alex carney

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About

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Alex Carney © 2020

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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 2020

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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 Science and Technology Studies, I'm interested in music technology, and how experimental artistic techniques and technological advances influence each other. I wrote my masters thesis, *Telematics: A Case Study in the Co-creation of Music and Technology* at UCL under the direction of Dr. Jack Stilgoe (2014). More broadly, while I research primarily in math these days, I try to keep an awareness of how the math, science and technology I work on interact with society, and still maintain a strong interest in music and music technology.

As a part of my Master of Advanced Study in Mathematics (more commonly called 'Part III') at Cambridge (2013), I wrote the essay *Uniform Boundedness of Rational Points*, supervised by Dr. Tom Fisher. This essay describes the work of several authors establishing that Lang's conjecture (for rational points on varieties of general type) implies the existence of a uniform bound on the number of rational points on a curve of genus g>1.

Hayden, AT; Lamb, MP; Carney, AJ, Similar curvature empiricisms for channels, channel belts, and valleys: No evidence for wider rivers on early Marsin progress.

Sinuous features on Earth and Mars have long been interpreted as rivers on the basis of matching long-standing empirical relationships between their curvature and width, yet the implicit hypothesis that this relationship is unique to rivers is untested. We developed a new automated tool to measure width, radius of curvature, and wavelength along sinuous features given their edges in order to test this hypothesis. This tool avoids biases of prior methods stemming from scale of digitization and manual measurement, and allows adjustment of digitization at different scales. We found that channels, channel belts, and valleys all have radius of curvature 2-4 times their width, and wavelength 4-5 times their width. This stems from a simple geometric reason—width serves to smooth out many smaller curves into fewer larger ones. Because channels and channel belts are fractal, the wavelength-width relationship for all these features is well explained by fractal geometry. We further highlight the fractal nature of channels and channel belts by approximating them with a random walk. Together, this indicates that relying on curvature-width relationships to interpret sinuous features as individual rivers might be in error. In particular, this puts into question prior interpretations of sinuous ridges on Mars as paleochannels, as well as analyses based on these interpretations, such as the finding that channels on Mars were larger than those on Earth. It is more likely that these sinuous features all represent channel belts or valleys and require no extraordinary situation. Because paleohydrology on Mars is often interpreted on the basis that ridge or valley width is equal to paleochannel width, these interpretations are likely too high. We thus conclude that the instantaneous volume of liquid-water flow on Mars is lower than previously determined, and that instead longer-duration flows are required to explain the paleochannel evidence.

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.

A. Carney, *The Hodge-index theorem for arithmetic intersections over function fields*, submitted 2019.

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, 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.

*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.

*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.

*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 © 2020

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Tuesday, Wednesday 3:30-5pm, 1015 Hylan Hall

Math 164, Multivariable Calculus

Math 282, Intro To Complex Variables

Math 143, Calculus III, MW 3:25-4:40

Since registration is currently full, and UofR rolled out their new URStudent system before it has waitlist functionality, I'm maintaining a waitlist here. It will be kept up-to-date as spots open up. Unfortunately I can't promise how many or if any spots will open up, and I doubt much will change before the start of fall term. Please email me to be added to the list, then I'll email you back with a tropical bird, and your bird will appear on the waitlist below to keep things anonymous. Number 1 is first off the list.

Hyacinth Macaw

Roseate Spoonbill

Capped Heron

Bare-Faced Curassow

Blue and Yellow Macaw

Chestnut-Eared Araçari

Yellow-Billed Cardinal

Jabiru Stork

Toco Toucan

Monk Parakeet

Math 140, Foundations of Calculus, MW 4:50-6:05

Alex Carney © 2020

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Sections 205,211,214 with Prof. Alexander Paulin

**Office Hours, 1065 Evans Hall:**

Tuesday, 1-3pm, Wednesday 10-11am.

Homework is due in section every Thursday, and Quizzes will be given in section every other week. I'll drop your lowest quiz score and lowest two homework scores, and quizzes will be curved as needed. No late homework or makeup quizzes will be accepted except in extreme circumstances, which must be arranged as far in advance as possible.

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Sections 202,203 with Prof. James Sethian

**Office Hours, 1065 Evans Hall:**

Thursday 10am-12pm

Homework is due in section every Monday, and we'll have short quizzes at the beginning of section every Friday. I don't accept late work or do makeup quizzes, but I will drop your lowest two scores at the end of term.

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Sections 203,205 with Prof. Ming Gu

Course Webpage and Course Schedule

**Office Hours, 1065 Evans Hall:**

Monday 9:30-10:30am, Thursday 11am-12pm

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Tuesday 3:30-5pm, room 732 (The hexagon room)

Since many of us are attending Arizona Winter School later this term, the goal is to use this seminar to prepare, focusing particularly on the Shimura Varieties and Adic Spaces sessions. Each talk should dedicate a good amount of time to interesting examples and problems. After AWS we may continue to explore further topics in these areas, or head in a new direction TBD.

Talk notes are linked when available

1/31 - Adic Spaces (Sander)

2/6 - Formal Schemes (Zixin and Alex)

2/13 - Perfectoid Spaces (Joe)

2/20 - Shimura Varieties I

2/27 - Shimura Varieties II

3/6 - Shimura Varieties III

3/13 - AWS (no seminar)

3/20 and beyond - TBD

AWS Course Outlines and notes (More likely to appear)

Florent Martin, Adic Spaces (good short summary)

PASTA (a similar seminar currently running at Emory)

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Tuesday 3:30-4:30, Room TBD

9/6 - Classical Theory of Modular Forms (Rahul)

9/13 - Automorphic Forms for SL_2(R) Part 1 (Archit)

9/20 - Automorphic Forms for SL_2(R) Part 2 (Rahul)

9/27 - Adelic Automorphic Forms (Joey)

10/4 - Representations of GL_2 over local fields (Alex "Big Frankie" Youcis)

10/11 - Representations of GL_2(A) (Alex "Don't call me Frank" Youcis)

10/18 - Hecke Theory/Jacquet Langlands (Sander)

10/25 - Jacquet Langlands Part 2 (Dylan)

11/1 - Trace Formula Part 1 (Alex Carney)

11/8 - Trace Formula Part 2 (Aaron Brookner)

11/15 - Automorphic Forms for Quaternion Algebras (Dylan)

11/15+ Overflow + Additional Topics?

Automorphic Forms on Adele Groups (Gelbart)

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Tuesday 1-2pm, room 740

2/22 - Basic definitions II - Rahul

2/29 - Basic properties of maps - Carney

3/7 - Classical rigid varieties + connection - Dylan

3/14 - No talk (Arizona Winter School)

3/21 - No talk (Spring break)

3/28 - Uniformization of abelian varieties - Ben

4/4 - Intro to formal schemes - Ian

4/11 - Étale site - Bertie

4/18 - Perfectoid spaces - Youcis

4/25 - Tilting I - Sunny

5/2 - Tilting II - Bertie

5/9 - Applications - Youcis

[Sch] - Lectures on p-adic Geometry

[Sch 2] - Perfectoid Spaces

[Sch 3] - Perfectoid Spaces and their Applications

[Con] - Seminaire Scholze

[Con 2] Several approaches to rigid analytic geometry

[Hub 1] - Continuous Valuations

[Hub 2] - A generalization of formal schemes and rigid analytic varieties

[Hub 3] - Étale Cohomology of Rigid Analytic Varieties and Adic Spaces (Available on link.springer)

[Ill] - FGA Explained, Part 4 (can give you a copy if you need it)

[Wed] Introduction to Adic Spaces

[Ked] Topics in Algebraic Geometry (rigid analytic geometry)

[Tate] - Rigid Analytic Spaces

[Bosch] - Lectures on Formal and Rigid Geometry (available on link.springer)

[Bosch 2] - A mini-course on formal and rigid geometry

[Mum] - An analytic construction of degenerating abelian varieties over complete rings

[Pap] Rigid-analytic geometry and the uniformization of abelian varieties

[Sai] Perfectoid spaces and the weight-monodromy conjecture, after Peter Scholze

[Car] The Weight-Monodromy Conjecture

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Tuesday 3:30-4:30pm, room 740

9/3 Alex Carney - Introduction and Overview of Iwasawa Theory.

9/10 Shelly - Class Field Theory.

9/17 Derek Hollowood - Leopoldt's Conjecture.

9/24 Sander - P-adic L-functions.

10/1 Eugenia

10/8 Alex Bertoloni

10/15 Rahul

Future talks to be assigned

Introduction to Cyclotomic Fields - Washington

Introduction to Iwasawa Theory for Elliptic Curves - Ralph Greenberg

Iwasawa Theory for Elliptic Curves - Ralph Greenberg

Iwasawa Theory - Past and Present - Ralph Greenberg

Why is an ideal class group a Tate-Schaferevich group? - Kevin Buzzard

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Service

I 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

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 © 2020

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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.

Take a look at a simple spectrogram visualization tool I built to go along education and bioacoustics research projects in the Brazilian Pantanal.

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.

Alex Carney © 2020

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Contact

Alex Carney

Office 1015, Hylan Hall

University of Rochester

Alex Carney © 2020