solution empirically used in LAC in 2020, we recommend more dispersed mega-site locations that result in a 28% reduction in travel inconvenience and avert an additional 1,000 infections.
We explore the tropical analog of spinors by representing tropical geometries as foliated Riemann surfaces endowed with degenerate complex structures. We investigate tropical limits of the Laplace-Beltrami operator and explicitly construct its square root, which defines a tropical Dirac operator. We find that the tropical Clifford algebra is classified as a degenerate Clifford algebra with nilpotent generators. The nilpotent generator allows us to work with a new kind of representation that allows for Grassmann odd numbers, effectively supersymmetrizing the tropical spin bundle. We show through Dirac-Bergmann's quantization procedure, that the corresponding tropicalized quantum field theories enjoy a purely fermionic topological symmetry which can be expected to give a new class of path integral localization that we call tropical localization similar to the alternative localization method recently constructed by Choi and Takhtajan. We also discuss how the tropical Dirac operator, when twisted by gauge fields, obeys a tropical version of the Lichnerowicz identity, thereby demonstrating how some elements of Yang-Mills curvature should arise in the tropical limit.
With the increasing interest in deploying Artificial Intelligence in medicine, we previously introduced HAIM (Holistic AI in Medicine), a framework that fuses multimodal data to solve downstream clinical tasks. However, HAIM uses data in a task-agnostic manner and lacks explainability. To address these limitations, we introduce xHAIM (Explainable HAIM), a novel framework leveraging Generative AI to enhance both prediction and explainability through four structured steps: (1) automatically identifying task-relevant patient data across modalities, (2) generating comprehensive patient summaries, (3) using these summaries for improved predictive modeling, and (4) providing clinical explanations by linking predictions to patient-specific medical knowledge. Evaluated on the HAIM-MIMIC-MM dataset, xHAIM improves average AUC from 79.9% to 90.3% across chest pathology and operative tasks. Importantly, xHAIM transforms AI from a black-box predictor into an explainable decision support system, enabling clinicians to interactively trace predictions back to relevant patient data, bridging AI advancements with clinical utility.
The Oxford English Dictionary defines precision medicine as "medical care designed to optimize efficiency or therapeutic benefit for particular groups of patients, especially by using genetic or molecular profiling." It is not an entirely new idea: physicians from ancient times have recognized that medical treatment needs to consider individual variations in patient characteristics. However, the modern precision medicine movement has been enabled by a confluence of events: scientific advances in fields such as genetics and pharmacology, technological advances in mobile devices and wearable sensors, and methodological advances in computing and data sciences. This chapter is about bandit algorithms: an area of data science of special relevance to precision medicine. With their roots in the seminal work of Bellman, Robbins, Lai and others, bandit algorithms have come to occupy a central place in modern data science ( Lattimore and Szepesvari, 2020). Bandit algorithms can be used in any situation where treatment decisions need to be made to optimize some health outcome. Since precision medicine focuses on the use of patient characteristics to guide treatment, contextual bandit algorith
The last decade has seen an explosion in models that describe phenomena in systems medicine. Such models are especially useful for studying signaling pathways, such as the Wnt pathway. In this chapter we use the Wnt pathway to showcase current mathematical and statistical techniques that enable modelers to gain insight into (models of) gene regulation, and generate testable predictions. We introduce a range of modeling frameworks, but focus on ordinary differential equation (ODE) models since they remain the most widely used approach in systems biology and medicine and continue to offer great potential. We present methods for the analysis of a single model, comprising applications of standard dynamical systems approaches such as nondimensionalization, steady state, asymptotic and sensitivity analysis, and more recent statistical and algebraic approaches to compare models with data. We present parameter estimation and model comparison techniques, focusing on Bayesian analysis and coplanarity via algebraic geometry. Our intention is that this (non exhaustive) review may serve as a useful starting point for the analysis of models in systems medicine.
Global warming imposes us to reflect on the way we carry research, embarking on the obligation to minimize the environmental impact of our research programs, with the reduction of our travel footprint being one of the easiest actions to implement, thanks to the advance of digital technology. The X-ray Integral Field Unit (X-IFU), the cryogenic spectrometer of the Athena space X-ray observatory of the European Space Agency will be developed by a large international consortium. The travel footprint associated with the development of the X-IFU is to be minimized. For that purpose, a travel footprint calculator has been developed and first released to the X-IFU consortium members. The calculator uses seven different emission factors and methods differing by up to a factor of ~5 for the same flying distance. The observed differences illustrate the lack of standards and regulations for computing the footprint of flight travels and are explained primarily, though partly, by different accountings of non-CO2 effects. The calculator enables us to compute the travel footprint of a large set of travels and can help identify a meeting place that minimizes the overall travel footprint for a larg
In this paper we generalize correspondence theorems of Mikhalkin and Nishinou-Siebert providing a correspondence between algebraic and parameterized tropical curves. We also give a description of a canonical tropicalization procedure for algebraic curves motivated by Berkovich's construction of skeletons of analytic curves. Under certain assumptions, we construct a one-to-one correspondence between algebraic curves satisfying toric constraints and certain combinatorially defined objects, called "stacky tropical reductions", that can be enumerated in terms of tropical curves satisfying linear constraints. Similarly, we construct a one-to-one correspondence between elliptic curves with fixed $j$-invariant satisfying toric constraints and "stacky tropical reductions" that can be enumerated in terms of tropical elliptic curves with fixed tropical $j$-invariant satisfying linear constraints. Our theorems generalize previously published correspondence theorems in tropical geometry, and our proofs are algebra-geometric. In particular, the theorems hold in large positive characteristic.
We show that the asymptotic behavior of the two main competing notions of rank of a linear series on a tropical curve is governed by asymptotic invariants, closely paralleling the theory of volumes in algebraic geometry. We introduce and study tropical notions of volume associated to both divisors and tropical modules. We prove optimal asymptotic results for each case. In addition, we show that the tropical volume is compatible with the tropicalization of curves.
This paper provides an overview of recent progress on the interplay between tropical geometry and non-archimedean analytic geometry in the sense of Berkovich. After briefly discussing results by Baker, Payne and Rabinoff in the case of curves, we explain a result by Cueto, Häbich and the author comparing the tropical Grassmannian of planes to the analytic Grassmannian. We also give an overview of the general higher-dimensional theory developed by Gubler, Rabinoff and the author. In particular, we explain the construction of generalized skeleta in which are polyhedral substructures of Berkovich spaces lending themselves to comparison with tropicalizations. We discuss the slope formula for the valuation of rational functions and explain two results on the comparison between polyhedral substructures of Berkovich spaces and tropicalizations.
We model the impact of local vaccine mandates on the spread of vaccine-preventable infectious diseases, which in the absence of vaccines will mainly affect children. Examples of such diseases are measles, rubella, mumps and pertussis. To model the spread of the pathogen, we use a stochastic SIR (Susceptible, Infectious, Recovered) model with two levels of mixing in a closed population, often referred to as the household model. In this model individuals make local contacts within a specific small subgroup of the population (e.g.\ within a household or a school class), while they also make global contacts with random people in the population at a much lower rate than the rate of local contacts. We consider what happens if schools are given freedom to impose vaccine mandates on all of their pupils, except for the pupils that are exempt from vaccination because of medical reasons. We investigate how such a mandate affects the probability of an outbreak of a disease and the probability that a pupil that is medically exempt from vaccination, gets infected during an outbreak. We show that if the population vaccine coverage is close to the herd-immunity level then both probabilities may in
Vaccination is essential for the management of infectious diseases, many of which continue to pose devastating public health and economic challenges across the world. However, many vaccines are imperfect having only a partial protective effect in decreasing disease transmission and/or favouring recovery of infected individuals, and possibly exhibiting trade-off between these two properties. Furthermore, population turnover, that is the rate at which individuals enter and exit the population, is another key factor determining the epidemiological dynamics. While these factors have yet been studied separately, we investigate the interplay between the efficiency and property of an imperfect vaccine and population turnover. We build a mathematical model with frequency incidence rate, a recovered compartment, and an heterogeneous host population with respect to vaccination. We first compute the basic reproduction number $\mathcal{R}_0$ and study the global stability of the equilibrium points. Using a sensitivity analysis, we then assess the most influential parameters determining the total number of infected and $\mathcal{R}_0$ over time. We derive analytically and numerically conditions
This paper discusses two main themes. First, it investigates the formation of a spatiotemporal cognitive map (mental image) of a road network in travelers memory, which entails the travelers global conceptual understanding of congestion or the degree of crowding of the network. Second, it tries to investigate how latent learning of travelers from previous experiences shapes parts of the mental image, even for the parts of the network with which the travelers are unfamiliar. An experiment of route choice experiences was conducted among 90 participants in order to gain insight into the formation of a cognitive map and latent learning. In this experiment, the following independent variables are connected to the formation of a mental image of the network and the quality of the generalization of the unfamiliar parts of the network: (i) dispersion of the links travel time throughout the network, (ii) number of trips the traveler makes, (iii) travelers gender, (iv) travelers driving experiences, (v) travelers natural level of optimism or pessimism, (vi) salient or noticeable features on the network, and (vii) the presence of traffic signals. Several nonparametric (distribution-free) tests
The success of precision medicine requires computational models that can effectively process and interpret diverse physiological signals across heterogeneous patient populations. While foundation models have demonstrated remarkable transfer capabilities across various domains, their effectiveness in handling individual-specific physiological signals - crucial for precision medicine - remains largely unexplored. This work introduces a systematic pipeline for rapidly and efficiently evaluating foundation models' transfer capabilities in medical contexts. Our pipeline employs a three-stage approach. First, it leverages physiological simulation software to generate diverse, clinically relevant scenarios, particularly focusing on data-scarce medical conditions. This simulation-based approach enables both targeted capability assessment and subsequent model fine-tuning. Second, the pipeline projects these simulated signals through the foundation model to obtain embeddings, which are then evaluated using linear methods. This evaluation quantifies the model's ability to capture three critical aspects: physiological feature independence, temporal dynamics preservation, and medical scenario d
Tropical circuits are circuits with Min and Plus, or Max and Plus operations as gates. Their importance stems from their intimate relation to dynamic programming algorithms. The power of tropical circuits lies somewhere between that of monotone boolean circuits and monotone arithmetic circuits. In this paper we present some lower bounds arguments for tropical circuits, and hence, for dynamic programs.
In this paper, we present a unified study of the moduli space of tropical curves and Outer space which we link via period maps to the moduli space of tropical abelian varieties and the space of positive definite quadratic forms. Our work is a first step towards exhibiting Outer space and the space of positive definite quadratic forms as analogues of Teichmüller space and Siegel space, respectively, in tropical geometry. All these spaces and the maps among them are described within the category of ideal stacky fans, which we describe in detail.
We show that the weights on a tropical variety can be recovered from the tropical scheme structure proposed by the Giansiracusas in arXiv:1308.0042, so there is a well-defined Hilbert-Chow morphism from a tropical scheme to the underlying tropical cycle. For a subscheme of projective space given by a homogeneous ideal I we show that this tropical scheme structure contains the same information as the set of valuated matroids of the vector spaces I_d for d \geq 0. We also give a combinatorial criterion to determine whether a given relation is in the congruence defining the tropical scheme structure.
We study the geometry of tropical extensions of hyperfields, including the ordinary, signed and complex tropical hyperfields. We introduce the framework of 'enriched valuations' as hyperfield homomorphisms to tropical extensions, and show that a notable family of them are relatively algebraically closed. Our main results are hyperfield analogues of Kapranov's theorem and the Fundamental theorem of tropical geometry. Utilising these theorems, we introduce fine tropical varieties and prove a structure theorem for them in terms of their initial ideals.