Number Theory and Representation Theory (June 6 - July 15, 2022)

Projects

• Arithmetic functions over function fields
• Arithmetic statistics
• Modular forms
• Partitions
• Sato-Tate distributions for K3 surfaces

Number Theory and Representation Theory Participants 

• Quanlin Chen
• Catherine Cossaboom (2024 Goldwater Scholar)
• Sanjana Das
• Henry Glunz 
• Annemily Hoganson (2023 Goldwater Scholar)
• Daniel Hu 
• Hari Iyer (2023 Goldwater Scholar)
• Faye Jackson (2023 Alice T. Schafer Prize winner; 2024 Morgan Prize winner)
• Thomas Jaklitsch
• Hana Lang 
• Misheel Otgonbayar
• Sasha Shashkov  (2023 Goldwater Scholar, 2023 Churchill Scholar)
• Eric Shen 
• Hamilton Wan 
• Nancy Xu 
• Sharon Zhou

Number Theory Colloquia Speakers (Thursdays at 3pm)

• June 9th: Bjorn Poonen (MIT)
  "Ranks of elliptic curves
• June 14th: Robert Schneider (Michigan Tech)
  "Making music with logarithms."
• June 16th: James Maynard (Oxford) 
  "Patterns in the primes"
• June 23rd: Lillian Pierce (Duke) 
  "Superorthogonality"
• June 30th: Jennifer Balakrishnan (Boston U) 
  "Mordell's conjecture and the last 100 years."
• July 7th: Ken Ribet (UC Berkeley) 
  "A meandering history of Fermat's Last Theorem for UVA REU students"
• July 12th at 2:00pm: Jacob Tsimerman (U Toronto)
  "The Ax-Grothendieck Theorem"
• July 12th at 3:15pm: Robert Schneider (Michigan Tech)
  "Multiplicative theory of integer partitions"
• July 14th: 2:00pm Toni Bluher (NSA) 
  “Difference sets, Galois theory, and an accidental isomorphism”
• July 14th: 3:00pm Amanda Folsom (Amherst) 
  "Quantum Jacobi forms and applications in number theory, topology, and mathematical physics"

Mini-Courses (1:00pm-2:00pm)

• June 13, 15, 17: Prof. Valia Gazaki "Abelian Varieties"
• June 20, 22, 24: Prof. You Qi "Representation Theory of the Symmetric Group"
• June 27, 29, July 1: Prof. Charlotte Ure "Beyond the Mordell-Weil Theorem"

Number Theory Papers

1. Quanlin Chen and Eric Shen, On effective Sato-Tate distributions for surfaces arising from products of elliptic curves, submitted for publication.
2. Catherine Cossaboom and Sharon Zhou, Hecke nilpotency for modular form mod 2 and an application to partition numbers, Ramanujan J., recommended for publication.
3. Sanjana Das, Asymptotic properties of maximal p-core p'-partitions, J. Combinatorial Theory, Series A, 193 Volume 193, (2023), 105685.
4. Sanjana Das, Hana Lang, Hamilton Wan, and Nancy Xu, The distribution of k-free effective divisors and the summatory totient function in function fields, submitted for publication.
5. Sanjana Das, Hana Lang, Hamilton Wan, and Nancy Xu,The distribution of error terms of smoothed summatory totient functions, Proceedings of the American Mathematical Society, accepted for publication.
6. Hana Lang, Hamilton Wan, and Nancy Xu, Distributions of hook lengths divisible by 2 or 3, J. Number Theory, 246 (2023), 227-251.
7. Henry Glunz, Significant digits of primes in subsets, Integers, 23 (2023), Article 46.
8. Annemily Hoganson and Thomas Jaklitsch, On certain McKay numbers for symmetric groups, Ramanujan Journal, accepted for publication.
9. Daniel Hu, Hari Iyer and Shasha Shashkov, Modular forms and an explicit Chebotarev variant of the Brun-Titchmarsh theorem, Research in Number Theory, 9 (2023), Art. 46.
10. Faye Jackson and Misheel Otgonbayar, Journal of Number Theory, Parts in k-indivisible Partitions Always Display Biases between Residue Classes,submitted for publication  for publication.
11. Faye Jackson and Misheel Otgonbayar,Unexpected biases between congruence classes for parts in k-indivisible partitions, submitted for publication.
12. Faye Jackson and Misheel Otgonbayar, Parts in k-indivisible partitions always display biases between residue classes, Journal of Number Theory, accepted for publication.

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Topology (June 6 - July 29, 2022)

Topology Participants

• Mahalo Blackburn
• Isaiah Dailey 
• Grace Davis
• Clara Huggins 
• Lindsay Marjanski
• Fredrick Mooers 
• Semir Mujevic 
• Andrew Niu
• Chloe Shupe
• Vincent Solon 
• Frank Zheng
• Kathleen Zopff

Topology Colloquia

• June 21 and 23: Inna Zakharevich (Cornell) (remote)
• July 13: Diana Hubbard (Brooklyn College) (remote)
• Sang-hyun Kim (Korea Institute for Advanced Studies) (remote)

Mini Courses

• Walker Stern: "A crash course on categories"
• Mark Pengitore: "Topological and geometric models of groups" 
• William Balderrama: "Simplicial complexes and simplicial homology" 

Professional development/social activities

• Latex Course
• Graduate School and Job panel
• Diversity panel

Topology Papers

1. Lindsay Marjanski, Vincent Solon, Frank Zheng, and Kathleen Zopff, Geodesic growth of numbered graph products, Journal of Groups, Complexity, Cryptology, Vol. 14, Issue 2 (2023).

Number Theory and Representation Theory Participants 

• Mateo Attanasio 
• Caroline Choi
• Yunseo Choi (2021 Regeneron Science Talent Search 1st Prize; 2024 Goldwater Scholar; 2026 Morgan Prize winner)
• Kevin Gomez       
• Alexandra Hoey (2022 Alice T. Schafer Prize, Honorable Mention)
• Daniel Hu (2022 Goldwater Scholar)
• Jonas Iskander
• Steven Jin (2022 Churchill Scholar)
• Ikuya Kaneko
• Kaya Lakein
• Anne Larsen
• Sean Li (2021 Davidson Fellow)
• Jerry Lu
• Andrei Mandelshtam (2021 Davidson Fellow, Honorable Mention; 2024 Goldwater Scholar)
• Spencer Martin
• Margaret Michael
• Apoorva Panidapu (2021 Davidson Laureate)
• Carl Schildkraut
• Casia Siegel
• Fernando Trejos Suárez         

Topology Participants

• Rebecca Bell
• Katherine Betts
• Ethan Clelland
• Allison Eckert
• Alexandra Emmons
• Calvin Godfrey
• Luz Grisales
• Troy Larsen
• Ryan Pesak
• Avery Schweitzer
• Jeffrey Utley
• Avalon Vanis

Number Theory and Representation Theory (June 7- July 16, 2021)

Projects

• Brauer groups for elliptic curves
• Divisors of Hecke trace modular forms and complex multiplication
• Elliptic curve Tamagawa products: Arithmetic statistics over number fields
• Lehmer-type conjectures for modular forms
• Mertens-type speculations for Mobius sums over number fields

Invited Speakers (Thursdays at 3pm)

• June 10th: Jennifer Balakrishnan (Boston University)
  "A tale of three curves"
• June 17th: Kannan Soundararajan (Stanford University)
  "Distribution of prime numbers"
• June 24th: Melanie Matchett Wood (Harvard University)
  "Distributions of class groups and unramified extensions"
• July 1st: Andrei Okounkov (Columbia University)
  "Partitions, q-series, and multiple q-zeta values"
• July 8th: Jordan Ellenberg (University of Wisconsin)
  “Heights of rational points on stacks”
• July 12th: Toni Bluher (NSA, Not on a Thursday)
  “Dickson polynomials, Chebyshev polynomials, and Wilson-like theorems”

Mini-Courses (1:00pm-2:00pm)

• Abelian Varieties  (June 14, 16, 18; Instructor: Prof. Evangelia Gazaki)
• Diophantine Approximation (June 21, 23, 25; Instructor: Postdoc Jinbo Ren)
• Representation Theory of S_n (June 28, 30, July 2; Instructor: Prof. You Qi)

Topology (June 7-July 30, 2021)

 Projects

• Equivariant finite spaces and topological complexity (Mentor: Walker Stern)
• Alexander invariants of trisected 2-knots (Mentor: William Olsen)
• The tri-pants graph of the twice punctured torus (Mentor: Filippo Mazzoli)

Colloquia

• Tarik Aougab (Haverford)
• Jessica Purcell (Monash)
• Sang-hyun Kim (KIAS)
• Henry Segerman (Oklahoma State), June 16th at 4:00pm
  "Triangulating the figure 8 knot complement"
• Wojciech Chachólski (KTH), July 14th at 10:00am
  "How to enable a machine to sense geometry"

Mini Courses

• An introduction to the fundamental group (Walker Stern)
• Curves on surfaces (Filippo Mazzoli)
• A crash course in knot theory (William Olsen)

Professional development/social activities

• Social mixer
• Latex Course
• Graduate School and Job panel
• Diversity panel

Topology Papers

1. Rebecca Bell, Allison N. Eckert, Ryan M. Pesak, and Avery Schweitzer, A finite equivariant generalization of motion planning and topological complexity, submitted for publication.
2. Katherine Betts, Troy Larsen, Jeffrey Utley, and Avalon Vanis, The Tri-Pants Graph of the Twice-Punctured Torus, submitted for publication.

Number Theory Papers

1. Mateo Attanasio, Caroline Choi, Andrei Mandelshtam, and Charlotte Ure, Symbol Length in Brauer Groups of CM Elliptic Curves, Proceedings of the American Mathematical Society, accepted for publication.
2. Yunseo Choi, Sean Li, Apoorva Panidapu, and Casia Siegel, Tamagawa products of elliptic curves over number fields, Journal de Théorie des Nombres de Bordeaux, accepted for publication.
3. Kevin Gomez, Kaya Lakein, and Anne Larsen, Supersingular loci from traces of Hecke operators, Proceedings of the American Mathematical Society, accepted for publication.
4. Alexandra Hoey, Jonas Iskander, Steven Jin and Fernando Trejos Suarez, An unconditional explicit bound on the error term in the Sato-Tate conjecture, Quarterly Journal of Mathematics (Oxford), accepted for publication.
5. Daniel Hu, Ikuya Kaneko, Spencer Martin, and Carl Schildkraut, Order of zeros of Dedekind zeta functions, Proceedings of the American Mathematical Society, accepted for publication.
6. Daniel Hu, Ikuya Kaneko, Spencer Martin, and Carl Schildkraut, On the Mertens Conjecture over Number fields , Mathematical Proceedings of the Cambridge Philosophical Society, accepted for publication.
7. Ikuya Kaneko and Jesse Thorner,Highly uniform Prime Number Theorems, submitted for publication.
8. Kaya Lakein and Anne Larsen, Some remarks on small values of Ramanujan's tau(n), Archiv der Mathematik (Basel), accepted for publication.
9. Kaya Lakein and Anne Larsen, A proof of Merca's conjectures on sums of odd divisor functions, Bulletin of the Australian Mathematical Society, accepted for publication.

2020 Participants

• Malik Amir
• Aaron Benda
• Talia Blum
• Caroline Choi
• Spencer Dembner
• Kristen Fukunaga
• Mitsuki Hanada
• Alexandra Hoey (2022 Alice T. Schafer Prize, Honorable Mention)
• (Carina) Letong Hong (2023 Morgan Prize Winner, 2022 Alice T. Schafer Prize Winner, 2022 Rhodes Scholar)
• Jonas Iskander
• Vanshika Jain (2021 Alice T. Schafer Prize, Honorable Mention)
• Tanmay Khale
• Kaya Lakein
• Dounia Lazreq
• Nathan Li
• Rachana Madhukara
• Chloe Marcum
• Thomas Martinez
• Sarah Nicholls
• Andrew Nguyen
• Cooper O'Kuhn
• Apoorva Panidapu
• Madelynn Roche
• Alec Sun
• Sabrina Traver
• Devin Vanyo
• Chase Wilson
• Shengtong Zhang

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2020 Activities

June 8, 2020: First Day of the Program

June 10, 2020: Black Lives Matter Observance

Geometry and Topology Research Projects

• 2D-TQFTs
• Subgroups of PSL_2(R)
• Braided disks in 4-space

Number Theory Research Projects

• Distribution of Primes
• Class Numbers and Elliptic curves (Lower bounds for class numbers)
• Modular forms (Variants of Lehmer's Conjecture)

REU Invited Speakers

• Ken Ribet (UC Berkeley): June 16 (Tuesday): 4:00pm
  "A brief History of Fermat's Last Theorem for REU Students"
  Slides Here 
  Cloud Recording (Password: 0Y*t8g%!)
• Sang-hyun Kim (Korea Institute for Advanced Study): June 24 (Wednesday):  8:00pm
  "Acute triangulations of the sphere."
  Slides Here
• James Maynard (Oxford University): June 25 (Thursday): 10:00am
  "Patterns in Primes"
  Slides Here
  Cloud Recording (Password: 2Q.FhJW=)
• Lillian Pierce (Duke University): June 30 (Tuesday): 4:00pm
  "The method of Moments: Examples in Number Theory"
• Henri Darmon (McGill University): July 7 (Tuesday): 4:00pm
  "The theory of complex multiplication and the arithmetic of real quadratic fields”
  Slides Here
  Cloud Recording(coming soon)
• Maryna Viazovska (EPFL, Lausaunne): July 9 (Thursday), 10:00am
  "From uncertainty principles to sphere packings"
  Slides Here
  Cloud Recording (Password: 3R+5wHS.)

REU Mini-Courses (1:00pm - 2:00pm)

• Essentials of topology and geometry of manifolds (First week) 
  (Instructors: Julie Bergner, Thomas Koberda, Tom Mark)
• Beyond the Mordell-Weil Theorem (June 11, 12, 15)
  (Instructor: Charlotte Ure)
• Complex multiplication: Constructing Hilbert class fields (June 16, 17, 18)
  (Instructor: Jinbo Ren)
• Representation Theory and Symmetric Groups (June 19, 22, 23, 24, 25, 26)  
  (Instructor: You Qi)
• Brauer Groups: What are they and what are they good for? (June 29, 30, July 1, 2, 3)
  (Instructor: Evangelia Gazaki)

July 4, 2020:  4th of July Holiday

July 17, 2020: Research Symposium

 

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2020 Results  

1.  M. Amir and L. Hong, On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan Journal, accepted for publication.     
2. T. Blum, C.  Choi, A. Hoey,  J. Iskander, K. Lakein, and T. Martinez, On class numbers, torsion subgroups, and quadratic twists of elliptic curves, Transactions of the American Mathematical Society, 375 (2021), no.1, 351–368. 
3. S. Dembner and V. Jain, Hyperelliptic curves and newform coefficients,  Journal of Number Theory, 225 (2020), 214-239.  
4. S. Dembner and V. Jain, A note on congruences for weakly holomorphic modular forms, Proceedings of the American Mathematical Society, 149 (2021), 3683-3686. 
5. M. Hanada and R. Madhukara, Fourier coefficients of level 1 Hecke eigenforms, Acta Arithmetica, 200 (2021), no. 4, 371–388        
6. L. Hong and S. Zhang, Towards Heim and Neuhauser's unimodality conjecture on the Nekrasov-Okounkov polynomials, Research in Number Theory, 7 (2021), Art. 17.
7. L. Hong and S. Zhang, Proof of the Ballantine-Merca Conjecture and theta function identities modulo 2, Proceedings of the American Mathematical Society, accepted for publication.
8. T. Khale, C. O'Kuhn, A. Panidapu, A. Sun, and. S. Zhang, A Bombieri-Vinogradov Theorem for primes in short intervals and small sectors, Journal of Number Theory (229), 2021, 142-167.

9. A. Panidapu and J. Thorner, The CM Sato-Tate distribution in short intervals, Involve, accepted for publication.     

 

REU Management Team: Julie Bergner, Thomas Koberda, Thomas Mark, and Ken Ono

2020 Participants

• Malik Amir
• Aaron Benda
• Talia Blum
• Caroline Choi
• Spencer Dembner
• Kristen Fukunaga
• Mitsuki Hanada
• Alexandra Hoey
• (Carina) Letong Hong
• Jonas Iskander
• Vanshika Jain
• Tanmay Khale
• Kaya Lakein
• Dounia Lazreq
• Nathan Li
• Rachana Madhukara
• Chloe Marcum
• Thomas Martinez
• Sarah Nicholls
• Andrew Nguyen
• Cooper O'Kuhn
• Apoorva Panidapu
• Madelynn Roche
• Alec Sun
• Sabrina Traver
• Devin Vanyo
• Chase Wilson
• Shengtong Zhang

2020 Activities

June 8, 2020: First Day of the Program

June 10, 2020: Black Lives Matter Observance

Geometry and Topology Research Projects

• 2D-TQFTs
• Subgroups of PSL_2(R)
• Braided disks in 4-space

Number Theory Research Projects

• Distribution of Primes
• Class Numbers and Elliptic curves (Lower bounds for class numbers)
• Modular forms (Variants of Lehmer's Conjecture)

REU Invited Speakers

• Ken Ribet (UC Berkeley): June 16 (Tuesday): 4:00pm
  "A brief History of Fermat's Last Theorem for REU Students"
  Slides Here 
  Cloud Recording (Password: 0Y*t8g%!)
• Sang-hyun Kim (Korea Institute for Advanced Study): June 24 (Wednesday):  8:00pm
  "Acute triangulations of the sphere."
  Slides Here
• James Maynard (Oxford University): June 25 (Thursday): 10:00am
  "Patterns in Primes"
  Slides Here
  Cloud Recording (Password: 2Q.FhJW=)
• Lillian Pierce (Duke University): June 30 (Tuesday): 4:00pm
  "The method of Moments: Examples in Number Theory"
• Henri Darmon (McGill University): July 7 (Tuesday): 4:00pm
  "The theory of complex multiplication and the arithmetic of real quadratic fields”
  Slides Here
  Cloud Recording(coming soon)
• Maryna Viazovska (EPFL, Lausaunne): July 9 (Thursday), 10:00am
  "From uncertainty principles to sphere packings"
  Slides Here
  Cloud Recording (Password: 3R+5wHS.)

 

2020 Results  

1.  M. Amir and L. Hong, On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan Journal, recommended for publication.     
2. T. Blum, C.  Choi, A. Hoey,  J. Iskander, K. Lakein, and T. Martinez, On class numbers, torsion subgroups, and quadratic twists of elliptic curves, submitted for publication.         
3. S. Dembner and V. Jain, Hyperelliptic curves and newform coefficients, submitted for publication.         
4. S. Dembner and V. Jain, A note on congruences for weakly holomorphic modular forms, submitted for publication.         
5. M. Hanada and R. Madhukara, Fourier coefficients of level 1 Hecke eigenforms, submitted for publication.         
6. L. Hong and S. Zhang, Towards Heim and Neuhauser's unimodality conjecture on the Nekrasov-Okounkov polynomials, submitted for publication.      
7. T. Khale, C. O'Kuhn, A. Panidapu, A. Sun, and. S. Zhang, A Bombieri-Vinogradov Theorem for primes in short intervals and small sectors, coming soon.     
8. A. Panidapu and J. Thorner, The CM Sato-Tate distribution in short intervals, coming soon.     

 

REU Management Team: Julie Bergner, Thomas Koberda, Thomas Mark, and Ken Ono

2019 Research Experiences for Undergraduates at Emory University

Number Theory
Dates: June 10 - July 19, 2019.
Math Building, Rooms E406 and E408

Instructors

• Ken Ono, Asa Griggs Candler Professor.
• Larry Rolen, Assistant Professor (Vanderbilt University).
• Jesse Thorner, NSF Postdoctoral Fellow (Stanford University)
• Madeline Locus Dawsey, 4th year Emory PhD Student.
• Ian Wagner, 4th year Emory PhD Student.

Basic Information

Ken Ono has been organizing REU programs since 2003 (formerly at U. Wisconsin (Madison) from 2003-2009). He has advised over 100 students including 6 Morgan Prize Winners and 6 Schafer Prize winners. The REU alums have won numerous other honors including Marshall Scholarships, NSF Graduate Fellowships (over 30), etc... 

Thanks to funding from the NSF, NSA and private donors, we will be organizing a summer research experience in mathematics for the summer of 2019 on the beautiful campus of Emory University (adjacent to the Centers for Disease Control and Prevention (CDC)). We will seek to fill 8-16 openings in the REU. Most of the participants will be US citizens or permanent residents who are presently enrolled in a US undergraduate institution or high school. We have offered openings to high school students who are adequately prepared for the program. NSF supported participants will receive a $5250 stipend and also free accomodations.

2019 Project areas

• Analytic Study of High Performance Swimming
• Elliptic curves and Galois representations
• Mock modular and quantum modular forms
• Additive Number Theory
• Distribution of Primes
• Moonshine

2019 Participants

• Erin Bevilacqua
• Yunseo Choi (2021 Regeneron Science Talent Search 1st place Winner; 2026 Morgan Prize winner)
• Kapil Chandran
• Nate Gillman (Goldwater Scholar)
• Jonas Iskander
• Vanshika Jain (2021 Alice T. Schafer Prize (Honorable Mention))
• Caleb Ji
• Josh Kazdan
• Michael Kural
• Alice Lin (Churchill Scholar)
• Vaughan McDonald
• Eleanor McSpirit
• Sven Mehisovic
• Alex Pascadi
• Junyao Peng
• Nadia Riverso
• Ashwin Sah (Goldwater Scholar, 2021 Morgan Prize winner, and 2020 Morgan Prize Runner Up)
• Victoria Talvola
• Adit Vishnu
• Zoe Walker

Their results

• E. Bevilacqua, K. Chandran, and Y. Choi, Ramanujan congruences for fractional partition functions, Proceedings of the American Mathematical Society, 150 (2022), 2755-2770. 
• Y. Choi, Congruences in fractional partition functions, Integers, 21 (2021), Art. A10.
• N. Gillman, M. Kural, A. Pascadi, J. Peng, and A. Sah, Patterns of primes in the Sato-Tate conjecture, Research in Number Theory, 6 (2020), Art. 9.
• J. Iskander, V. Jain, and V. Talvola, Exact formulae for the fractional partition functions, Research in Number Theory, 6 (2020), Art. 20.
• C. Ji, J. Kazdan, and V. McDonald, Primes with Beatty and Chebotarev conditions,  Journal of Number Theory, 216 (2020), 307-334.
• M. Kural, V. McDonald, and A. Sah, Mobius formulas for densities of sets of prime ideals, Archiv der Mathematik, 115 (2020), 53-66.
• A. Lin, E. McSpirit, and A. Vishnu, Algebraic relations between partition functions and the j-function, Research in Number Theory, 6 (2020), Art. 6.

Application Materials (Deadline for Completed Applications: February 15, 2019)

Note. In accordance with the new REU consortium rule, applicants will not be required to accept or decline an offer before early March 2019. 

Apply here. 

A complete application consists of:

• Cover sheet (automatically generated by MathPrograms.org)
• CV (clearly indicate citizenship)
• Two letters of recommendation
• Undergraduate Transcripts (unofficial ok)
• Personal/Research Statement: Please explain your interest in arithmetic geometry/number theory and describe your previous research experience (if any).

2018 Research Experiences for Undergraduates at Emory University

Number Theory

Dates: June 4 - July 13, 2018.

Math Building, Rooms E406 and E408

Instructors

• John Duncan, Assistant Professor.
• Ken Ono, Asa Griggs Candler Professor.
• Larry Rolen, Visiting Assistant Professor (Georgia Tech).

Basic Information

Ken Ono has been organizing REU programs since 2003 (formerly at U. Wisconsin (Madison) from 2003-2009). He has advised over 100 students including 6 Morgan Prize Winners and 5 Schafer Prize winners. The REU alums have won numerous other honors including Marshall Scholarships, NSF Graduate Fellowships (over 30), etc... 

We are organizing a summer Research Experience in Mathematics for the summer of 2018 on the beautiful campus of Emory University (adjacent to the Centers for Disease Control and Prevention (CDC)). We seek to fill 10-14 openings in the REU. Most of the participants will be US citizens or permanent residents who are presently enrolled in a US undergraduate institution or high school. We have offered openings to high school students who are adequately prepared for the program. NSF supported participants will receive a $5000 stipend and also free accomodations.

2018 Project areas

• Analytic Study of High Performance Swimming
• Elliptic curves and Galois representations
• Mock modular and quantum modular forms
• Additive Number Theory
• Distribution of Primes
• Moonshine

2018 Participants

• Ryan Chen (2018 Goldwater Scholar, 2019 Churchill Fellow )
• Katherine Gallagher (2018 Goldwater Scholar, 2019 Marshall Fellow)
• Nate Gillman (2019 Goldwater Scholar)
• Xavier Gonzalez (special participant, 2018 Rhodes Scholar))
• Lucia Li
• Samuel Marks
• Sven Mehisovic
• Fiona Muir
• Matthew Schoenbauer (2018 Goldwater Scholar)
• Naomi Sweeting (2018 Goldwater Scholar, 2019 Schafer Prize Winner)
• Matthew Tyler
• Katja Vassilev
• Katy Woo (2019 Goldwater Scholar, 2020 Schafer Prize Runner up)

Their results

• Ryan Chen, Samuel Marks, and Matthew Tyler, p-adic properties of Hauptmoduln with applications to moonshine, SIGMA, 15 (2019), Art. 33.
• Madeline Locus Dawsey, Jon Howell, Sven Mehisovic, Fiona Muir, Ken Ono, Sage Ono, and Larry Rolen, Case study of elite breaststrokers using inertial measurement units, submitted for publication.
• Nate Gillman, Xavier Gonzalez, Ken Ono, Larry Rolen, and Matthew Schoenbauer, From partitions to Hodge numbers of Hilbert schemes of surfaces, Philosophical Transactions of the Royal Society, Series A, 2020;378 (2163)
• Katie Gallagher, Lucia Li, Naomi Sweeting, Katja Vassilev, Katharine Woo, Generating functions for power moments of elliptic curves over Fp, Journal of Number Theory, 201 (2019), 53-67.
• Katie Gallagher, Lucia Li, and Katja Vassilev, Lacunarity of Han-Nekrasov-Okounkov q-series, Annals of Combinatorics, accepted for publication.
• Nate Gillman, Xavier Gonzalez, and Matthew Schoenbauer, Exact formulas for invariants of Hilbert schemes, Research in Number Theory 4 (2018), Art. 39.
• Naomi Sweeting and Katy Woo, Formulas for Chebotarev densities of Galois extensions of number fields, Research in Number Theory (2019), 5, No. 4.
• Naomi Sweeting and Katy Woo, On the zeros of a class of modular functions, Annals of Combinatorics, 23 (2019), 417-422.

2017 Research Experiences for Undergraduates at Emory University

Number Theory

Dates: June 5 - July 14, 2017.

Math Building, Rooms E406 and E408

Instructors

• Hannah Larson, (2017 Schafer Prize Winner, Harvard University)
• John Duncan, Assistant Professor.
• Jackson Morrow, Emory PhD Student.
• Jesse Thorner, NSF Postdoc, Stanford University.
• David Zureick-Brown, Assistant Professor.
• Ken Ono, Asa Griggs Candler Professor.

Basic Information

Ken Ono has been organizing REU programs since 2003 (formerly at U. Wisconsin (Madison) from 2003-2009). He has advised over 100 students including 5 Morgan Prize Winners and 5 Schafer Prize winners. The REU alums have won numerous other honors including Marshall Scholarships, NSF Graduate Fellowships (over 30), etc... 

We are organizing a summer Research Experience in Mathematics for the summer of 2017 on the beautiful campus of Emory University (adjacent to the Centers for Disease Control and Prevention (CDC)). We seek to fill 10-14 openings in the REU. Most of the participants will be US citizens or permanent residents who are presently enrolled in a US undergraduate institution or high school. We have offered openings to high school students who are adequately prepared for the program. NSF supported participants will receive a $5000 stipend and also free accomodations.

2017 Project areas

• Elliptic curves and Galois representations
• Mock modular and quantum modular forms
• Additive Number Theory
• Distribution of Primes
• Moonshine

2017 Participants

• Ryan Alweiss
• Tessa Cotron
• Dean Cureton
• Sam DeHority
• Sanath Devalapurkar
• Xavier Gonzalez (2017 Rhodes Scholar)
• John Halliday
• Sammy Luo
• Anya Michaelsen
• Emily Stamm
• Neekon Vafa
• Roger Van Peski
• Sameera Vemulapalli (2017 Alice T.Schafer Runner Up)
• Danielle Wang (2019 Alice T. Schafer Runner Up)
• Catherine Yeo
• Weitao Zhu

Their results

• Ryan Alweiss and Sammy Luo, Bounded gaps between primes in short intervals, Research in Number Theory, 4 (2018), Art. 15.
• Tessa Cotron, Anya Michaelsen, Emily Stamm, and Weitao Zhu, Lacunary eta-quotients modulo powers of primes, Ramanujan Journal (2020), https://doi.org/10.1007/s11139-020-00257-y.
• Dean Cureton and Catherine Yeo, Polynomials that behave like the Riemann zeta-function, High School J. Math., January 2018.
• Sam DeHority, Xavier Gonzalez, Neekon Vafa, and Roger Van Peski, Moonshine for all finite groups, Research in the Mathematical Sciences, 5 (2018), 14.
• Sanath Devalapurkar and John Halliday, The Dieudonne modules and Ekedahl-Oort types of Jacobians of hyperelliptic curves in odd characteristic, submitted.
• Sameera Vemulapalli and Danielle Wang, Uniform bounds for the number of rational points on symmetric squares of curves with low Mordell-Weil rank, submitted.

2016 Research Experiences for Undergraduates at Emory University

Number Theory

Dates: June 6 - July 15, 2016

Math Building, Rooms E406 and E408

Instructors

• Olivia Beckwith, NSF Graduate Fellow
• David Zureick-Brown, Assistant Professor.
• Ken Ono, Asa Griggs Candler Professor.
• Michael Mertens Postdoctoral Fellow, (Emory)
• Jesse Thorner, (Emory) Graduate Student.

Basic Information

Ken Ono has been organizing REU programs since 2003 (formerly at U. Wisconsin (Madison) from 2003-2009). He has advised over 100 students including 4 Morgan Prize Winners and 4 Schafer Prize winners. The REU alums have won numerous other honors including Marshall Scholarships, NSF Graduate Fellowships (over 30), etc... 

We are organizing a summer Research Experience in Mathematics for the summer of 2016 on the beautiful campus of Emory University (adjacent to the Centers for Disease Control and Prevention (CDC)). We seek to fill 10-14 openings in the REU. Most of the participants will be US citizens or permanent residents who are presently enrolled in a US undergraduate institution or high school. We have offered openings to high school students who are adequately prepared for the program. NSF supported participants will receive a $5000 stipend and also free accomodations.

2016 Project areas

• Elliptic curves and Galois representations
• Mock modular and quantum modular forms
• Additive Number Theory
• Distribution of Primes
• Moonshine

Participants

• Asra Ali
• Tessa Cotron
• Robert Dicks
• Sarah Fleming (2017 Schafer Prize, Runner-Up)
• Elaine Hou
• Meena Jagadeesan (2018 Goldwater Scholar)
• Aaron Landesman (2017 Morgan Prize, Runner-Up)
• Yang Liu
• Nitya Mani (2018 Goldwater Scholar, 2019 Schafer Prize, Honorable Mention, 2019 Hertz Fellow)
• Peter Park
• Zhuo Qun (Alex) Song
• Ashvin Swaminathan (2018 Morgan Prize Winner)
• James Tao
• Yujie Xu

Their results

• Asra Ali and Nitya Mani, Infinite Product Exponents for Modular forms, Research in Number Theory, 2 (2016), 21.
• Asra Ali and Nitya Mani, Shifted convolution L-series values for elliptic curves, Archiv der Mathematik, 110 (3) (2018), 225-244.
• Tessa Cotron, Robert Dicks, and Sarah Fleming, Asymptotics and congruences for partition functions which arise from finitary permutation groups, Research in Number Theory, 2(2016), 20.
• Tessa Cotron, Robert Dicks, and Sarah Fleming, Congruences between word lengths statistics for finitary alternating and symmetric groups, Arch. Math. (Basel), 109 (2017), 201-214.
• Elaine Hou and Meena Jagadeesan, Dyson's Partition Ranks and their Multiplicative Extensions, Ramanujan Journal, Ramanujan J. 45 (2018), 817-839.
• Aaron Landesman, Ashvin Swaminathan, James Tao, and Yujie Xu, Lifting Subgroups of Symplectic Groups over ℤ/ℓℤ, Research in Number Theory, 3 (2017), 14.
• Aaron Landesman, Ashvin Swaminathan, James Tao, and Yujie Xu, Surjectivity of Galois representations in rational families of Abelian varieties, Algebra and Number Theory, 13 (2019), 995-1038.
• Aaron Landesman, Ashvin Swaminathan, James Tao, and Yujie Xu, Hyperelliptic curves with maximal Galois action on the torsion points of their Jacobians, Indiana University Math. J., accepted for publication.
• Yang Liu, Peter S. Park, and Zhuo Qun Song, Bounded Gaps Between Products of Distinct Primes, Research in Number Theory, 3, (2017) 26.
• Yang Liu, Peter S. Park, and Zhuo Qun Song, The "Riemann Hypothesis" is True for Period Polynomials of Almost All Newforms, Research in the Mathematical Sciences, 3 (2016), 31.

Research Experiences for Undergraduates at Emory University

Number Theory

Dates: June 8- July 17, 2015.

Math Building, Rooms E406 and E408

Instructors

• David Zureick-Brown, Assistant Professor.
• Ken Ono, Asa Griggs Candler Professor.
• Michael Mertens Postdoctoral Fellow, (Emory)
• Jesse Thorner, (Emory) Graduate Student.
• Sarah Trebat-Leder, NSF Graduate Fellow, (Emory)

Basic Information

The PI has been organizing REU programs since 2003 (formerly at U. Wisconsin (Madison) from 2003-2009). He has advised over 100 students including 4 Morgan Prize Winners and 4 Schafer Prize winners. The REU alums have won numerous other honors including Marshall Scholarships, NSF Graduate Fellowships (over 30), etc... 

Thanks to the generous support of the National Science Foundation and Emory University, we are again organizing a summer Research Experience in Mathematics for the summer of 2014 on the beautiful campus of Emory University (adjacent to the Centers for Disease Control and Prevention (CDC)). We seek to fill 8-10 openings in the REU. Most of the participants will be US citizens or permanent residents who are presently enrolled in a US undergraduate institution or high school. We have offered openings to high school students who are adequately prepared for the program. NSF supported participants will receive a $3600 stipend and free accomodations.

2015 Project areas

• Elliptic curves and Galois representations
• Mock modular and quantum modular forms
• Additive Number Theory
• Distribution of Primes
• Moonshine

Participants

• Lea Beneish (Indiana University)
• Evan Chen (2019 Morgan Prize, Runner-Up) (MIT)
• Claire Frechette (Brown University)
• Aaron Landesman (Harvard University): 2017 Morgan Prize, Runner-Up
• Hannah Larson (Harvard University), 2017 Schafer Prize Winner, 2024 Breakthrough Mirzakhani New Frontiers Prize
• Maddie Locus (University of Georgia)
• Peter Park (Princeton University)
• Peter Ruhm (Stanford University)
• Ashvin Swaminathan (Harvard University) (2018 Morgan Prize Winner)
• Robin Zhang (Stanford University)

Their results

• E. Alwaise, R. Dicks, J. Friedman, L. Gu, Z. Harner, H. Larson, M. Locus, I. Wagner, and J. Weinstock, Shifted distinct-part partition identities in arithmetic progressions, Annals of Combinatorics, 21 (2017), 479-494.
• L. Beneish and C. Frechette, p-adic properties of certain half-integral weight modular forms, J. Number Theory 168 (2016), 413-432.
• E. Chen, P. Park, and A. Swaminathan, Linnik's Theorem for Sato-Tate laws on elliptic curves with complex multiplication, Research in Number Theory 1 (2015), 28.
• E. Chen, P. Park, and A. Swaminathan, On logarithmically Benford sequences, Proceedings of the American Mathematical Society 144 (2016), 4599-4608.
• E. Chen, P. Park, and A. Swaminathan, Elliptic curve variants of the least quadratic nonresidue problem and Linnik's Theorem, Int. J. Number Th. 14 (2018), 255-288.
• A. Landesman, P. Ruhm, and R. Zhang, Spin canonical rings of log stacky curves, Annales de L'Insitut Fourier 66 (2016), 2339-2383.
• A. Landesman, P. Ruhm, and R. Zhang, Section rings of Q-divisors on minimal rational surfaces, Mathematical Research Letters, 25 (2015), 1329-1357.
• H. Larson, Generalized Andrews-Gordon style identities, Research in Number Theory 1(2015), 13.
• H. Larson, Modular units from quotients of Rogers-Ramanujan style q-series Proceedings of the American Mathematical Society 144 (2016), 4169-4182.
• H. Larson, Proof of a conjecture regarding the level of Rose's generalized sum-of-divisor functions, Research in Number Theory 1 (2015), 16.
• H. Larson, Coefficients of McKay-Thompson series and distributions of the moonshine module, Proceedings of the American Mathematical Society, 144 (2016), 4183-4197.

Research Experiences for Undergraduates at Emory University

Number Theory

2014 Research Experiences for Undergraduates at Emory University

Number Theory

Dates: June 9 - July 18, 2014

Math Building, Rooms E406 and E408

Instructors

• David Zureick-Brown, Assistant Professor.
• Ken Ono, Asa Griggs Candler Professor.
• Michael Griffin, NSF Graduate Fellow, (Emory)
• Jesse Thorner, (Emory) Graduate Student.
• Sarah Trebat-Leder, NSF Graduate Fellow, (Emory)

Basic Information

The PI has been organizing REU programs since 2003 (formerly at U. Wisconsin (Madison) from 2003-2009). He has advised over 100 students including 4 Morgan Prize Winners and 4 Schafer Prize winners. The REU alums have won numerous other honors including Marshall Scholarships, NSF Graduate Fellowships (over 30), etc... 

Thanks to the generous support of the National Science Foundation and Emory University, we are again organizing a summer Research Experience in Mathematics for the summer of 2014 on the beautiful campus of Emory University (adjacent to the Centers for Disease Control and Prevention (CDC)). We seek to fill 8-10 openings in the REU. Most of the participants will be US citizens or permanent residents who are presently enrolled in a US undergraduate institution or high school. We have offered openings to high school students who are adequately prepared for the program. NSF supported participants will receive a $3600 stipend and free accomodations.

2014 Project areas

• Elliptic curves and Galois representations
• Mock modular and quantum modular forms
• Additive Number Theory
• Distribution of Primes

Participants

• Lea Beneish (Indiana University)
• Lynn Chua (MIT)
• Claire Frechette (Brown University)
• Benjamin Gunby (MIT)
• Hannah Larson (Harvard University): 2017 Schafer Prize Winner, 2024 Breakthrough Mizakhani New Frontiers Prize
• Maddie Locus (University of Georgia)
• Evan O'Dorney (Harvard University): 2016 Morgan Prize Runner-Up
• Soohyun Park (MIT)
• Alexander Smith (Princeton University)
• Geoffrey Smith (Yale University)
• David Yang (MIT): 2017 Morgan Prize Winner
• Lynnelle Ye (Stanford, special participant)
• Allen Yuan (Harvard University)

Their results

• L. Beneish and H. Larson, Traces of singular values of hauptmoduln, International Journal of Number Theory, 11 (2015), pages 1027-1048.
• L. Chua, B. Gunby, S. Park, and A. Yuan, Proof of a conjecture of Guy on class numbers, International Journal of Number Theory, 11 (2015), pages 1345-1355.
• B. Gunby, A. Smith, and A. Yuan, Irreducible canonical representations in positive characteristic, Research in Number Theory, 1 (2015), 1.
• C. Frechette and M. Locus, Combinatorial properties of Rogers-Ramanujan type identities arising from Hall-Littlewood polynomials, Annals of Combinatorics 20 (2016), 345-360.
• M. Jameson, J. Thorner, and L. Ye, Benford's Law for coefficients of newforms, International Journal of Number Theory 12 (2016), 483-494.
• L. Chua, S. Park, and G. Smith, Bounded gaps between primes in special sequences, Proceedings of the American Mathematical Society 143 (2015), 4597-4611.
• E. O'Dorney, Canonical rings of Q-divisors on P^1, Annals of Combinatorics 19 (2015), 765-784.
• G. Smith and L. Ye, The partition function modulo 3 in arithmetic progressions, Ramanujan Journal 39 (2016), 603-608.
• D. Yang, S_n equivariant sheaves and Kozsul cohomology, Research in the Mathematical Sciences, 1 (2014), 13.

Research Experiences for Undergraduates at Emory University

Number Theory

2013 Research Experiences for Undergraduates at Emory University

Number Theory

Dates: June 3, 2013- July 26, 2013

Math Building, Room E406

Instructors

• David Zureick-Brown, Assistant Professor.
• Ken Ono, Asa Griggs Candler Professor.
• Robert Lemke Oliver,NSF postdoc (Stanford)
• Larry Rolen,NSF Graduate Fellow, (Emory)

2013 Project areas

• Elliptic curves and Galois representations
• Mock modular and quantum modular forms
• Additive Number Theory
• Distribution of Primes

Participants

• Ping Ngai (Brian) Chung (MIT)
• Eric Larson* (Harvard): 2014 Frank and Brennie Morgan Prize Winner
• Hannah Larson* (S. Eugene High School): 2013 Davidson Fellow, 2024 Breakthrough Mirzakhani New Frontiers Prize
• Shiyu Li (UC Berkeley): 2014 Honorable Mention for Schafer Prize
• Akhil Mathew (Harvard)
• Sarah Peluse (U. Chicago): 2014 Schafer Prize Winner, 2022 Breakthrough Mirzakhani New Frontiers Prize
• Sarah Pitman (Emory U.)
• Jesse Silliman (U. Chicago)
• Geoffrey Smith (Yale): Goldwater Scholar
• Isabel Vogt (Harvard)

Note. Special participant.

Their results

• Ping Ngai (Brian) Chung and Shiyu Li, On the residue classes of pi(n) modulo t, Integers, 13(2013), A79.
• Ping Ngai (Brian) Chung and Shiyu Li, Bounded gaps between products of special primes, Mathematics, 2 (2014), pages 37-52.
• Hannah Larson and Geoffrey Smith, Congruence properties of Taylor coefficients of modular forms, International Journal of Number Theory, 10 (2014), pages 1501-1518.
• Sarah Peluse, Irreducible representations of SU(n) with prime power degree, Seminaire Lotharingien de Combinatoire 71 (2013/14), Art. B71d.
• Sarah Peluse, On zeros of Eichler integrals, Archiv der Mathematik, 102 (2014), pages 71-81.
• Sarah Pitman, 3F2-hypergeometric functions and supersingular elliptic curves, Involve, 8(2015), pages 481-490.
• Jesse Silliman and Isabel Vogt, Powers in Lucas sequences via Galois representations, Proceedings of the American Mathematical Society, 143 (2015), pages 1027-1041.

Research Experiences for Undergraduates at Emory University

Number Theory

Instructors

• David Zureick-Brown, Assistant Professor.
• Ken Ono, Asa Griggs Candler Professor.
• Robert Lemke Oliver, Emory PhD student.

Description of the 2012 project areas.

Participants

• Jennifer Bryson (Texas A&M University)
• David Corwin (Princeton University)
• Tony Feng (Harvard University)
• Zane Kun Li (Princeton University)
• Keenan Monks (Harvard University)
• Sarah Peluse (University of Chicago): 2014 Schafer Prize Winner
• Sarah Pitman (Emory University)
• Sarah Trebat-Leder (Princeton University)
• Lynnelle Ye (Stanford University)

Their results

• J. Bryson, K. Ono, S. Pitman, R. C. Rhoades, Unimodal sequences and quantum and mock modular forms, Proc. Natl. Acad. Sci. USA, 109 No. 40 (2012), pages 16063-16067.
• D. Corwin, T. Feng, Z. Li, and S. Trebat-Leder, Elliptic curves with full 2-torsion and maximal adelic Galois representations, Math. Comp., 83 (2014), pages 2925-2951.
• K. Monks, S. Peluse, L. Ye, Strings of special primes in arithmetic progressions, Arkiv Math.,101 (2013), pages 219-234.
• K. Monks, S. Peluse, L. Ye, Congruence properties of Borcherds product exponents, Int. J. Numb. Th., 9 (2013), pages 1563-1578.
• K. Monks and L. Ye, Congruences of concave composition functions, submitted for publication.
• S. Peluse, An infinite family of lacunary series, Ramanujan J., accepted for publication.

Research Experiences for Undergraduates at Emory University

Number Theory: Number fields and modular forms

Participants

• Eva Belmont (Harvard)
• Alexander Carney (University of Michigan): 2012 Marshall Scholar
• Anastassia Etropolski (Bard)
• Eric Larson (Harvard): 2014 Frank and Brennie Morgan Prize Winner
• Holden Lee (MIT): 2013 Gates Scholar
• Soek Hyeong Lee (Stanford)
• Alexandra Musat (Cal Tech)
• Gyujin Oh (Stanford)
• Sarah Pitman (Emory University)
• Larry Rolen (University of Wisconsin)
• Sarah Trebat-Leder (Princeton)

Note: Some participants were funded by non-NSF sources.

Some pictures

• Group photo
• Group photo
• Sarah Trebat-Leder, Alexandra Musat, Eva Belmont, Holden Lee
• Anastassia Etropolski, Alex Carney, Sarah Pitman
• Larry Rolen and Eric Larson
• Eugene Oh and Sean Lee

Their results

• E. Belmont, H. Lee, A. Musat, and S. Trebat-Leder, l-adic properties of partition functions, Monatshefte fur Mathematik, 173 (2013), pages 219-234.
• A. Carney, A. Etropolski, S. Pitman, Powers of the eta-function and Hecke operators, International Journal of Number Theory, 8 (2012), pages 599-611.
• E. Larson and L. Rolen, Upper bounds for the number of number fields with alternating Galois group,  Proceedings of the American Mathematical Society, 141 (2013), pages 499-503.
• E. Larson and L. Rolen, https://uva.theopenscholar.com/files/ken-ono/files/reu2011larson_rolen2.pdf, 5 (2012), pages 91-97.
• E. Larson and L. Rolen, Integrality properties of the CM values of certain weak Maass forms, Forum Mathematicum, 27 (2015), pages 961-972.
• S. H. Lee and G. Oh, On the distribution of cyclic number fields of prime degree, International Journal of Number Theory, 8 (2012), pages 1463-1475.

Research Experiences for Undergraduates at University of Wisconsin-Madison

Number Theory: Investigating Elliptic curves, Hypergeometry, and Modular Forms

Participants

• Zana Chan (Bard College)
• Rebecca Hoberg (University of Chicago)
• Eric Larson (Harvard University)
• Pak Hin Lee (Stanford University)
• Keenan Monks (Hazleton HS, Pa): '10 Siemens Semi-Finalist, '10 International S.-T. Yau Math, Honorable Mention (top 3 from US), 2011 Intel STS (6th place)
• Ying-Ying Tran (Cal Tech)
• Dmitry Vaintrob (Harvard University)
• Mckenzie West (St. Olaf)
• Alexandr Zamorzaev (MIT)

Note: Some participants are funded by non-NSF sources.

Their results

• R. Hoberg, G. Tran, and M. West, Subbarao's Conjecture on the parity of the partition function.
• E. Larson and D. Vaintrob, On the surjectivity of Galois representations associated to elliptic curves over number fields, Bulletin of the London Mathematical Society, 46 (2014), 197-209.
• E. Larson and D. Vaintrob, Determinants of subquotients of Galois representations associated to abelian varieties, Journal of the Institute of Mathematics of Jussieu, 13 (2014), 517-559.
• P.-H. Lee and A. Zamorzaev, Parity of the partition function and traces of singular moduli, International Journal of Number Theory, 8 (2012), pages 395-409.
• K. Monks, On supersingular elliptic curves and hypergeometric functions, Involve, 5 (2012), pages 99-113.
• Y.-Y. Tran, Generalization of Atkin's orthogonal polynomials and supersingular elliptic curves, Proceedings of the American Mathematical Society, 141 (2013), pages 1135-1141.

Research Experiences for Undergraduates at University of Wisconsin-Madison

Number Theory: Continued fractions, probability, modular forms, elliptic curves and L-functions

Participants

• Jeremy Booher (Harvard University): 2010 Herschel Smith Fellow (Cambridge).
• George Boxer (Princeton University): 2010 Churchill Fellow (Cambridge).
• Peter Diao (Princeton University)
• Anastassia Etropolski (Bard College)
• Amanda Hittson (Bryn Mawr College)
• Rachel Kirsch (University of Maryland)
• Elise McCall (MIT)
• Alex Perry (Columbia University)

Special Participants

• Adam Goldberg (Home school, Berkeley, California)
• Edward Hou (High school student in Appleton, Wisconsin)
• Tony Feng (Phillips Andover, Massachusetts)

Their results

• J. Booher, A. Etropolski, and A. Hittson, Evaluations of cubic twisted Kloosterman sheaves, International Journal of Number Theory, 6 (2010), pages 1349-1365.
• G. Boxer and P. Diao, Effective nonvanishing of canonical Hecke L-functions,  Proceedings of the American Mathematical Society, 138 (2010), pages 3891-3897.
• G. Boxer and P. Diao, 2-Selmer groups of quadratic twists of elliptic curves, Proceedings of the American Mathematical Society, 138 (2010), pages 1969-1978.
• E. Dummit, A. Goldberg, and A. Perry, A conjecture of Evans on sums of Kloosterman sums, Proceedings of the American Mathematical Society, 138 (2010), pages 3047-3056.
• T. Feng, R. Kirsch, E. McCall, and M. Wage, Birth death processes and q-continued fractions Transactions of the American Mathematical Society, 364 (2012), pages 2703-2721.

Research Experiences for Undergraduates at University of Wisconsin-Madison

Number Theory: Modular forms and Number Theory

Participants

• Emily Clader (Columbia University): 2009 Columbia Valedictorian!
• Dan Collins (Cornell University)
• Yvonne Kemper (UC Berkeley)
• Daniel Le (Stanford University): Fulbright Scholar, and NSF Graduate Fellow
• Shelly Manber (MIT)
• Shrenik Shah (Harvard University): 2008 Mumford Prize (Harvard), and 2008 Hoopes Senior Thesis Prize (Harvard), NSF Graduate Fellowship
• Ameya Velingker (Harvard University)

Special Participants

• Matt Wage (Appleton High School, 2008 Intel Science Talent Search Finalist)
• Sally Wolfe (Memorial High School/UW Madison Special Student)
• Maria Monks (MIT, short term visitor from Joe Gallian's REU): 2009 Alice T. Schafer Prize Winner, 2010 Morgan Prize runner-up, 2011 Morgan Prize Winner

Their results

• E. Clader, Y. Kemper, and M. Wage, Lacunarity of certain partition theoretic generating functions, 
  Proceedings of the American Mathematical Society, 137 2009, pages 2959-2968.
• D. Collins and S. Wolfe, Congruences for Han's generating function, Involve, 2 (2009), pages 225-236.
• D. Le, S. Manber, and S. Shah, On p-adic properties of twisted traces of singular moduli, International Journal of Number Theory, 6 (2010), pages 625-653.
• M. Monks, Number theoretic properties of generating functions related to Dyson's rank for partitions into distinct parts, Proceedings of the American Mathematical Society, 138 (2010), pages 481-494.
• A. Velingker, An exact formula for the coefficients of Han's generating function, Annals of Combinatorics, accepted for publication.

Research Experiences for Undergraduates at University of Wisconsin-Madison

Number Theory: Drinfeld modules and function fields

Participants

• Shanshan Ding (Columbia University)
• Doris Dobi (MIT): 2009 Alice T. Schafer Prize runner up, and NDSEG and NSF Graduate Fellow
• Samuel Lichtenstein (Harvard University): 2008 Goldwater Scholar, NSF Graduate Fellow 
• Adam Merberg (Brown University): NSF Graduate Fellow
• Noam Tanner (Princeton University)
• Elena Udovina (Harvard University): 2008 Herschel Smith Fellow (Cambridge) 
• Nicholas Wage (Harvard University)
• Irena Wang (Harvard University)

Special Participants

• Matthew Wage (Appleton High School): 2008 Intel STS Finalist
• Sally Wolfe (Memorial High School, Madison)

Their results

• Shanshan Ding, Smallest irreducible of the form x^2-dy^2, International Journal of Number Theory, 5 (2009), pages 449-456.
• Doris Dobi, Nick Wage, and Irena Wang, Supersingular rank two Drinfeld modules and analogs of Atkin's orthogonal polynomials, International Journal of Number Theory, 5(2009), pages 885-895.
• Sam Lichtenstein, https://uva.theopenscholar.com/files/ken-ono/files/2007reu_sam.pdf, Proceedings of the American Mathematical Society, 136 (2008), pages 3419-3428.
• Adam Merberg, Divisibility of class numbers of imaginary quadratic function fields, Involve, 1 (2008), pages 47-58.
• Noam Tanner, Strings of consecutive primes in function fields, International Journal of Number Theory, 5 (2009), pages 81-88.
• Matt Wage, On Lehmer-type questions for special classes of arithmetic functions, International Journal of Modern Mathematics, 3 (2008), pages 245-260.
• Sally Wolfe, Parity of the partition function and the modular discriminant, Involve, 1 (2008), pages 9-19.
• Elena Yudovina, Diophantine equations and congruences over function fields, Proceedings of the American Mathematical Society, 136 (2008), pages 3839-3850.

Research Experiences for Undergraduates at University of Wisconsin-Madison

Number Theory: Arithmetic Dynamics and Modular Forms

Participants

• Yaim Cooper (MIT):    Runner Up, 2007 Alice T. Schafer Prize
• Carl Erickson (Stanford University): 2007 Churchill Fellow (Cambridge)
• Richard Gottesman (Brown University)
• Alison Miller (Harvard University): 2008 Alice T. Schafer Prize Co-Winner, 2008 Churchill Fellow, 2008 Hoopes Prize (Harvard)
• Yusra Naqvi (Swarthmore College)
• Aaron Pixton (Princeton University): 2008 Churchill Fellow (Cambridge) and 2009 Morgan Prize Winner
• Brian Rice (Harvey Mudd College): 2007 Goldwater Scholar
• KwokFung Tang (University of Chicago)
• Irena Wang (Harvard University)

Special Participants

• Nick Wage (Appleton East High School, Wisconsin)
• David Hansen (Watchung Hills Regional High School, New Jersey)
• Daniel Kane (MIT): 2007 Frank and Brennie Morgan Prize Winner.

Their results

• Y. Cooper, N. Wage, and I. Wang,  Congruences for modular forms of non-positive weight, International Journal of Number Theory,  4 (2008), pages 1-13.
• C. Erickson, A. Miller, and A. Pixton,  Orders at infinity of modular forms with Heegner divisor,  Proceedings of the American Mathematical Society, 135 (2007), pages 3115-3126.
• A. Miller and A. Pixton,  Arithmetic traces of non-holomorphic modular invariants, International Journal of Number Theory,  6 (2010), pages 69-87.
• R. Gottesman and K. Tang,  Quadratic recurrences with a positive density of prime divisors, International Journal of Number Theory, 6 (2010), pages 1027-1045.
• D. Hansen and Y. Naqvi, Shimura lifts of half-integral weight modular forms arising from theta functions, Ramanujan Journal,  17 (2008), pages 343-354.
• B. Rice, Primitive prime divisors of first-order polynomial recurrence sequences, Integers, 7 , #A26.

Research Experiences for Undergraduates at University of Wisconsin-Madison

Number Theory: Modular Forms and q-Series

Participants

• Zajj Daugherty (Harvey Mudd)
• Po-Ru Loh (CalTech)
• Robert Rhoades (Bucknell)
• Rhiannon Schayer (Northwestern)

Special Participants

• Sam Lachterman (Wisconsin)
• Brendan Younger (Wisconsin)
• Nick Wage

Their results

• Z. Daugherty, Integrality properties of quotients of Wronskians of the Andrews-Gordon series, Integers, A10, 2006.
• P.-R. Loh and R. Rhoades, p-adic and combinatorial properties of modular form coefficients, International Journal of Number Theory, 2 (2006), pages 305-328.
• N. Wage, (Appleton High School) Character sums and Ramsey properties of generalized Paley graphs, Integers, A18, 2006.

Research Experiences for Undergraduates at University of Wisconsin-Madison

Number Theory: Investigating Elliptic Curves, Modular Forms and q-Series

Participants

• Stephanie Basha (Santa Clara University)
• Daniel Corson (MIT) 
• David Favero (Brown University)
• Jayce Getz (Harvard University)
• Katie Liesinger (University of Chicago)
• Harris Nover (CalTech)
• Emma Smith (MIT)
• Sarah Zubairy (University of Rochester)

Their results

• S. Basha, J, Getz, H, Nover and E. Smith, Systems of orthogonal polynomials arising from the modular j-function, J. Math. Anal. Appl. 289 (2004), pages 336-354.
• D. Corson, D. Favero, K. Liesinger and S. Zubairy, Characters and q-series in Q(sqrt(2)), J. Number Th. 107 (2004), pages 392-405.