Publications (68)
Metrized graphs, electrical networks, and Fourier analysis
Matthew Baker, Xander Faber
A metrized graph is a finite weighted graph whose edges are thought of as line segments. In this expository paper, we study the Laplacian operator on a metrized graph and some impo…
Chip-firing games, potential theory on graphs, and spanning trees
Matthew Baker, Farbod Shokrieh
We study the interplay between chip-firing games and potential theory on graphs, characterizing reduced divisors (-parking functions) on graphs as the solution to an energy (or…
Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory
Spencer Backman, Matthew Baker, Chi Ho Yuen
Let be a regular matroid. The Jacobian group of is a finite abelian group whose cardinality is equal to the number of bases of . This group generalizes th…
Riemann-Roch and Abel-Jacobi theory on a finite graph
Matthew Baker, Serguei Norine
It is well-known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of l…
The foundation of generalized parallel connections, 2-sums, and segment-cosegment exchanges of matroids
Matthew Baker, Oliver Lorscheid, Zach Walsh +1
We show that, under suitable hypotheses, the foundation of a generalized parallel connection of matroids is the relative tensor product of the foundations. Using this result, we sh…
Matroids over partial hyperstructures
Matthew Baker, Nathan Bowler
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this,…
Elliptic matroids and modular curves
Matthew Baker
For , let be the rank-3 matroid on whose bases are the three-element non-zero-sum subsets. Let denote the open subscheme of t…
Descartes' rule of signs, Newton polygons, and polynomials over hyperfields
Matthew Baker, Oliver Lorscheid
We develop a theory of multiplicities of roots for polynomials over hyperfields and use this to provide a unified and conceptual proof of both Descartes' rule of signs and Newton's…
Lift theorems for representations of matroids over pastures
Matthew Baker, Oliver Lorscheid
Pastures are a class of field-like algebraic objects which include both partial fields hyperfields and have nice categorical properties. We prove several lift theorems for represen…
A lower bound for average values of dynamical Green's functions
Matthew Baker
We provide a Mahler/Elkies-style lower bound for the average values of dynamical Green's functions on the projective line over an arbitrary valued field, and give some dynamical an…
On Channel Model to Bridge the Gap between MIMO Design and Performance Requirements in 3GPP
Lynda Berrah, Raphael Visoz, Didier Le Ruyet +5
Accurate channel modeling has become critical for evaluating multiple-input multiple-output (MIMO) performance, especially as 5G standardization matures and efforts toward 6G begin…
Robert F. Coleman 1954-2014
Matthew Baker, Barry Mazur, Ken Ribet
Robert F. Coleman, a highly original mathematician who has had a profound influence on modern number theory and arithmetic geometry, passed away on March 24, 2014. We give an overv…
Necklaces, permutations, and periodic critical orbits for quadratic polynomials
Matthew Baker, Andrea Chen, Sophie Li +1
Let denote the Gleason polynomial, whose roots correspond to parameters such that the critical point is periodic of exact period under iteration of $…
Global discrepancy and small points on elliptic curves
Matthew Baker, Clayton Petsche
Let E be an elliptic curve defined over a number field k. In this paper, we define the ``global discrepancy'' of a finite set Z of algebraic points on E which in a precise sense me…
Automorphisms of
Matthew Baker, Yuji Hasegawa
We determine the automorphism group of the modular curve for all prime numbers .
Lower bounds for the canonical height on elliptic curves over abelian extensions
Matthew Baker
Let K be a number field and let E/K be an elliptic curve. If E has complex multiplication, we show that there is a positive lower bound for the canonical height of non-torsion poin…
The sesquicentennial of the prime number
Matthew Baker
The year 2026 marks the 150th anniversary of a remarkable achievement in mathematics: in 1876, the French mathematician Édouard Lucas showed that the 39-digit number $M_{127} := 2^…
The Bernardi process and torsor structures on spanning trees
Matthew Baker, Yao Wang
Let G be a ribbon graph, i.e., a connected finite graph G together with a cyclic ordering of the edges around each vertex. By adapting a construction due to O. Bernardi, we associa…
A finiteness theorem for canonical heights attached to rational maps over function fields
Matthew Baker
Let K be a function field, let f be a rational function of degree d at least 2 defined over K, and suppose that f is not isotrivial. In this paper, we show that a point P in P^1(Kb…
The moduli space of matroids
Matthew Baker, Oliver Lorscheid
In the first part of the paper, we clarify the connections between several algebraic objects appearing in matroid theory: both partial fields and hyperfields are fuzzy rings, fuzzy…
Uniform structures and Berkovich spaces
Matthew Baker
A uniform space is a topological space together with some additional structure which allows one to make sense of uniform properties such as completeness or uniform convergence. Mot…
Foundations of matroids -- Part 2: Further theory, examples, and computational methods
Matthew Baker, Oliver Lorscheid, Tianyi Zhang
In this sequel to "Foundations of matroids - Part 1", we establish several presentations of the foundation of a matroid in terms of small building blocks. For example, we show that…
Hodge theory in combinatorics
Matthew Baker
George Birkhoff proved in 1912 that the number of proper colorings of a finite graph G with n colors is a polynomial in n, called the chromatic polynomial of G. Read conjectured in…
Finiteness results for modular curves of genus at least 2
Matthew Baker, Enrique Gonzalez-Jimenez, Josep Gonzalez +1
A curve X over the field Q of rational numbers is modular if it is dominated by X_1(N) for some N; if in addition the image of its jacobian in J_1(N) is contained in the new subvar…
Matroid bingo
Matthew Baker, Hope Dobbelaere, Brennan Fullmer +1
We investigate some natural probability distributions associated with the game of matroid bingo.
On the Existence of Balanced Generalized de Bruijn Sequences
Matthew Baker, Bhumika Mittal, Haran Mouli +1
A balanced generalized de Bruijn sequence with parameters is a cyclic sequence of bits such that (a) the number of 0's equals the number of 1's, and (b) each substrin…
Equidistribution of small subvarieties of an abelian variety
Matthew Baker, Su-ion Ih
We prove an equidistribution result for small subvarieties of an abelian variety which generalizes the Szpiro-Ullmo-Zhang theorem on equidistribution of small points.
Lorentzian polynomials and matroids over triangular hyperfields 1: Topological aspects
Matthew Baker, June Huh, Mario Kummer +1
Lorentzian polynomials serve as a bridge between continuous and discrete convexity, connecting analysis and combinatorics. In this article, we study the topology of the space $\mat…
On the structure of hyperfields obtained as quotients of fields
Matthew Baker, Tong Jin
We determine all isomorphism classes of hyperfields of a given finite order which can be obtained as quotients of finite fields of sufficiently large order. Using this result, we d…
A Lower Bound for the Canonical Height on Abelian Varieties over Abelian Extensions
Matthew Baker, Joseph Silverman
Let A be an abelian variety defined over a number field K, and consider the canonical height function attached to a symmetric ample line bundle L on A. We prove that there is a pos…
The Jacobian of a Sixth-Root-of-Unity Matroid
Matthew Baker, Changxin Ding, Xu Zhuang
The Jacobian group (also called the sandpile group, Picard group, or critical group) of a graph or, more generally, of a regular matroid has been well studied. Sixth-root-of-unity…
Harmonic morphisms and hyperelliptic graphs
Matthew Baker, Serguei Norine
We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-…
Matroids over hyperfields
Matthew Baker, Nathan Bowler
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. We call the resulting objects…
Canonical representatives for divisor classes on tropical curves and the Matrix-Tree Theorem
Yang An, Matthew Baker, Greg Kuperberg +1
Let be a compact tropical curve (or metric graph) of genus . Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective…
Equidistribution of small points, rational dynamics, and potential theory
Matthew Baker, Robert Rumely
If phi(z) is a rational function on P^1 of degree at least 2 with coefficients in a number field k, we compute the homogeneous transfinite diameter of the v-adic filled Julia sets…
Canonical Heights, Transfinite Diameters, and Polynomial Dynamics
Matthew Baker, Liang-Chung Hsia
Let phi(z) be a polynomial of degree at least 2 with coefficients in a number field K. Iterating phi gives rise to a dynamical system and a corresponding canonical height function,…
Special curves and postcritically-finite polynomials
Matthew Baker, Laura DeMarco
We study the postcritically-finite (PCF) maps in the moduli space of complex polynomials . For a certain class of rational curves in , we characte…
Preperiodic points and unlikely intersections
Matthew Baker, Laura DeMarco
In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points of one-dimensional complex dynamical systems. We show that for any fixed complex n…
On the structure of nonarchimedean analytic curves
Matthew Baker, Sam Payne, Joseph Rabinoff
Let K be an algebraically closed, complete nonarchimedean field and let X be a smooth K-curve. In this paper we elaborate on several aspects of the structure of the Berkovich analy…
A modern perspective on Tutte's homotopy theorem
Matthew Baker, Tong Jin, Oliver Lorscheid
We begin with a review of Tutte's homotopy theory, which concerns the structure of certain graph associated to a matroid (together with some extra data). Concretely, Tutte's path t…
Lifting harmonic morphisms II: tropical curves and metrized complexes
Omid Amini, Matthew Baker, Erwan Brugallé +1
In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a…
On a theorem of Lafforgue
Matthew Baker, Oliver Lorscheid
We give a new proof, along with some generalizations, of a folklore theorem (attributed to Laurent Lafforgue) that a rigid matroid (i.e., a matroid with indecomposable basis polyto…
Weight functions on Berkovich curves
Matthew Baker, Johannes Nicaise
Let be a curve over a complete discretely valued field . We give tropical descriptions of the weight function attached to a pluricanonical form on and the essential skel…
Galois theory and torsion points on curves
Matthew Baker, Kenneth A. Ribet
In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of A. Tamagawa and the present au…
The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves
Matthew Baker, Joseph Rabinoff
Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let Îbe its value group. Given a smooth, proper, connected K-c…
The Jacobian of a regular orthogonal matroid and torsor structures on spanning quasi-trees of ribbon graphs
Matthew Baker, Changxin Ding, Donggyu Kim
Previous work of Chan--Church--Grochow and Baker--Wang shows that the set of spanning trees in a plane graph is naturally a torsor for the Jacobian group of . Informally, th…
Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta
Omid Amini, Matthew Baker, Erwan Brugallé +1
Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are con…
Analysis and dynamics on the Berkovich projective line
Robert Rumely, Matthew Baker
This is a set of expanded lecture notes from the Berkovich Space seminar held at the University of Georgia during Spring, 2004. The purpose of the notes is to provide a non-technic…
Harmonic analysis on metrized graphs
Matthew Baker, Robert Rumely
We study the Laplacian on a metrized graph, and its eigenfunctions.
Fusion rules for pastures and tracts
Matthew Baker, Tianyi Zhang
Baker and Bowler defined a category of algebraic objects called tracts which generalize both partial fields and hyperfields. They also defined a notion of weak and strong matroids…
Uncountable sets and an infinite real number game
Matthew Baker
We give a short proof of the well-known fact that the unit interval [0,1] is uncountable by means of a simple infinite game. We also show using this game that a (non-empty) perfect…
On some notions of rank for matrices over tracts
Matthew Baker, Noah Solomon, Tianyi Zhang
Given a tract in the sense of Baker and Bowler and a matrix with entries in , we define several notions of rank for . In this way, we are able to unify and find conce…
Representability of orthogonal matroids over partial fields
Matthew Baker, Tong Jin
Let be nonnegative integers, and let . For a matroid of rank on the finite set and a partial field in the sense of Sempl…
5G NR Positioning Enhancements in 3GPP Release-18
Hyun-Su Cha, Gilsoo Lee, Amitava Ghosh +3
New radio (NR) positioning in the Third Generation Partnership Project (3GPP) Release 18 (Rel-18) enables 5G-advanced networks to achieve ultra-high accuracy positioning without de…
Reciprocity via Reciprocants
Matthew Baker
The determinant of a skew-symmetric matrix has a canonical square root given by the Pfaffian. Similarly, the resultant of two reciprocal polynomials of even degree has a canonical…
Nonarchimedean geometry, tropicalization, and metrics on curves
Matthew Baker, Sam Payne, Joseph Rabinoff
We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical mu…
Linear series on metrized complexes of algebraic curves
Omid Amini, Matthew Baker
A metrized complex of algebraic curves is a finite metric graph together with a collection of marked complete nonsingular algebraic curves, one for each vertex, the marked points b…
New building blocks for -geometry: bands and band schemes
Matthew Baker, Tong Jin, Oliver Lorscheid
We develop and study a generalization of commutative rings called bands, along with the corresponding geometric theory of band schemes. Bands generalize both hyperrings, in the sen…
Zolotarev's Magical Proof of Quadratic Reciprocity
Matthew Baker
We present a creative reimagining of Zolotarev's classical proof of the Law of Quadratic Reciprocity.
A finiteness property of torsion points
Matthew Baker, Su-Ion Ih, Robert Rumely
Let k be a number field, let E/k be an elliptic curve, and let S be a finite set of places of k contianing the archimedean places. Let F be an algebraic closure of k. We prove that…
Degeneration of Linear Series From the Tropical Point of View and Applications
Matthew Baker, David Jensen
In this survey, we discuss linear series on tropical curves and their relation to classical algebraic geometry, describe the main techniques of the subject, and survey some of the…
Contrastive Predictive Coding with Compression for Enhanced Channel State Feedback in Wireless Networks
Ahmed Y. Radwan, Fahad Syed Muhammad, Matthew Baker +1
Accurate and timely channel state information (CSI) is essential for next-generation wireless systems, yet existing works treat CSI compression and CSI prediction as separate probl…
The Reproducible Research Platform establishes a unified open science environment bridging data and software lifecycles across disciplines, from proposal to publication
Andreas P. Cuny, Henry Lütcke, Andrei-Valentin PlamadÄ +5
Many research groups aspire to make data and code FAIR and reproducible, yet struggle because the data and code life cycles are disconnected, executable environments are often miss…
Specialization of linear systems from curves to graphs
Matthew Baker
We investigate the interplay between linear systems on curves and graphs in the context of specialization of divisors on an arithmetic surface. We also provide some applications of…
Representation theory for polymatroids
Matthew Baker, June Huh, Donggyu Kim +2
We develop a theory of representations of (discrete) polymatroids over tracts in terms of Plücker coordinates and suitable Plücker relations. As special cases, we recover polymat…
Foundations of matroids I: Matroids without large uniform minors
Matthew Baker, Oliver Lorscheid
The foundation of a matroid is a canonical algebraic invariant which classifies representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures, a…
Lorentzian polynomials and matroids over triangular hyperfields 2: Analytic aspects
Matthew Baker, June Huh, Mario Kummer +1
Brändén and Huh showed that Lorentzian polynomials unify Hodge-Riemann relations in combinatorics: their supports are M-convex, and every M-convex set supports a Lorentzian polyn…
Metric Properties of the Tropical Abel-Jacobi Map
Matthew Baker, Xander Faber
Let X be a tropical curve (or metric graph), and fix a base point p on X. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(X) is canonic…