alex carney

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Alex Carney © 2021

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I'm currently a Visiting Assistant Professor at the University of Rochester. Before coming here, I did my undergrad in math at University of Michigan, completed masters degrees in mathematics (Part III) at Cambridge and in science and technology studies at University College London while studying in the UK as a Marshall Scholar, and most recently completed my PhD in math at UC Berkeley in 2019, advised by Xinyi Yuan.

Additionally, I serve as communications director for the Juara Foundation, and spend my summers teaching music and leading education, health, and sustainable technology projects in Mato Grosso, Brazil. I can be found teaching and performing on violin in both the US and Brazil (and occasionally further afield).

Outside of academic and foundation work, I enjoy distance running, playing music, photography, and working on a 1994 VW Kombi I've converted to run on solar power.

Content and photos © Alex Carney 2021

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My current research is primarily in arithmetic geometry and arithmetic dynamics. I use heights and Arakelov theory to study the dynamics of rational points on algebraic varieties, and am interested in dynamical versions of well-known theorems and conjectures about rational points on abelian varieties. You can find more details in the introduction to my thesis here.

In addition to mathematics, I do research in music education and in science and technology studies, focused on music and sound. I've developed an interdisciplinary workshop, *Pantanal Sonora*, which combines musical composition and improvisation with environmental education, ecology, math, and physics, and which I've run in Brazil and the US, in-person and virtually.

I also occasionally wade into other fields where math expertise can be useful. I'm currently contributing to a project modeling the geometry of ancient rivers on Mars, have advised a number of engineering field research projects in the past through my work with the Juara Foundation, and serve as a pro-bono technical expert consultant for Public Defenders in Alameda and Contra Costa Counties in California through PDQuery.

I've mentored a number of undergraduate research projects, including in the Brazilian Pantanal alongside my work with the Juara Foundation. Students interested in getting involved can contact me here.

A. Carney* Specialization of canonical heights on abelian varieties*, preprint, 2021.

Given a family of abelian varieties over a quasiprojective smooth curve T^0 over a global field and a point P on the generic fiber, we show that the Néron-Tate canonical height h_{X_t}(P_t) of P_t along each fiber is exactly equal to a Weil height h_{\bar M}(t) given by an adelic metrized line bundle \bar M on the unique smooth projective curve T containing T^0. As a consequence, we show that a conjecture of Zhang on the finiteness of small-height specializations of P is equivalent to \bar M being big.

A. Carney, W. Hindes, T. Tucker* Integral points in orbits in characteristic p*, preprint, 2021.

We prove a characteristic p version of a theorem of Silverman on integral points in orbits over number fields and establish a primitive prime divisor theorem for polynomials in this setting. We provide some applications of these results, including a finite index theorem for arboreal representations coming from quadratic polynomials over function fields of odd characteristic.

Hayden, AT; Lamb, MP; Carney, AJ, *Similar curvature-to-width ratios for channels and channel belts: Implications for paleo-hydraulics of fluvial ridges on Mars, *Geology, v. 49, p. 837-841, 2021.

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.

A. Carney, *Heights and arithmetic dynamics over finitely generated fields*, preprint, 2020.

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.

A. Carney, *The arithmetic Hodge index theorem and rigidity of dynamical systems over function fields*, Pac. J. Math., 309(1):71-102, 2020

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.

E. Shirley, A. Carney, M. Alves Pereira, *Teaching soundscapes in the Brazilian Pantanal: Benefits of integrating music and science education*, Proceedings of the 34th International Society of Music Educators World Conference, Helsinki, Finland (2020), 499–506.

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.

A. Carney, R. Hortsch, M. Zieve, *Near-injectivity of polynomial functions on number fields, *in progress.

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.

E. Shirley, A. Carney, C. Hannaford, G. Ewing, *Using music to teach ecology and conserva- tion: a pedagogical case study from the Brazilian Pantanal*, Proceedings of the 33rd International Society of Music Educators World Conference, Baku, Azerbaijan (2018), 169–176.

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.

A. Carney, A. Etropolski, S. Pitman, *Powers of the eta-function and Hecke operators*, International Journal of Number Theory, 8 (2012), no. 3, 599–611.

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.

A. Carney, A, Khodkar, *Signed edge k-domination numbers in graphs*, Bulletin of the Institute of Combinatorics and its Applications 62 (2011), 66–78.

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.

A. Carney, *The arithmetic Hodge-index theorem and rigidity of algebraic dynamical systems overfunction fields*, PhD dissertation, UC Berkeley, 2019.

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.

A. Carney, *Telematics: A Case Study in the Co-creation of Music and Technology*, Master's dissertation, University College London, 2014.

A. Carney, *Uniform Boundedness of Rational Points*, Master's dissertation (Part III), Churchill College, University of Cambridge, 2013.

Berkeley Center for Law and Technology, Law and Technology Speaker Series, March 17, 2022.

*Torsion points in families and big adelic line bundles*, Upstate NY Number Theory Conference, Union college, October 23, 2021.

Given a family of abelian varieties A → T over a quasiprojective curve with a point P on the generic fiber, I’ll show that the Néron-Tate height of P_t on each fiber A_t is a height on T given by a metrized line bundle \bar M_P . I’ll then show that this makes a conjecture of Zhang on specializations of small height equivalent to the bigness of \bar M_P .

*A more geometric approach to specialization of canonical heights,* University of Rochester Algebra and Number Theory Seminar, September 23, 2021

Canonical heights, for example the N\’eron-Tate height on an abelian variety, are one of the key tools for studying both abelian varieties and algebraic dynamical systems. Given a family $X\to T$ of dynamical systems with a section $P:T\to X$, the content of several theorems and conjectures is essentially that the canonical height of $P_t$ on each fiber should be given by a height $h_P$ on $T$. I’ll show how, using adelic line bundles, this problem can be translated into arithmetic intersection theory, and prove the corresponding statement for a one-parameter family of abelian varieties.

*Variation of Canonical Heights via Adelic Line Bundles*, Arakelov Theory Conference, Regensburg, Germany, September 7, 2021.

In this talk I’ll explain how adelic line bundles provide a geometric realization of canonical height functions, then use this to show that, given a family of abelian varieties over a curve, the canonical height of a section across each fiber produces a Weil height on the base. I’ll then show how this provides partial results for more general families of dynamical systems, and discuss the possibilities and difficulties of this approach to higher dimensional families.

*Canonical heights and vector heights in families*, Harvard Arithmetic Dynamics Seminar, April 9, 2021.

Canonical heights are a standard tool of arithmetic dynamics over global fields. When studying families of dynamical systems, or systems over larger fields, however, there are significant geometric obstacles to constructing canonical heights. Either Northcott fails to hold, or the construction requires a model for the family with such strict properties (good reduction, minimality, extensions of rational maps,…) that such a model is unlikely to exist outside of very special settings. Instead, I’ll show how to resolve this problem using vector-valued heights, first introduced in characteristic zero by Yuan and Zhang. These generalize R-valued heights, produce canonical heights for any polarized dynamical system, and exhibit Northcott finiteness conditional on a strong non-isotriviality condition, generalizing work of Lang-Neron, Baker, and Chatzidakis-Hrushovski. This is achieved by working simultaneously over a system of models with much weaker requirements. As part of ongoing work, I’ll show how these arithmetic methods can produce results that hold over any field, and discuss how this can extend to quasi-projective varieties as well as projective varieties.

*Rational polynomials are (almost) always injective.*, AMS Spring Sectional Meeting, Special Session on Algebraic Geometry in Dynamics, Providence, RI, March 20-21, 2021.

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.

*Reaching new heights: measuring complexity in geometry and dynamics
*Astronaut Scholarship Foundation Scientific Connections Conference, March 11-12, 2021.

Aritmetic Dynamics International Online Seminar (ADIOS), February 17, 2021.

*Heights, dynamics, and isotriviality over finitely generated fields*, Joint Mathematics Meeting, AMS Special Session on Current Trends in Arithmetic Dynamics, January 9, 2021.

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.

*Birdsong, Composition, and Improvisation*, online workshop with Nashville Symphony Education Program, October, 2020.

*Heights and dynamics over arbitrary fields*, Princeton University/Institute for Advanced Study Number Theory Seminar, October 15, 2020.

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: an elaboration on birdsong to teach contemporary music techniques and science*, ISME world conference, PRIME special interest group video presentation and live presentation, September 2nd, 2020.

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 arithmetic Hodge-index theorem and dynamical systems*, AMS Spring Central and Western Joint Sectional Meeting, Special section on arithmetic dynamics, Manoa, Hawaii, March 23, 2019.

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.

*Near injectivity of polynomial functions on number fields*, AMS Spring Central and Western Joint Sectional Meeting, Special section on algebraic points, Manoa, Hawaii, March 23, 2019.

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.

*Absolute Hodge cycles on abelian varieties of CM-type*, UC Berkeley Number Theory Seminar, March 13, 2019.

*The arithmetic Hodge-index theorem and dynamical systems*, Rutgers Junior Number Theory Days, Newark, NY, November 2, 2018.

*The arithmetic Hodge-index theorem and dynamical systems*, Morningside Center of Mathematics, Chinese Academy of Sciences, Beijing, China, October 19, 2018.

*Using music to teach ecology and conservation: a pedagogical case study from the Brazilian Pantanal*, 33rd International Society of Music Educators World Conference, Baku, Azerbaijan July 17, 2018.

*Technology and Intercultural Exchange in a Sustainable Model for Music Education*, International Society of Music Educators World Conference, Porto Alegre, Rio Grande do Sul, Brazil, July 23, 2014.

*Hecke actions on powers of the Dedekind eta-function and vector partitions*, Hawai'i Conference in Algebraic Number Theory, Arithmetic Geometry and Modular Forms, Honolulu, Hawaii, March 7, 2012.

Alex Carney © 2021

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Monday 2-3pm, 1015 Hylan

Tuesday 5-6pm, on Zoom (link on Blackboard)

Math 210, Introduction to Financial Mathematics

Math 233, Introduction to Cryptology

Math 210, Introduction to Financial Mathematics

Math 164, Multivariable Calculus. Course info on Blackboard

Math 233, Introduction to Cryptology. Course info on Blackboard. Computational examples in Sage

Alex Carney © 2021

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Service

I am a cofounder, and serve as communications director on the board of directors of the Juara Foundation, a 501(c)3 non-profit dedicated to education and conservation. Our work is primarily based in the Brazilian Pantanal and surrounding areas. In addition to running our online presence and photography, I lead our Pantanal Music Exchange project, and teach and perform on violin throughout Mato Grosso in partership with Instituto Ciranda, a local youth music organization. You can find out more about my work and our other projects at juarafound.org

I cofounded, and serve as a technical expert for PDQuery, a project which connects public defenders with grad students and early career specialists for consultations on technical evidence. We aim to better equip public defenders to handle a variety of scientific and technical evidence, to defend against forensic evidence that lacks scientific merit, and to support the accused's 6th ammendment rights.

I am a reviewer for the American Mathematical Society's Mathematical Reviews on MathSciNet.

At UC Berkeley I served for several years on the boards of the Math Grad Student Association (MGSA) and Unbounded Representation (URep). MGSA organizes social and mentoring events for math grad students, and URep promotes diversity in the math department as well as healthy department culture.

Alex Carney © 2021

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Etcetera

You can vew some of my photos by clicking here. Most are either from travels or from my summer work in Brazil with the Juara Foundation.

A spectrogram visualization tool I built to go along with music education and bioacoustics research projects.

Summer and Fall 2020 I led a small team doing data analysis to predict poll worker shortages for the 2020 presidential election, and worked with Power the Polls to help adivse their new poll worker recruiting efforts. This graphic shows the potential impact of new worker recruitment around the country.

In 2014, the '94 VW Kombi Juara Foundation we had been driving around Brazil finally broke down completely. So we did what any reasonable person would do: tore out the motor, took a grinder to the roof, and put in four solar panels and a questionably-legally-imported 20hp electric motor. It's still a work in progress, but we've already won several bets and taken trips around Pocone. You can see test-drive footage on our youtube page.

See me performing a Beatles Medley with a piano quartet back at TED^{x}UofM back in 2012.

Alex Carney © 2021

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Contact

Alex Carney

Office 1015, Hylan Hall

University of Rochester

Alex Carney © 2021