Matt Baker | Georgia Institute of Technology (original) (raw)

Papers by Matt Baker

University Lecture Series, 2008

Lecture Series 45 (2008), published by the American Mathematical Society. The author was supporte... more Lecture Series 45 (2008), published by the American Mathematical Society. The author was supported by NSF grants DMS-0602287 and DMS-0600027 during the writing of these lecture notes. The author would like to thank Xander Faber, David Krumm, and Christian Wahle for sending comments and corrections on an earlier version of the notes, Michael Temkin and the anonymous referees for their helpful comments, and Clay Petsche for his help as the author's Arizona Winter School project assistant. Finally, the author would like to express his deep gratitude to Robert Rumely for his inspiration, support, and collaboration. 6 The alternate construction presented here is adapted from Berkovich's paper [Ber95].

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces... more We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids, as well as phased matroids in the sense of Anderson-Delucchi. We call the resulting objects matroids over hyperfields. In fact, there are (at least) two natural notions of matroid in this context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, Grassmann-Plücker functions, and dual pairs, and establish some basic duality theorems. We also show that if F is a doubly distributive hyperfield then the notions of weak and strong matroid over F coincide.

A metrized graph is a finite weighted graph whose edges are thought of as line segments. In this ... more 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 important functions related to it, including the ``j-function'', the effective resistance, and eigenfunctions of the Laplacian. We discuss the relationship between metrized graphs and electrical networks, which provides some

Contemporary Mathematics, 2006

A metrized graph is a weighted graph whose edges are viewed as line segments, or alternatively, i... more A metrized graph is a weighted graph whose edges are viewed as line segments, or alternatively, it is a singular Riemannian 1-manifold. In this expository paper, we study the Laplacian operator on a metrized graph and some important objects related to it, including the "j-function", the effective resistance, and the "canonical measure". We discuss the relationship between metrized graphs and electrical networks, which provides some physical intuition for the concepts being dealt with. We also explain the relation between the Laplacian on a metrized graph and the combinatorial Laplacian matrix. Finally, we obtain a new proof of Foster's network theorem.

Journal of Algebraic Combinatorics, 2011

Let Γ be a tropical curve (or metric graph), and fix a base point p ∈ Γ. We define the Jacobian g... more Let Γ be a tropical curve (or metric graph), and fix a base point p ∈ Γ. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(Γ) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for Γ. This result is useful for reducing certain questions about the Abel-Jacobi map Φp : Γ → J(Γ), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2-connected component of G are commensurable over Q. As an application of our direct limit theorem, we derive some local comparison formulas between ρ and Φ * p (ρ) for three different natural "metrics" ρ on J(Γ). One of these formulas implies that Φp is a tropical isometry when Γ is 2-edge-connected. Another shows that the canonical measure µ Zh on a metric graph Γ, defined by S. Zhang, measures lengths on Φp(Γ) with respect to the "sup-norm" on J(Γ).

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

Let K be a function field, let ϕ ∈ K(T ) be a rational map of degree d ≥ 2 defined over K, and su... more Let K be a function field, let ϕ ∈ K(T ) be a rational map of degree d ≥ 2 defined over K, and suppose that ϕ is not isotrivial. In this paper, we show that a point P ∈ P 1 (K) has ϕ-canonical height zero if and only if P is preperiodic for ϕ. This answers affirmatively a question of Szpiro and Tucker, and generalizes a recent result of Benedetto from polynomials to rational functions. We actually prove the following stronger result, which is a variant of the Northcott finiteness principle: there exists ε > 0 such that the set of points P ∈ P 1 (K) with ϕ-canonical height at most ε is finite. Our proof is essentially analytic, making use of potential theory on Berkovich spaces to study the dynamical Green's functions g ϕ,v (x, y) attached to ϕ at each place v of K. For example, we show that every conjugate of ϕ has bad reduction at v if and only if g ϕ,v (x, x) > 0 for all x ∈ P 1 Berk,v , where P 1 Berk,v denotes the Berkovich projective line over the completion ofK v .

International Mathematics Research Notices, 2009

We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between ... more 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-Hurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a 2-edge-connected graph G which is not a cycle, there is at most one involution ι on G for which the quotient G/ι is a tree. We also show that the number of spanning trees in a graph G is even if and only if G admits a non-constant harmonic morphism to the graph B2 consisting of 2 vertices connected by 2 edges. Finally, we use the Riemann-Hurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.

Duke Mathematical Journal, 2011

In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points... more 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 a, b ∈ C, and any integer d ≥ 2, the set of c ∈ C for which both a and b are preperiodic for z d + c is infinite if and only if a d = b d . This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if rational functions ϕ, ψ ∈ C(z) have infinitely many preperiodic points in common, then all of their preperiodic points coincide (and in particular the maps must have the same Julia set). This generalizes a theorem of Mimar, who established the same result assuming that ϕ and ψ are defined overQ. The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with non-archimedean Berkovich spaces playing an essential role. for helpful discussions, Daniel Connelly for computations related to Conjecture 1.6, and the anonymous referees for useful suggestions. We also thank Sarah Koch for generating the images in . Finally, we thank AIM for sponsoring the January 2008 workshop that inspired the results of this paper.

Canadian Journal of Mathematics, 2007

This paper studies the Laplacian operator on a metrized graph, and its spectral theory.

Algebra & Number Theory, 2008

Let k be a number field, and let G be either the multiplicative group G m /k or an elliptic curve... more Let k be a number field, and let G be either the multiplicative group G m /k or an elliptic curve E/k. Let S be a finite set of places of k containing the archimedean places. We prove that if α ∈ G(k) is nontorsion, then there are only finitely many torsion points ξ ∈ G(k) tors which are S-integral with respect to α. We also formulate conjectural generalizations for dynamical systems and for abelian varieties.

Algebra & Number Theory, 2008

We investigate the interplay between linear systems on curves and graphs in the context of specia... more 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 our results to graph theory, arithmetic geometry, and tropical geometry.

Advances in Mathematics, 2007

It is well-known that a finite graph can be viewed, in many respects, as a discrete analogue of a... more 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 linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann-Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph.

Annales de l’institut Fourier, 2006

Given a dynamical system associated to a rational function ϕ(T ) on P 1 of degree at least 2 with... more Given a dynamical system associated to a rational function ϕ(T ) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k,

University Lecture Series, 2008

Lecture Series 45 (2008), published by the American Mathematical Society. The author was supporte... more Lecture Series 45 (2008), published by the American Mathematical Society. The author was supported by NSF grants DMS-0602287 and DMS-0600027 during the writing of these lecture notes. The author would like to thank Xander Faber, David Krumm, and Christian Wahle for sending comments and corrections on an earlier version of the notes, Michael Temkin and the anonymous referees for their helpful comments, and Clay Petsche for his help as the author's Arizona Winter School project assistant. Finally, the author would like to express his deep gratitude to Robert Rumely for his inspiration, support, and collaboration. 6 The alternate construction presented here is adapted from Berkovich's paper [Ber95].

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces... more We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids, as well as phased matroids in the sense of Anderson-Delucchi. We call the resulting objects matroids over hyperfields. In fact, there are (at least) two natural notions of matroid in this context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, Grassmann-Plücker functions, and dual pairs, and establish some basic duality theorems. We also show that if F is a doubly distributive hyperfield then the notions of weak and strong matroid over F coincide.

A metrized graph is a finite weighted graph whose edges are thought of as line segments. In this ... more 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 important functions related to it, including the ``j-function'', the effective resistance, and eigenfunctions of the Laplacian. We discuss the relationship between metrized graphs and electrical networks, which provides some

Contemporary Mathematics, 2006

A metrized graph is a weighted graph whose edges are viewed as line segments, or alternatively, i... more A metrized graph is a weighted graph whose edges are viewed as line segments, or alternatively, it is a singular Riemannian 1-manifold. In this expository paper, we study the Laplacian operator on a metrized graph and some important objects related to it, including the "j-function", the effective resistance, and the "canonical measure". We discuss the relationship between metrized graphs and electrical networks, which provides some physical intuition for the concepts being dealt with. We also explain the relation between the Laplacian on a metrized graph and the combinatorial Laplacian matrix. Finally, we obtain a new proof of Foster's network theorem.

Journal of Algebraic Combinatorics, 2011

Let Γ be a tropical curve (or metric graph), and fix a base point p ∈ Γ. We define the Jacobian g... more Let Γ be a tropical curve (or metric graph), and fix a base point p ∈ Γ. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(Γ) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for Γ. This result is useful for reducing certain questions about the Abel-Jacobi map Φp : Γ → J(Γ), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2-connected component of G are commensurable over Q. As an application of our direct limit theorem, we derive some local comparison formulas between ρ and Φ * p (ρ) for three different natural "metrics" ρ on J(Γ). One of these formulas implies that Φp is a tropical isometry when Γ is 2-edge-connected. Another shows that the canonical measure µ Zh on a metric graph Γ, defined by S. Zhang, measures lengths on Φp(Γ) with respect to the "sup-norm" on J(Γ).

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

Let K be a function field, let ϕ ∈ K(T ) be a rational map of degree d ≥ 2 defined over K, and su... more Let K be a function field, let ϕ ∈ K(T ) be a rational map of degree d ≥ 2 defined over K, and suppose that ϕ is not isotrivial. In this paper, we show that a point P ∈ P 1 (K) has ϕ-canonical height zero if and only if P is preperiodic for ϕ. This answers affirmatively a question of Szpiro and Tucker, and generalizes a recent result of Benedetto from polynomials to rational functions. We actually prove the following stronger result, which is a variant of the Northcott finiteness principle: there exists ε > 0 such that the set of points P ∈ P 1 (K) with ϕ-canonical height at most ε is finite. Our proof is essentially analytic, making use of potential theory on Berkovich spaces to study the dynamical Green's functions g ϕ,v (x, y) attached to ϕ at each place v of K. For example, we show that every conjugate of ϕ has bad reduction at v if and only if g ϕ,v (x, x) > 0 for all x ∈ P 1 Berk,v , where P 1 Berk,v denotes the Berkovich projective line over the completion ofK v .

International Mathematics Research Notices, 2009

We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between ... more 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-Hurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a 2-edge-connected graph G which is not a cycle, there is at most one involution ι on G for which the quotient G/ι is a tree. We also show that the number of spanning trees in a graph G is even if and only if G admits a non-constant harmonic morphism to the graph B2 consisting of 2 vertices connected by 2 edges. Finally, we use the Riemann-Hurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.

Duke Mathematical Journal, 2011

In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points... more 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 a, b ∈ C, and any integer d ≥ 2, the set of c ∈ C for which both a and b are preperiodic for z d + c is infinite if and only if a d = b d . This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if rational functions ϕ, ψ ∈ C(z) have infinitely many preperiodic points in common, then all of their preperiodic points coincide (and in particular the maps must have the same Julia set). This generalizes a theorem of Mimar, who established the same result assuming that ϕ and ψ are defined overQ. The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with non-archimedean Berkovich spaces playing an essential role. for helpful discussions, Daniel Connelly for computations related to Conjecture 1.6, and the anonymous referees for useful suggestions. We also thank Sarah Koch for generating the images in . Finally, we thank AIM for sponsoring the January 2008 workshop that inspired the results of this paper.

Canadian Journal of Mathematics, 2007

This paper studies the Laplacian operator on a metrized graph, and its spectral theory.

Algebra & Number Theory, 2008

Let k be a number field, and let G be either the multiplicative group G m /k or an elliptic curve... more Let k be a number field, and let G be either the multiplicative group G m /k or an elliptic curve E/k. Let S be a finite set of places of k containing the archimedean places. We prove that if α ∈ G(k) is nontorsion, then there are only finitely many torsion points ξ ∈ G(k) tors which are S-integral with respect to α. We also formulate conjectural generalizations for dynamical systems and for abelian varieties.

Algebra & Number Theory, 2008

We investigate the interplay between linear systems on curves and graphs in the context of specia... more 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 our results to graph theory, arithmetic geometry, and tropical geometry.

Advances in Mathematics, 2007

It is well-known that a finite graph can be viewed, in many respects, as a discrete analogue of a... more 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 linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann-Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph.

Annales de l’institut Fourier, 2006

Given a dynamical system associated to a rational function ϕ(T ) on P 1 of degree at least 2 with... more Given a dynamical system associated to a rational function ϕ(T ) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k,