arXiv subjects

Noah Giansiracusa

Publications and source records attributed to Noah Giansiracusa.

At least 19 recordsLinked to original sources

arXiv2024

Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal

A "tropical ideal" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal ideal to a tropical ideal. We call this the Macaulay tropical ideal. It has a universal property: any other extension of the given principal ideal to a tropical ideal with the expected Hilbert function is a weak image of the Macaulay tropical ideal. For each $n\geq 2$ and $d\geq 1$ our construction yields a non-realizable degree $d$ hypersurface scheme in $\mathbb{P}^n$. Maclagan-Rincón produced a non-realizable line in $\mathbb{P}^n$ for each $n$, and for $(d,n)=(1,2)$ the two constructions agree. An appendix by Mundinger compares the Macaulay construction with another method for canonically extending ideals to tropical ideals.

math.AG↗

arXiv2022

The universal tropicalization and the Berkovich analytification

Given an integral scheme X over a non-archimedean valued field k , we construct acuniversal closed embedding of X into a k-scheme equipped with a model over the field with one element (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of X by earlier work of the authors, and we show that the set-theoretic tropicalization of X with respect to this universal embedding is the Berkovich analytification of X. Moreover, using the scheme-theoretic tropicalization, we obtain a tropical scheme $Trop_{univ}(X)$ whose T-points give the analytification and which canonically maps to all other scheme-theoretic tropicalizations of X. This makes precise the idea that the Berkovich analytification is the universal tropicalization. When X = spec A is affine, we show that $Trop_{univ}(X)$ is the limit of the tropicalizations of X with respect to all embeddings in affine space, thus giving a scheme-theoretic enrichment of a well-known result of Payne. Finally, we show that $Trop_{univ}(X)$ represents the moduli functor of semivaluations on X, and when X = spec A is affine there is a universal semivaluation on A taking values in the idempotent semiring of regular functions on the universal tropicalization.

math.AG↗

arXiv2021

Point configurations, phylogenetic trees, and dissimilarity vectors

In 2004 Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map with a weighted variant, we show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter--Speyer envisioned. Moreover, we provide a geometric interpretation in terms of configurations of points on rational normal curves and construct a finite tropical basis that yields an explicit characterization of weighted dissimilarity vectors.

math.AG↗

arXiv2019

Chow quotients of Grassmannians by diagonal subtori

The literature on maximal torus orbits in the Grassmannian is vast; in this paper we initiate a program to extend this to diagonal subtori. Our main focus is generalizing portions of Kapranov's seminal work on Chow quotient compactifications of these orbit spaces. This leads naturally to discrete polymatroids, generalizing the matroidal framework underlying Kapranov's results. By generalizing the Gelfand-MacPherson isomorphism, these Chow quotients are seen to compactify spaces of arrangements of parameterized linear subspaces, and a generalized Gale duality holds here. A special case is birational to the Chen-Gibney-Krashen moduli space of pointed trees of projective spaces, and we show that the question of whether this birational map is an isomorphism is a specific instance of a much more general question that hasn't previously appeared in the literature, namely, whether the geometric Borel transfer principle in non-reductive GIT extends to an isomorphism of Chow quotients.

math.AG↗

arXiv2019

Equations for point configurations to lie on a rational normal curve

The parameter space of $n$ ordered points in projective $d$-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in $(\mathbb{P}^d)^n$. The resulting variety was used to study the birational geometry of the moduli space $\overline{\mathrm{M}}_{0,n}$ of $n$-tuples of points in $\mathbb{P}^1$. In this paper we turn to a more classical question, first asked independently by both Speyer and Sturmfels: what are the defining equations? For conics, namely $d=2$, we find scheme-theoretic equations revealing a determinantal structure and use this to prove some geometric properties; moreover, determining which subsets of these equations suffice set-theoretically is equivalent to a well-studied combinatorial problem. For twisted cubics, $d=3$, we use the Gale transform to produce equations defining the union of two irreducible components, the compactified configuration space we want and the locus of degenerate point configurations, and we explain the challenges involved in eliminating this extra component. For $d \ge 4$ we conjecture a similar situation and prove partial results in this direction.

math.AG↗

arXiv2019

A module-theoretic approach to matroids

Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces as modules over an idempotent semifield. All together, this provides bridges between the combinatorics of matroids, the algebra of idempotent modules, and the geometry of tropical linear spaces. The goal of this paper is to strengthen and expand these bridges by systematically developing the idempotent module theory of matroids. Applications include a geometric interpretation of strong matroid maps and the factorization theorem; a generalized notion of strong matroid maps, via an embedding of the category of matroids into a category of module homomorphisms; a monotonicity property for the stable sum and stable intersection of tropical linear spaces; a novel perspective of fundamental transversal matroids; and a tropical analogue of reduced row echelon form.

math.AG↗

arXiv2019

Fibonacci, golden ratio, and vector bundles

There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the exceptional group $G_2$ case of these bundles in three different ways, we derive a family of summation formulas for Fibonacci numbers in terms of the golden ratio.

math.AG↗

arXiv2018

Computational geometry and the U.S. Supreme Court

We use the United States Supreme Court as an illuminative context in which to discuss three different spatial voting preference models: an instance of the widely used single-peaked preferences, and two models that are more novel in which vote outcomes have a strength in addition to a location. We introduce each model from a formal axiomatic perspective, briefly discuss practical motivation for each in terms of judicial behavior, prove mathematical relationships among the voting coalitions compatible with each model, and then study the two-dimensional setting by presenting computational tools for working with the models and by exploring these with judicial voting data from the Supreme Court.

cs.CG↗

arXiv2018

Matroidal representations of groups

We develop the rudiments of a finite-dimensional representation theory of groups over idempotent semifields by considering linear actions on tropical linear spaces. This can be considered a tropical representation theory, a characteristic one modular representation theory, or a matroidal representation theory---and we draw from all three perspectives. After some general properties and constructions, including a weak tropical analogue of Maschke's theorem, we turn to a study of the regular representation of a finite group and its tropicalization. For abelian groups we find an interesting interplay between elementary number theory and matroid theory---even cyclic groups are surprisingly rich---and we conclude with some possible first steps toward a tropical character theory.

math.RT↗

arXiv2018

Spatial analysis of U.S. Supreme Court 5-to-4 decisions

While the U.S. Supreme Court is commonly viewed as comprising a liberal bloc and a conservative bloc, with a possible swing vote or median justice between them, surprisingly many case decisions are not explained by this simple model. We introduce a pair of spatial methods for conceptualizing many 5-to-4 voting alignments that have occurred on the Court and which defy the usual liberal/conservative dichotomy. These methods, utilizing higher order Voronoi diagrams and halving lines (k-sets), are based on the geometry of the two-dimensional ideal space locations obtained from applying multidimensional scaling to voting data. We also introduce a two-dimensional metric method for determining the crucial fifth vote in each 5-to-4 ruling and for determining the median justice in any collection of terms within a natural court.

cs.CY↗

arXiv2017

Persistence Terrace for Topological Inference of Point Cloud Data

Topological data analysis (TDA) is a rapidly developing collection of methods for studying the shape of point cloud and other data types. One popular approach, designed to be robust to noise and outliers, is to first use a smoothing function to convert the point cloud into a manifold and then apply persistent homology to a Morse filtration. A significant challenge is that this smoothing process involves the choice of a parameter and persistent homology is highly sensitive to that choice; moreover, important scale information is lost. We propose a novel topological summary plot, called a persistence terrace, that incorporates a wide range of smoothing parameters and is robust, multi-scale, and parameter-free. This plot allows one to isolate distinct topological signals that may have merged for any fixed value of the smoothing parameter, and it also allows one to infer the size and point density of the topological features. We illustrate our method in some simple settings where noise is a serious issue for existing frameworks and then we apply it to a real data set by counting muscle fibers in a cross-sectional image.

stat.ME↗

arXiv2017

Persistent homology machine learning for fingerprint classification

The fingerprint classification problem is to sort fingerprints into pre-determined groups, such as arch, loop, and whorl. It was asserted in the literature that minutiae points, which are commonly used for fingerprint matching, are not useful for classification. We show that, to the contrary, near state-of-the-art classification accuracy rates can be achieved when applying topological data analysis (TDA) to 3-dimensional point clouds of oriented minutiae points. We also apply TDA to fingerprint ink-roll images, which yields a lower accuracy rate but still shows promise, particularly since the only preprocessing is cropping; moreover, combining the two approaches outperforms each one individually. These methods use supervised learning applied to persistent homology and allow us to explore feature selection on barcodes, an important topic at the interface between TDA and machine learning. We test our classification algorithms on the NIST fingerprint database SD-27.

stat.ML↗

arXiv2017

From Poland to "Petersburg": The Banach-Tarski Paradox in Bely's Modernist Novel

Andrei Bely's novel "Petersburg," first published in 1913, was declared by Vladimir Nabokov one of the four greatest masterpieces of 20th-century prose. The Banach-Tarski Paradox, published in 1924, is one of the most striking and well-known results in 20th-century mathematics. In this paper we explore a potential connection between these two landmark works, based on various interactions with the Moscow Mathematical School and passages in the novel itself.

math.HO↗

arXiv2017

Modular interpretation of a non-reductive Chow quotient

The space of n distinct points and a disjoint parameterized hyperplane in projective d-space up to projectivity---equivalently, configurations of n distinct points in affine d-space up to translation and homothety---has a beautiful compactification introduced by Chen-Gibney-Krashen. This variety, constructed inductively using the apparatus of Fulton-MacPherson configuration spaces, is a parameter space of certain pointed rational varieties whose dual intersection complex is a rooted tree. This generalizes $\overline{M}_{0,n}$ and shares many properties with it. In this paper, we prove that the normalization of the Chow quotient of $(\mathbb{P}^d)^n$ by the diagonal action of the subgroup of projectivities fixing a hyperplane, pointwise, is isomorphic to this Chen-Gibney-Krashen space $T_{d,n}$. This is a non-reductive analogue of Kapranov's famous quotient construction of $\overline{M}_{0,n}$, and indeed as a special case we show that $\overline{M}_{0,n}$ is the Chow quotient of $(\mathbb{P}^1)^{n-1}$ by an action of a semidirect product of the additive and multiplicative group.

math.AG↗

arXiv2017

Geometry in the Courtroom

There has been a recent media blitz on a cohort of mathematicians valiantly working to fix America's democratic system by combatting gerrymandering with geometry. While statistics commonly features in the courtroom (forensics, DNA analysis, etc.), the gerrymandering news raises a natural question: in what other ways has pure math, specifically geometry and topology, been involved in court cases and legal scholarship? In this survey article, we collect a few examples with topics ranging from the Pythagorean formula to the Ham Sandwich Theorem, and we discuss some jurists' perspectives on geometric reasoning in the legal realm. One of our goals is to provide math educators with engaging real-world instances of some abstract geometric concepts.

math.HO↗

arXiv2017

A Grassmann algebra for matroids

We introduce an idempotent analogue of the exterior algebra for which the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces. The top wedge power of a tropical linear space is its Plucker vector, which we view as a tensor, and a tropical linear space is recovered from its Plucker vector as the kernel of the corresponding wedge multiplication map. We prove that an arbitrary d-tensor satisfies the tropical Plucker relations (valuated exchange axiom) if and only if the d-th wedge power of the kernel of wedge-multiplication is free of rank one. This provides a new cryptomorphism for valuated matroids, including ordinary matroids as a special case.

math.AG↗

arXiv2017

Mathematical Symbolism in a Russian Literary Masterpiece

Andrei Bely's modernist novel "Petersburg," first published in 1913, is considered a pinnacle of the Symbolist movement. Nabokov famously ranked it as one of the four greatest masterpieces of 20th-century prose. The author's father, Bugaev, was an influential mathematician and for 12 years served as the president of the Moscow Mathematical Society; he was also a source of inspiration for one of the main characters in the son's novel. While the philosophical views and political leanings of the mathematicians surrounding Bely, and their impact on "Petersburg," have been a topic of recent academic interest, there has not yet been a direct investigation of the surprisingly frequent and sophisticated mathematical passages in the book itself. We attempt here to rectify this gap in the scholarly literature, and in doing so find a rich tapestry of mathematical ideas and allusions.

math.HO↗

arXiv2016

The dual complex of $\bar{M}_{0,n}$ via phylogenetics

The moduli space $\bar{M}_{0,n}$ of stable rational n-pointed curves has divisorial boundary with simple normal crossings. In this brief note I observe that the dual complex is a flag complex; that is, a collection of irreducible boundary divisors has nonempty intersection if and only if the pairwise intersections are nonempty. Rather than proving this directly, I translate the statement to a setting in phylogenetics where it is widely used and multiple explicit proofs have been written. It appears this result is known by experts but lacks a detailed reference in the literature, except recently for $n=7$.

math.AG↗