Articles

Title With Status
PDF Creating a Dynamic Database of Finite Groups Lewis Combes, John Jones, Jennifer Paulhus, Manami Roy, Sam Schiavone In preparation
PDF arXiv Hypergeometric L-functions in average polynomial time II Edgar Costa, Kiran Kedlaya Accepted to ANTS XVI
PDF arXiv DOI Counterexamples to a Conjecture of Ahmadi and Shparlinski Taylor Dupuy, Kiran Kedlaya, Christelle Vincent Experimental Mathematics 23 (2023) no. 3, pp 540-544
PDF arXiv DOI Commutative character sheaves and geometric types for supercuspidal representations Clifton Cunningham Annales Henri Lebesgue 4 (2021), pp 1389-1420
arXiv DOI Computing classical modular forms Alex Best, Jonathan Bober, Andrew Booker, Edgar Costa, John Cremona, Maarten Derickx, Min Lee, David Lowry-Duda, Andrew Sutherland, John Voight Arithmetic Geometry, Number Theory, and Computation, Simons Symp, Springer, 2021, pp 131-213
PDF arXiv DOI Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB Taylor Dupuy, Kiran Kedlaya, Christelle Vincent Arithmetic Geometry, Number Theory, and Computation, Simons Symp, Springer, 2021, pp 375-448
PDF DOI Zen and the art of database maintenance Edgar Costa Arithmetic Geometry, Number Theory, and Computation, Simons Symp, Springer, 2021, pp 277-292
PDF arXiv DOI Hypergeometric L-functions in average polynomial time Edgar Costa, Kiran Kedlaya Proceedings of the Fourteenth Algorithmic Number Theory Symposium (ANTS-XIV), Open Book Series 4, Math. Sci. Pub., Berkeley, 2020, pp 143-159
PDF arXiv DOI The Inverse Galois Problem for p-adic fields Proceedings of the Thirteenth Algorithmic Number Theory Symposium (ANTS-XIII), Open Book Series 2, Math. Sci. Pub., Berkeley, 2019, pp 393-409
PDF arXiv DOI From the function-sheaf dictionary to quasicharacters of p-adic tori Clifton Cunningham Journal of the Institute of Mathematics of Jussieu 17 (2018) no. 1, pp 1-37
PDF arXiv DOI ZpL: a p-adic precision package Xavier Caruso, Tristan Vaccon Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation. ACM, New York, 2018, pp 119-126
PDF arXiv DOI Characteristic polynomials of p-adic matrices Xavier Caruso, Tristan Vaccon Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation. ACM, New York, 2017, pp 389-396
PDF arXiv DOI Division and slope factorization of p-adic polynomials Xavier Caruso, Tristan Vaccon Proceedings of the 2016 ACM on International Symposium on Symbolic and Algebraic Computation. ACM, New York, 2016, pp 159-166.
PDF arXiv DOI Rectifiers and the local Langlands correspondence: the unramified case Moshe Adrian Mathematical Research Letters 23 (2016) no. 3, pp 593-619
PDF arXiv DOI p-adic Stability in Linear Algebra Xavier Caruso, Tristan Vaccon Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation. ACM, New York, 2015, pp 101-108.
PDF arXiv DOI The canonical measure on a reductive p-adic group is motivic Julia Gordon Annales scientifiques de l'École normale supérieure 50 (2015) no. 2, pp 345-355
PDF arXiv DOI The 3-adic eigencurve at the boundary of weight space International Journal of Number Theory 10 (2014), no. 7, 1791-1806.
PDF arXiv DOI Tracking p-adic precision Xavier Caruso, Tristan Vaccon LMS Journal of Computation and Mathematics 17 (Special issue A) (2014), 274-294.
PDF arXiv DOI Bounding Picard numbers of surfaces using p-adic cohomology Timothy Abbott, Kiran Kedlaya Arithmetic, Geometry and Coding Theory (AGCT 2005), Séminaires et Congrès 21, Societé Mathématique de France, 2009, 125-159. Magma source is available.
PDF arXiv Constructing local L-packets for tame unitary groups

Theses

Title Notes
PDF arXiv The Local Langlands Correspondence for Tamely Ramified Groups Ph.D. Thesis under Benedict Gross. Defended April 19, 2011
PDF arXiv Formal Groups, Complex Cobordism and Quillen's Theorem Minor thesis, a Ph.D. requirement at Harvard

Software

Mathematical Databases

Project Active Contributions
browse code paper
2008-2023 1,480 commits 105,323 ++ 103,649 --
The LMFDB has been a major focus of my work at MIT since 2018, and I now serve on its editorial board. My contributions fall into two categories: improving LMFDB's infrastructure and adding mathematical content to the site. On the infrastructure front, I converted the backend database from MongoDB to PostgreSQL, spinning off a Python-SQL interface library, Psycodict, in the process. I have also worked on centralizing functionality (such as code for laying out search pages) that had been duplicated in many sections, easing the process of adding new site-wide functionality and adding new sections. My mathematical contributions are described below.
Abelian varieties over finite fields browse code paper
2016-2020 44 commits 11,913 ++ 8,168 --
In part 1 of the project to build a database of abelian varieties over finite fields within the LMFDB (joint with Dupuy, Kedlaya and Vincent), we used the Honda-Tate theorem to enumerate isogeny classes and compute many quantities that are invariant across isogeny classes.
Classical modular forms browse code paper
2018-2021 51 commits 852 ++ 233 --
As part of a large group effort, we rebuilt the classical modular form section of the LMFDB from scratch, using code from Magma, Pari and Sage. My contributions were mostly on front-end code, so are reflected in the commits to the LMFDB respository more than the CMF repository.
Abelian varieties over finite fields browse code
2019-2023 105 commits 2,554 ++ 1,205 --
In part 2 of the project to build a database of abelian varieties over finite fields within the LMFDB (joint with Costa, Dupuy, Marseglia, Vincent and West) we are using Deligne modules and work of Centeleghe-Stix to compute isomorphism classes and polarizations within many of the isogeny classes.
Finite groups browse code paper
2019-2023 948 commits 51,143 ++ 22,368 --
As part of a collaboration with Combes, Jones, Paulhus, Roy, and Schiavone, we have taken existing databases of finite groups, computed various quantities about them (most notably subgroup lattices, character tables and abstract isomorphism classes for permutation groups) and created a searchable database of groups, subgroups, and characters.
Modular curves browse code
2022-2023 396 commits 43,795 ++ 8,823 --
Building on two workshops, we are building a database of modular curves using the bijection with finite index subgroups of GL(2, Zhat). I have been helping coordinate work on finding models, maps between them, and rational points.

Open Source Computation

Project Active Contributions
try code
2006-2023 500 commits 34,949 ++ 16,871 --
I have been involved in Sage since Sage Days 1. Much of my effort has gone toward creating the p-adics module, but I have also worked on finite fields, elliptic curves, number fields, the transition to git and github, the coercion system and many other parts of the software.

Infrastructure & Collaboration

Project Active Contributions
browse code paper
2020-2022 374 commits 22,028 ++ 7,647 --
With the start of the COVID-19 pandemic, many research seminars moved online. Together with Costa, Poonen and Sutherland, I created an online index of these seminars which helps researchers worldwide find talks of interest that are newly accessible. It also provides useful tools for creating lightweight homepages for seminars and conferences.
try code paper
2021-2022 17 commits 12,276 ++ 402 --
The pandemic made it harder to form collaborations between students in undergraduate classes to work on homework. This site allows instructors at MIT to opt in and help their students find study groups. The project is mostly driven by Sutherland (and the linked paper is written by him), but I helped with the initial setup and the matching algorithm.
code
2019-2022 118 commits 18,542 ++ 11,399 --
The most popular interface to PostgreSQL from Python is psycopg, which allows for the execution of SQL commands from python. While robust and fast, it requires users to know SQL. For the LMFDB, I developed a layer on top of psycopg that constructs queries from dictionaries and returns the output in dictionaries as well. In support of researchseminars and psetpartners, this part of the LMFDB was extracted out into its own library.

Talks

Please e-mail me to learn more about talks where I did not use slides.

Title Dates Venue Paper Description
PDF How to Win the Lottery July 1, 2024 MathRoots Talk on Cash Winfall, the lottery game that was actually profitable to play
PDF Finite groups and K3 surfaces in the LMFDB Jun 25, 2025 Computational Geometry Talk on finite groups and K3 surfaces in the LMFDB
PDF Modular curves in the LMFDB Sep 21, 2023 Modular curves and Galois representations Talk on modular curves in the LMFDB
PDF Modular curves and Finite Groups: Building Connections via Computation Mar 2, 2023 COUNT Talk on modular curves and finite groups in the LMFDB
PDF Modular curves and Finite Groups: Building Connections via Computation Jan 11, 2023 2023 Simons collaboration annual meeting Talk on modular curves and finite groups in the LMFDB
PDF The L-functions and modular forms database May 22, 2022 Big Data in Pure Mathematics Introduction to the LMFDB
PDF p-adic precision May 7, 2021 Simons collaboration monthly meeting arXiv Different approaches to p-adic precision, including lattice and lazy methods.
PDF Algebraic structures in Sage February 15, 2021 Sage-Oscar Days Introduction to Sage, with a view toward finite fields
PDF A database of p-adic tori March 18, 2020 MIT Lie groups seminar Talk on database of tori
PDF A database of p-adic tori November 21, 2019 University of British Columbia number theory seminar Talk on database of tori
PDF The inverse Galois problem for p-adic fields September 10, 2019 University of Bordeaux number theory seminar arXiv
PDF A database of p-adic tori September 2, 2019 p-Adic Langlands Correspondence Conference Talk on database of tori
PDF The inverse Galois problem for p-adic fields August 30, 2019 University of Utrecht number theory seminar arXiv
PDF A database of p-adic tori June 8, 2019 2019 CMS Summer Meetings Talk on database of tori
A database of abelian varieties June 6, 2019 Arithmetic of Low-Dimensional Abelian Varieties Lightning talk on database of abelian varieties
PDF The inverse Galois problem for p-adic fields July 17, 2018 ANTS 13 arXiv
PDF Using lattices to track p-adic precision February 20, 2018 Numerical Methods for Algebraic Curves arXiv Lattice-based precision for p-adic computation.
Computing with higher genus curves, abelian varieties and K3 surfaces December 7, 2017 Pitt algebra seminar Introduction to goals of the Simons Collaboration
PDF The inverse Galois problem for p-adic fields November 13, 2017 BU number theory seminar arXiv
How to Win the Lottery September 14, 2017 Reed Colloquium Talk on Cash Winfall, the lottery game that was actually profitable to play
An introduction to p-adics in Sage August 22, 2017 Sage Days 88
p-adics in Sage July 17, 2017 Sage Days 87 Introduction to Sage and p-adics.
Makdisi's algorithm for Jacobians of p-adic curves June 5, 2017 Sage Days 86.5 Application of p-adic precision to computing with divisors
Computing with Modular Forms May 22-26, 2017 UNCG Summer School Summer school on computational modular forms; I'm speaking on overconvergent modular symbols
Modular Forms and Modular Symbols in Sage February 8, 2017 CLap-CLap Introduction to using Sage for computations of modular forms
Commutative Character Sheaves December 2, 2016 American University arXiv Talk on the new function-sheaf dictionary paper.
The Inverse Galois Problem for p-adic Fields September 29, 2016 University of Calgary Colloquium arXiv Talk on algebraic tori database project
Algebraic Tori and a Computational Inverse Galois Problem September 17, 2016 PANTS XXVI arXiv Talk on algebraic tori database project
PDF Overcongergent Modular Symbols in Sage March 20, 2016 Sage Days 71 Introduction to Overconvergent Modular Symbols in Sage
Elliptic Curves, Triangles and L-functions March 11, 2016 Tufts University Colloquium Colloquium on ranks of elliptic curves, the congruent number problem and the Birch and Swinnerton-Dyer conjecture
PDF Algebraic Tori and a Computational Inverse Galois Problem January 26, 2016 University of Pittsburgh arXiv Talk on algebraic tori database project
PDF Overconvergent Modular Symbols in Sage September 30, 2015 RIMS Workshop on Computer Algebra Systems Talk on Sage and Overconvergent Modular Symbols
Positive slope pieces of the eigencurve via interpolation July 30, 2015 Computational Representation Theory in Number Theory Talk on joint work with Ander Steele
Positive slope pieces of the eigencurve via interpolation May 28, 2015 Explicit Methods for Abelian Varieties Talk on joint work with Ander Steele
PDF A function-sheaf dictionary for tori over local fields June 17, 2014 CNTA XIII arXiv Talk on my function-sheaf dictionary paper
PDF A function-sheaf dictionary for tori over local fields May 24, 2014 FRG on periods of automorphic forms and applications to L-functions arXiv Half-hour talk on my function-sheaf dictionary paper
PDF Numerical Methods in p-adic Linear Algebra February 3, 2014 Lethbridge Number Theory and Combinatorics Seminar arXiv
PDF A function-sheaf dictionary for tori over local fields January 15, 2014 Joint Mathematics Meetings 2014 arXiv Shorter talk on my work with Clifton
PDF Quasicharacter Sheaves for Tori December 5, 2013 QVNTS arXiv Updated talk on my work with Clifton
PDF Plans for p-adics in Sage September 5, 2013 Sage Days 52
PDF Geometrizing Characters of Tori June 24, 2013 PRIMA 2013 arXiv Updated talk on my work with Clifton
PDF Geometrizing Characters of Tori May 12, 2013 Alberta Number Theory Days 2013 arXiv Updated talk on my work with Clifton
PDF Geometrizing Quasi-characters of Tori January 28, 2013 UCSC Number Theory Seminar arXiv Hour talk on my work with Clifton - some material has since been corrected
PDF Geometrizing Quasi-characters of Tori January 10, 2013 Joint Mathematics Meetings 2013 arXiv 20 minute talk on my work with Clifton
PDF The ABC Conjecture November 16, 2012 University of Calgary Colloquium Sketch of Mochizuki's work
PDF Introduction to Sage October 16, 2012 University of Calgary - PMAT 621 Most of the talk was this Sage notebook.
PDF The Local Langlands Correspondence and Character Sheaves May 19, 2012 Pacific Northwest Number Theory Conference 2012 PDF Similar to UW Number theory talk
PDF The Local Langlands Correspondence and Character Sheaves May 17, 2012 University of Washington Number Theory Seminar PDF Improved version of talk from Utah
PDF The Local Langlands Correspondence and Character Sheaves April 26, 2012 University of Utah number theory seminar PDF Thesis talk with a bit added on my work with Clifton
PDF Precision Models for Arithmetic in Local Fields March 1, 2012 UBC number theory seminar arXiv An improved version of my talk from Lyon in June
The State of p-Adics in Sage February 19, 2012 Sage Days 36
Coercions and Categories in Sage January 26, 2012 UW Sage Seminar Sage notebook of this talk available here
PDF The Local Langlands Correspondence for Tamely Ramified Groups December 8, 2011 PIMS Number Theory Seminar PDF Thesis talk, extended to a 2 hour format
PDF Zeta Functions with p-Adic Cohomology June 24, 2011 Geocrypt 2011 arXiv A modified version of the talk I gave at CRM in April 2010
PDF Precision Models for Arithmetic in Local Fields June 6, 2011 Théorie de Hodge p-adique, équations différentielles p-adiques et leurs applications arXiv Talk on my work with Xavier Caruso
PDF The Local Langlands Correspondence for Tamely Ramified Groups February 9, 2011 Harvard number theory seminar PDF Thesis talk, modified notes from my October 2010 talk.
The Local Langlands Correspondence for Tamely Ramified Groups January 20, 2011 University of Calgary number theory seminar PDF Thesis talk, similar notes to my October 2010 talk.
The Local Langlands Correspondence for Tamely Ramified Groups December 1, 2010 Universite Rennes Cryptography Seminar PDF Thesis talk, similar notes to my October 2010 talk.
PDF The Local Langlands Correspondence for Tamely Ramified Groups October 2, 2010 Quebec-Maine Number Theory 2010 PDF An overview of the Langlands correspondence for GL_n and an outline of my thesis work.
PDF Zeta Functions with p-adic cohomology April 20, 2010 Counting Points: Theory, Algorithms and Practice arXiv A presentation of the algorithm in my paper with Kiran and Tim.
Bruhat-Tits Theory April 2, 9, 16, 2010 Alcove Seminar An overview of the Bruhat-Tits building.
Coxeter Groups and 4-dimensional Polytopes February 25, 2010 Trivial Notions Coxeter groups and classification of polytopes, together with Zome models of polyhedra and the 120-cell.
Berkovich Spaces April 29, 2009 Alcove Seminar An overview of Berkovich spaces.
Weil Conjectures February 26, 2009 Trivial Notions Weil conjectures and Weil cohomology theories for non-number theorists.
Laplace to Langlands Sept 25, Sept 30, Oct 2, 2008 Alcove Seminar A presentation on Gross's paper From Laplace to Langlands via representations of orthogonal groups.
Embeddings of PSL_2(F_p) into semisimple linear algebraic groups May 19, 2008 STAGE A presentation of the results in Serre's Exemples de plongements des groupes PSL_2(F_p) dans des groupes de Lie simples, Inventiones mathematicae, 1996.
A Bound for the number of automorphisms of an arithmetic Riemann surface April 10, 2008 Trivial Notions Based on the Arizona Winter School talk.
PDF A Bound for the number of automorphisms of an arithmetic Riemann surface April 1, 2008 UW Number Theory Seminar Based on the Arizona Winter School talk.
PDF A Bound for the number of automorphisms of an arithmetic Riemann surface March 19, 2008 Arizona Winter School Joint work with Linda Gruendken, Guillermo Mantilla, Dermot McCarthy, Kate Stange, Ying Zong, Maryna Viazovska. Based on Belolipetsky and Jones' paper of the same name.
PDF Representations de GL_2(Q_p) et (phi, Gamma)-modules February 20, 27, March 5, 2008 Taylor's Seminar Talks on Colmez's paper of the same name. Joint talks with Ana Caraiani. List of results and index of notation: PDF.
Eigenvarieties December 10, 2007 STAGE A review based on Kevin Buzzard's lecture notes from the Eigenvarieties semester, which can be found here.
PDF p-Adic arithmetic in SAGE September 30, 2007 Sage Days 5 Future directions for p-adic support in Sage.
PDF p-Adics in SAGE June 14, 2007 Sage Days 4 Implementation notes for p-adics in Sage.
Algorithms for p-adic polynomials and matrices April 9, 2007 STAGE A discussion of some work of mine on algorithms for computing determinants and characteristic polynomials of p-adic matrices. I haven't written this material up yet.
Sums of Squares Formulas March 8, 2007 Trivial Notions Overview of the results in this paper of Dugger and Isaksen.
Vanishing cycles in the analytic case December 11, 2006 Weil II seminar Vanishing cycles and perverse sheaves as used in Deligne's proof of the Weil conjectures.
The eigencurve near the boundary of weight space for p=2 and p=3 November 13, 2006 STAGE arXiv Presentation of a paper of Buzzard's and a followup paper of mine

Conferences

I use MathMeetings.net to find conferences. Number Theory Web is also useful.

Past and Future Conferences Attended

A gold row indicates a conference that I organized, while a blue row indicates one where I spoke.

Dates Year Conference Location
October 7-112024SQuaRE on p-adic databasesAIM, Pasadena, CA
July 15-192024ANTS 16MIT, Cambridge, MA
July 8-122024The Mordell conjecture 100 years laterMIT, Cambridge, MA
June 24-282024Computational geometryBIRS, Banff, AB
April 22-262024Hypergeometric motivesMIT, Cambridge, MA
March 11-152024Modular curves workshop 3MIT, Cambridge, MA
February 14-172024K3 surfacesICERM, Providence, RI
February 5-92024Shimura curves in the LMFDBDartmouth, Hanover, NH
December 4-82023Open-source cyberinfrastructure supporting mathematics researchAIM, Pasadena, CA
September 18-222023Modular curves and Galois representationsUniversity of Zagreb
July 10-142023LucantICERM, Providence, RI
March 6-102023SQuaRE on p-adic databasesAIM, San Jose, CA
February 27-March 32023COUNTCIRM, Luminy, France
Feburary 6-102023Sage Days 117Le Teich, France
January 11-1220232023 Simons collaboration annual meetingSimons, New York, NY
November 5-92022Modular curves workshop 2MIT, Cambridge, MA
July 17-August 62022Number theory informed by computationPCMI, Park City, UT
May 23-272022SQuaRE on p-adic databasesAIM, San Jose, CA
May 21-222022Big Data in Pure MathematicsOnline
April 11-152022Explicit Methods in ModularityOnline
March 21-252022Modular curves workshopMIT, Cambridge, MA
January 13-1420222022 Simons collaboration annual meetingSimons, New York, NY
February 15-192021Sage/Oscar Days for Combinatorial Algebraic GeometryOnline
June 30-July 42020ANTS 14Online
January 9-1020202020 Simons collaboration annual meetingSimons, New York, NY
September 2-62019p-Adic Langlands Correspondence ConferenceUniversité Rennes 1, Rennes, France
June 7-1020192019 CMS Summer MeetingsRegina, SK
June 3-72019Arithmetic of Low-Dimensional Abelian VarietiesICERM, Providence, RI
January 10-1120192019 Simons collaboration annual meetingSimons, New York, NY
July 16-202018ANTS 13UW, Madison, WI
February 19-232018Numerical Methods for Algebraic CurvesRennes 1, Rennes, France
Sep 29-Oct 12017Open Source Computation and Algebraic SurfacesBIRS, Banff, AB
August 21-252017Sage Days 88IMA, Minneapolis, MN
July 17-212017Sage Days 87UVM, Burlington, VT
June 5-92017Sage Days 86.5Simon Frasier, Vancouver, BC
May 22-262017UNCG Summer School on Computational Number Theory: Modular FormsUNCG, Greensboro, NC
February 6-102017CLap-CLapUniversité Rennes 1, Rennes, France
September 17-182016PANTS XXVIUNCG, Greensboro, NC
May 16-202016The p-adic Langlands Program and Related TopicsIU, Bloomington, IN
May 9-132016AIM LMFDB WorkshopAIM, San Jose, CA
March 25-282016Brisol LMFDB WorkshopBristol, UK
March 20-242016Explicit p-adic methods in number theoryOxford, UK
December 4-72015CMS Winter Meeting: Representation TheoryMontreal, QC
Sep 30-Oct 22015Towards New Development of Mathematics via Computational Algebra SystemsRIMS, Kyoto, Japan
July 27-312015Computational Representation Theory in Number TheoryOSU, Corvallis, OR
May 25-292015Explicit Methods for Abelian VarietiesUC, Calgary, AB
April 6-102015The Kudla ProgrammeCRM, Montreal, QC
February 16-202015Regulators, Mahler measures, and special values of L-functionsCRM, Montreal, QC
December 1-52014Motivic Integration, Orbital Integrals, and Zeta-FunctionsBIRS, Banff, AB
June 23-July 42014Counting Arithmetic ObjectsCRM, Montreal, QC
June 16-202014CNTA XIIICarleton, Ottawa, ON
June 2-62014p-adic variation in number theoryBU, Boston, MA
May 23-252014Periods of automorphic forms and applications to L-functionsHarvard, Cambridge, MA
May 19-232014Representations of Reductive GroupsMIT, Cambridge, MA
April 18-202014Alberta Number Theory Days VIBIRS, Banff, AB
February 17-212014Hot Topics: Perfectoid Spaces and their ApplicationsMSRI, Berkeley, CA
January 15-182014Joint Mathematics MeetingsBaltimore, MD
December 6-92013CMS Winter MeetingOttawa, ON
September 2-62013Sage Days 52Université Rennes 1, Rennes, France
August 19-212013Workshop on Curves and ApplicationsUC, Calgary, AB
June 24-282013PRIMA 2013Jiao Tong University, Shanghai, China
May 10-122013Alberta Number Theory DaysBIRS, Banff, AB
March 25-292013Sage Days 47PSU, Portland, OR
Feb 26-Mar 22013Sage Days 46Oahu, HI
Jan 31-Feb 32013Sage Days 44UW, Madison, WI
January 9-122013Joint Mathematics MeetingsSan Diego, CA
December 7-102012CMS Winter MeetingMontreal, QC
May 24-292012Sage Days 40.5Wallace Falls, Seattle, WA
May 19-212012Pacific Northwest Number Theory ConferenceUniversity of Idaho, Moscow, ID
May 7-112012Geometrization of Smooth CharactersBIRS, Banff, AB
Mar 24-Apr 12012LMFDB workshopBristol, UK
March 11-162012Galois Representations, Shimura Varieties and Automorphic FormsFields Institute, Toronto, ON
Febuary 19-232012Sage Days 36UCSD, La Jolla, CA
September 15-182011Periods of Automorphic Forms and Applications to L-functionsColumbia, New York, NY
June 26-July 12011L-packetsBIRS, Banff, AB
June 18-222011GeoCrypt 2011Bastia, France
June 6-92011Théorie de Hodge p-adique, équations différentielles p-adiques et leurs applications Lyon, France
October 2-32010Quebec-Maine Number Theory ConferenceUniversité Laval, Quebec City, QC
June 2-52010Number Theory and Representation Theory: Dick Gross' 60th Birthday ConferenceHarvard, Cambridge, MA
April 19-232010Counting Points: Theory, Algorithms and PracticeCRM, Montreal, QC
December 1-42009Sage Days 18CMI, Cambridge, MA
May 16-212009Sage Days 15UW, Seattle, WA
Mar 29-Apr 72009p-Adic Methods in Arithmetic Algebraic GeometryHebrew University, Jerusalem, Israel
March 13-192009Arizona Winter School: Quadratic FormsUA, Tucson, AZ
January 20-252009Sage Days 12UCSD, San Diego, CA
June 23-272008LMFDB WorkshopUW, Seattle, WA
June 9-202008Sage Devel Days 1UW, Seattle, WA
February 5-92008Sage Days 7IPAM, Los Angeles, CA
November 10-142007Sage Days 6Bristol, UK
Sep 30-Oct 32007Sage Days 5CMI, Cambridge, MA
June 12-172007Sage Days 4UW, Seattle, WA
February 17-212007Sage Days 3IPAM, Los Angeles, CA
October 6-102006Sage Days 2UW, Seattle, WA
February 4-52006Sage Days 1UCSD, San Diego, CA

Curriculum Vitae   PDF

Education

Degree Subject University Completed
Doctor of Philosophy Mathematics Harvard 2011
Bachelor of Science Mathematics 2006
Bachelor of Science Literature 2006

Employment

Position University Duration
Research Scientist 2018-2024
Postdoctoral Fellow 2015-2017
Postdoctoral Fellow 2014-2015
Postdoctoral Fellow Calgary 2011-2014

Honors and Awards

Publications

Software

Talks

Teaching Experience

Course Institution Role Semesters
UROP Various projects Undergraduate research supervisor Summer 2022, 2023
18.782 Arithmetic Geometry Instructor Fall 2020
Ind. Study p-adic Analysis Instructor Spring 2019
Math 430 Introduction to Abstract Algebraic Systems Instructor Fall 2017
Math 220 Analytic Geometry & Calculus 1 Lecturer Fall 2017
Math 240 Analytic Geometry & Calculus 3 Lecturer Spring 2017
Math 220 Analytic Geometry & Calculus 1 Lecturer Fall 2016
Math 430 Introduction to Abstract Algebraic Systems Instructor Spring 2016
Math 220 Analytic Geometry & Calculus 1 Lecturer Fall 2015
Math 200 Multivariable Calculus Lecturer Fall 2014
Math 211 Linear Methods Calgary Lecturer Fall 2013
PMAT 527/627 Computational Number Theory Calgary Instructor Fall 2012
Math 223a Algebraic Number Theory Harvard Course assistant Fall 2008
QR 28 Magic of Numbers Harvard Teaching fellow Fall 2008
18.821 Project Lab in Mathematics Teaching assistant Spring 2006
18.02 Multivariable Calculus Instructor Fall 2003, Spring 2005, Fall 2005
Many short courses Mathcamp Counselor Summers 2003, 2006-2009, 2011, 2016

Conferences Organized

Conference With Budget Dates
Modular curves workshop 2 Edgar Costa, Andrew Sutherland, John Voight via Simons November 5-9, 2022
Explicit Methods in Modularity Eran Assaf, Edgar Costa, Brendan Hassett None April 11-15, 2022
Modular curves workshop Edgar Costa, Andrew Sutherland, John Voight via Simons March 21-25, 2022
Sage Days 87 Jen Balakrishnan, Xavier Caruso, Edgar Costa, Taylor Dupuy, William Stein, Christelle Vincent $47K July 17-22, 2017
Explicit p-adic methods in number theory Jen Balakrishnan $36K March 20-24, 2016
CMS Winter Session: Representation Theory Clifton Cunningham None December 4-7, 2015
Alberta Number Theory Days Brandon Fodden $10K May 10-12, 2013
Sage Days 44 Rob Harron, William Stein $10K January 31-February 3, 2013
Sage Days 40.5 William Stein $10K May 24-29, 2012
Sage Days 36 Kiran Kedlaya, William Stein $30K February 19-23, 2012

Other Activities

Served on grant review panel 2023
Created researchseminars.org 2020
Academic Coordinator for Mathcamp 2016 Mathcamp Summer 2016
Organizer of the Number theory video seminar Fall 2014
Organizer of the ABC seminar Calgary Fall 2012
Organizer of the Alcove seminar Harvard 2008-2011
Developer for the Sage project 2006-present
Volunteer teacher at Splash 2002-2009
Coach of two MATHCOUNTS teams Catlin Gabel , West Sylvan 1998-2002

Interests   PDF

Mathematical Databases

The LMFDB is a website that hosts dozens of collections of mathematical objects, from the eponymous L-functions and classical modular forms to elliptic curves, number fields, finite groups and Artin representations. Mathematically, it is motivated by the Langlands program, but many of the objects arise much more broadly.

These databases make a new kind of experimental mathematics possible, with a spectacular example occurring last year. While studying elliptic curves ordered by conductor, a group of mathematicians including an undergraduate student observed that the average point counts modulo primes displayed an unexpected oscillation. This "murmurations" phenomenon was only apparent when averaging over tens of thousands of curves, and has led to further theoretical exploration including a workshop at ICERM in July 2023, a proof of an analogue in the modular form setting, and investigations in non-arithmetic settings like Maass forms. As the use of data more broadly in society continues to grow, I believe that mathematical databases have a key role to play.

I have worked on most sections of the LMFDB, including major contributions to

p-adic Computation

p-adic numbers have played a central role in many advances in modern number theory, from Deligne's proof of the Weil conjectures to the deformation theory underlying Wiles' proof of Fermat's Last Theorem to Scholze's perfectoid spaces. They can be defined either analytically, via the completion process that constructs the real numbers from the rationals, or algebraically, as the inverse limit of finite rings ℤ/p^nℤ as n increases. In either perspective, they provide a link between divisibility and modular arithmetic on the one hand and topology and analysis on the other.

From a computational point of view, there are two main challenges in implementing p-adic arithmetic: tracking precision through a computation, and handling algebraic extensions, which are far more complicated than for real numbers. I have led the development of p-adic arithmetic in Sage, including a lattice method for tracking precision developed in a series of papers with Xavier Caruso and Tristan Vaccon ([1] [2] [3] [4] [5]).

I am interested in applications of p-adics to number theory and have worked on point counting algorithms for surfaces over finite fields and average polynomial time algorithms for computing L-functions of hypergeometric motives.

Application Materials - Principal Research Scientist at MIT

  • Curriculum Vitae
  • Research Statement
  • Statement on teaching, service and mentorship
  • Representative Publications:
    Title With Status
    PDF Creating a Dynamic Database of Finite Groups Lewis Combes, John Jones, Jennifer Paulhus, Manami Roy, Sam Schiavone In preparation
    PDF arXiv Hypergeometric L-functions in average polynomial time II Edgar Costa, Kiran Kedlaya Accepted to ANTS XVI
    PDF arXiv DOI Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB Taylor Dupuy, Kiran Kedlaya, Christelle Vincent Arithmetic Geometry, Number Theory, and Computation, Simons Symp, Springer, 2021, pp 375-448
    PDF arXiv DOI Tracking p-adic precision Xavier Caruso, Tristan Vaccon LMS Journal of Computation and Mathematics 17 (Special issue A) (2014), 274-294.

Resources

This page is aimed at students who are looking for resources beyond the textbook and homework problems.

General

  • MIT's OpenCourseWare offers videos, textbooks and courses which can provide a different perspective on a topic you might be struggling with.
  • Math Stack Exchange is a question and answer site aimed at calculus and beyond.
  • Math Overflow is a more advanced question and answer site aimed at research level mathematics.
  • Researchseminars is an index of online talks in math and other subjects, many of which are open to watch.

Calculus

Number theory

  • The L-functions and modular forms database has millions of examples of number theoretic objects, along with exposition to explain some of the associated mathematics.
  • This list of number theory software and tables has a lot of interesting content.

The society of p-adicts was founded in 2012 by Clifton Cunningham and David Roe. You can find the current run of T-shirts here. Note that orders are only fulfilled if enough T-shirts are ordered within a two week period (see the counter at the bottom of the page). If you place an order but it's not fulfilled, you won't be charged. Gather together other p-adicts to make sure you can get a shirt!

Contact Info

The best way to reach me is by e-mail.

E-mail

roed@mit.edu

Mailing Address

Department of Mathematics
Room 2-336
Massachusetts Institute of Technology
Cambridge, MA 02139-4307