We develop a theory of vector-valued heights and intersections defined relative to finitely generated extensions K/k. These generalize both number field and geometric heights. When k is Q or F_p, or when a non-isotriviality condition holds, we obtain Northcott-type results. We then prove a version of the Hodge Index Theorem for vector-valued intersections, and use it to prove a rigidity theorem for polarized dynamical systems over any field.

The surface of Mars contains abundant sinuous ridges that appear similar to river channels in planform, but they stand as topographic highs. Ridges have similar curvature-to-width ratios as terrestrial meandering rivers, which has been used to support the hypothesis that ridges are inverted channels that directly reflect channel geometry. Anomalously wide ridges, in turn, have been interpreted as evidence for larger rivers on Mars compared to Earth. An alternate hypothesis, however, is that ridges are exhumed channel-belt deposits—a larger zone of relatively coarse-grained deposits formed from channel lateral migration and aggradation. Here we measured landform wavelength, radius of curvature, and width to evaluate whether they can be used to distinguish channels, channel-belts, and martian ridges. We found that all three landforms follow similar scaling relations, in which radius of curvature-to-width ratios range from 1.7-7.3, and wavelength-to-width ratios range from 5.8-13. We interpret this similarity to be a geometric consequence of a sinuous curved line of finite width. Combined with observations of ridge-stacking patterns, our results suggest that wide ridges on Mars more likely indicate fluvial channel-belts that formed over significant time, rather than anomalously large rivers.

In one of the fundamental results of Arakelov's arithmetic intersection theory, Faltings and Hriljac (independently) proved the Hodge Index Theorem for arithmetic surfaces by relating the intersection pairing to the negative of the Néron-Tate height pairing. More recently, Moriwaki and Yuan-Zhang generalized this to higher dimension. In this work, we extend these results to projective varieties over transcendence degree one function fields. The new challenge is dealing with non-constant but numerically trivial line bundles coming from the constant field via Chow's K/k-trace functor. As an application of the Hodge Index Theorem, we also prove a rigidity theorem for the set of canonical height zero points of polarized algebraic dynamical systems over function fields. For function fields over finite fields, this gives a rigidity theorem for preperiodic points, generalizing previous work of Mimar, Baker-DeMarco, and Yuan-Zhang.

Musical training naturally introduces students to physical properties of sound such as frequency and amplitude as they learn how to control pitch, tone, and volume. This connection between music and science, however, is much more profound than might be apparent. The same ear training and audial-interpretive methods musicians employ to understand and create music are fundamental to modern, technologically advanced scientific studies that seek to better understand our environment by listening to it. We built a field course for musicians that integrates music education with the physics of waves and ecological cycles, taught at a scientific research station with instruction from professionals in diverse fields. Here, we detail the curriculum we use to teach so-called “core” subjects of biology and physics through music. Our program emphasizes the soundscape concept—a realization that all sound can be viewed as music. This approach allows exploration of how multiple sounds come together to form the world we perceive around us, producing mashes of different timbres and melodies whose complexity can be interpreted altogether or parsed apart. Through a survey with open response questions, we demonstrate some of the musical and scientific lessons learned by students. We then discuss the benefits of such a program that extend beyond the purely pedagogical: integration of music into core subjects improves music and science education, as well as providing opportunity for more data collection and processing for research and improving the accessibility of contemporary techniques and compositions. More broadly, this type of combined program builds up the arts as an important part of science, nature, and culture, rather than a secondary, non-scholastic pursuit.

We show that, for any f(X) in Q[X], the function Q->Q defined by c mapsto f(c) is at most 6-to-1 outside a finite set. We prove analogous results in which Q is replaced by any finitely-generated field K of characteristic zero, where the number 6 is replaced by an explicit constant N. These results may be viewed as analogues for the affine line of the results of Mazur and Merel on rational torsion on elliptic curves, and this interpretation suggests a common generalization to the setting of morphisms between arbitrary varieties.

The Pantanal Sonora Project is an ongoing outreach project that unites music and environmental education and highlights the simultaneous promotion of musical development, empowerment, interest in science, as well as the conservation agenda of a natural heritage region. Interdisciplinary projects of this nature are soundly rooted in theory, but have not been thoroughly described in the literature, which instead focuses on infusing song lyrics with images of nature to promote conservation. Here we provide a concise review of the literature on music education to promote empowerment and conservation, and justify our method of uniting the two seemingly separate subjects. We then describe the curriculum and materials from the Pantanal Sonora Project, which is based in the Pantanal region of Brazil, a priority area for conservation. We set out empirical goals for future projects and describe limitations to the method we employed, suggesting that these limitations can be overcome in future projects. We further contend that this type of music and environmental education project has the potential to empower rural community members, increase interest in science, and may be used in introductory music teaching in addition to work with more advanced students.

Half-integer weight Hecke operators and their distinct properties play a majorrole in the theory surrounding partition numbers and Dedekind’s eta-function. General-izing the work of Ono, here we obtain closed formulas for the Hecke images of allnegative powers of the eta-function. These formulas are generated through the use ofFaber polynomials. In addition, congruences for a large class of powers of Ramanujan’sDelta-function are obtained in a corollary. We further exhibit a fast calculation for manylarge values of vector partition functions.

The closed neighborhood N_G[e] of an edge e in a graph G is the set consisting of e and of all edges having an end-vertex in common with e. Let f be a function on E(G), the edge set of G, into the set {-1,1} and let k>0 be an integer. If the sum of f(x) over {x in {N[e]}} is >=k$ for each edge e \in E(G), then f is called a signed edge k-domination function (SEkDF) of G. The signed edge k-domination number gamma'_{sk}(G) of G is defined as \gamma'_{sk}(G) = the minimum sum of f(e) over all SEkDFs of G. In this paper we calculate the signed edge k-domination numbers for complete graphs and complete bipartite graphs. We then show that, for any simple graph G, gamma'_{sk}(G) >= |V(G)| - |E(G)| + k - 1, and characterize all graphs that achieve this lower bound.

In one of the fundamental results of Arakelov’s arithmetic intersection theory, Faltingsand Hriljac (independently) proved the Hodge-index theorem for arithmetic surfaces byrelating the intersection pairing to the negative of the Neron-Tate height pairing. Morerecently, Moriwaki and Yuan–Zhang generalized this to higher dimension. In this work, weextend these results to projective varieties over transcendence degree one function fields.The new challenge is dealing with non-constant but numerically trivial line bundles comingfrom the constant field via Chow’sK/k-image functor. As an application of the Hodge-index theorem to heights defined by intersections ofadelic metrized line bundles, we also prove a rigidity theorem for the set height zero pointsof polarized algebraic dynamical systems over function fields. In the special case of a globalfield, this gives a rigidity theorem for preperiodic points, generalizing previous work of Mimar, Baker–DeMarco, and Yuan–Zhang.

Abstract: Abstract: Any polynomial f in Q[X] induces a map Q->Q. We show that this map is usually 1-to-1 and always at most 6-to-1 over all but finitely many values. Analogous bounds hold over every number field K. If we interpret Mazur and Merel's theorems on rational torsion of elliptic curves as bounding the rational N-to-1 behavior for morphisms between genus one curves, then our result can be seen as a parallel which doesn't require the structure of an abelian variety. We formulate a conjecture about morphisms between arbitrary varieties which implies both our result and the uniform boundedness conjecture for rational torsion on abelian varieties, and discuss implications to other conjectures and obstructions to proving it.

Abstract: Expanding on work of Moriwaki and Yuan-Zhang, we show how to define vector-valued arithmetic intersections and heights relative to any finitely generated field extension K/k. When K=Q$ or K/k has transcendence degree one, these reproduce the usual number field and geometric heights, respectively. Letting X be a projective variety over K, we prove that these heights have a Northcott property, provided either that k=Q or F_q, or that X is totally non-isotrivial over k, a necessary condition which is slightly stronger than not isotrivial. This generalizes previous work of Baker, Chatzidakis-Hrushovski, and the Lang-Néron Theorem. Now let f,g:X->X be two polarizable dynamical systems on X. Using the above vector-valued intersections and the Northcott property for heights, we prove the following rigidity theorem for preperiodic points: If Prep(f) intersect Prep(g) is Zariski dense in X, then Prep(f)=Prep(g). By the Lefschetz principle, this holds over any field.

Classically, heights are defined over number fields or transcendence degree one function fields. This is so that the Northcott property, which says that sets of points with bounded height are finite, holds. Here, expanding on work of Moriwaki and Yuan-Zhang, we show how to define arithmetic intersections and heights relative to any finitely generated field extension K/k, and construct canonical heights for polarizable arithmetic dynamical systems f:X->X. These heights have a corresponding Northcott property when k is Q or F_q. When k is larger, we show that Northcott for canonical heights is conditional on the non-isotriviality of f:X->X, generalizing work of Lang-Neron, Baker, and Chatzidakis-Hrushovski. Additionally, we prove the Hodge Index Theorem for arithmetic intersections relative to K/k. Since, when Northcott holds, preperiodic points are the same as height zero points, this has applications to dynamical systems. By the Lefschetz principle, these results can be applied over any field.

Pantanal Sonora is a multi-day workshop combining music and environmental education. While typically held out in nature, this year we invited students—and invite you—to experience the Pantanal virtually, learn to analyze it sonically, and create compositions and improvisations reflecting on both the Pantanal and your own environment.

The Hodge-index theorem in classical algebraic geometry states that the signature of the intersection form on a surface is +1,-1,...,-1. In one of the fundamental results of Arakelov theory, Faltings and Hriljac extend this to arithmetic surfaces by relating the intersection pairing to the negative of the N\'eron-Tate height pairing. In this talk, I'll explain how Yuan and Zhang (in the number field case) and my work (in the function field case) generalize this to higher dimensional varieties. An important use of arithmetic intersection theory is in defining height functions of both points and subvarieties, and as an example I will show how the Hodge-index theorem is used to prove a rigidity theorem for preperiodic points on algebraic dynamical systems.

Any polynomial f in Q[X] induces a map Q->Q. We show that this map is at most 6-to-1 over all but finitely many values. Analogous bounds hold over every number field K, depending only on the number of roots of unity zeta such that zeta+zeta^-1 is in K. If we interpret Mazur and Merel's theorems on rational torsion of elliptic curves as bounding the rational N-to-1 behavior for morphisms between genus one curves, then our result can be seen as a parallel for the affine line. We formulate a conjecture about morphisms between arbitrary varieties which implies both our result and the uniform boundedness conjecture for rational torsion on abelian varieties, and discuss possible techniques and obstructions to proving additional cases.