Research
My primary research interests are analytic number theory and representation theory. More specifically, I work on problems in automorphic forms and representations of algebraic groups, and their generalizations on arithmetic covering groups.
This work is supported by NSF Grant DMS-2101392, and prior to this by DMS-1801527.
Below is a list of recent publications and preprints. Collaborative work with Dan Bump, including many supporting materials on multiple Dirichlet series, can also be found in the Papers section of his home page. His page contains some additional supporting materials, mostly for our older papers from the "aughts," so you might want to try there first.
- Crystal Invariant Theory I: Geometric RSK (with G. Frieden, P. Pylyavskyy, and T. Scrimshaw), preprint.
- Metaplectic Iwahori Whittaker functions and supersymmetric lattice models (with V. Buciumas, D. Bump, and H.P.A. Gustafsson), submitted for publication
- Frozen Pipes: Lattice models for Grothendieck polynomials (with C. Frechette, A Hardt, E. Tibor, K. Weber), submitted for publication
- Colored vertex models and Iwahori Whittaker functions (with V. Buciumas, D. Bump, and H.P.A. Gustafsson), submitted for publication
- Colored five-vertex models and Demazure atoms (with V. Buciumas, D. Bump, and H.P.A. Gustafsson), J. Combin. Theory Ser. A 178 48 pp. (2021)
- Vertex operators, solvable lattice models, and metaplectic Whittaker functions (with V. Buciumas, D. Bump, and H.P.A. Gustafsson), Comm. Math. Phys. 380 (2):535--579 (2020)
- Duality for metaplectic ice (with V. Buciumas, D. Bump, and N. Gray), See the arXiv link for full details. Also appearing as an appendix to Comm. in Number Theory and Physics 13 (1):101--148 (2019)
- A Yang-Baxter equation for metaplectic ice (with V. Buciumas and D. Bump), Comm. in Number Theory and Physics 13 (1):101--148 (2019)
- Hecke modules from metaplectic ice (with V. Buciumas, D. Bump, and S. Friedberg), Selecta Math. 24 (3):2523--2570 (2018)
- On Hamiltonians for six-vertex models (with A. Schultz), J. Combin. Theory Ser. A 55, 100--121 (2018)
- Matrix coefficients and Iwahori-Hecke algebra modules (with D. Bump and S. Friedberg), Advances in Math. 299 247--271 (2016)
- Whittaker coefficients of metaplectic Eisenstein series (with S. Friedberg), GAFA, 25 4:1180--1239 (2015)
- Deformations of the Weyl character formula for classical groups and the six-vertex model (with A. Schultz), J. Algebraic Combin., 42, 4:917--958 (2015)
- Iwahori Whittaker functions and Demazure operators (with D. Bump and A. Licata), Rallis memorial volume, J. Number Theory, 146, 41--68 (2015)
- Unique functionals and representations of Hecke algebras (with D. Bump and S. Friedberg), Pacific J. Math. 260, no. 2, 381--394 (2012)
- Eisenstein Series, Crystals, and Ice (expository paper with D. Bump and S. Friedberg), Notices of AMS 58 (2011), no. 11, 1563--1571.
- Metaplectic Ice (with Bump, Chinta, Friedberg, and Gunnells), in Multiple Dirichlet series, L-functions and automorphic forms, Progress in Math. 300, 65--92 (2012)
- Crystals of Type B and metaplectic Whittaker functions (with Bump, Chinta, and Gunnells), in Multiple Dirichlet series, L-functions and automorphic forms, Progress in Math. 300, 93--118 (2012)
- Coefficients of the n-fold Theta function and Weyl group multiple Dirichlet series (with Bump, Friedberg, and Hoffstein), in Contributions in Analytic and Algebraic Number Theory, Springer Proc. in Math., 9, 83--95. (2012)
- Schur polynomials and the Yang-Baxter equation (with D. Bump and S. Friedberg), Comm. Math. Phys. 308 (2011), no. 2, 281--301.
- A crystal definition for symplectic multiple Dirichlet series (with J. Beineke and S. Frechette), in Multiple Dirichlet series, L-functions and automorphic forms, Progress in Math. 300, 37--63 (2012)
- Weyl group multiple Dirichlet series of Type C (with J. Beineke and S. Frechette), Pacific J. Math. 254 (2011), no. 1, 11--46.
- Weyl group multiple Dirichlet series, Eisenstein series, and crystal bases (with D. Bump and S. Friedberg), Annals of Math. 173 1081--1120 (2011)
- Weyl group multiple Dirichlet series: Type A combinatorial theory (with D. Bump and S. Friedberg) -- Preliminary Version. Book available as: Annals of Math. Studies v. 175, Princeton Univ. Press (2011).
- Gauss sum combinatorics and metaplectic Eisenstein series (with D. Bump and S. Friedberg), In ''Automorphic forms and L-functions I. Global Aspects,'' Contemporary Mathematics v. 488 (2009) 61--82.
- Twisted Weyl group multiple Dirichlet series: the stable twisted case (with D. Bump and S. Friedberg), Eisenstein series and applications 1--26. Progr. Math., 258, Birkhauser.
- Weyl Group Multiple Dirichlet Series III: Eisenstein series and twisted unstable A_r (joint with D. Bump, S. Friedberg, and J. Hoffstein), Annals of Math. 166 (2007)
- Residues of Weyl Group Multiple Dirichlet Series associated to GL(n+1) (with Daniel Bump), Proc. Symp. Pure Math. 75 (2006)
- Weyl Group Multiple Dirichlet Series II: The Stable Case (joint with D. Bump and S. Friedberg), Invent. Math. 165 (2006), 325-355
- Weyl Group Multiple Dirichlet Series I (joint with D. Bump, G. Chinta, S. Friedberg, and J. Hoffstein), Proc. Symp. Pure Math. 75 (2006)
- On Kubota's Dirichlet Series (with Daniel Bump), J. Reine Angew. (2006) 598, 159-184
- Cubic twists of GL(2) automorphic L-functions, (joint with S. Friedberg and J. Hoffstein), Invent. Math. 160 (2005) no. 1, 31-58
- Non-vanishing twists of GL(2) automorphic L-functions, (joint with A. Bucur, G. Chinta, S. Frechette and J. Hoffstein), IMRN 78 (2004), 4211-4239
- Analytic Continuation for Cubic Multiple Dirichlet Series, Ph.D. Thesis, Brown University (2003).
Graduate Students at UMN
I have had five Ph.D. students graduate under my supervision at Minnesota:
- Heidi Goodson, graduated May 2016 (point counts for Calabi-Yau manifolds via hypergeometric functions), tenure track assistant professor at Brooklyn College. First job: visiting assistant professor at Haverford College. Heidi's thesis work and subsequent improvements:
- Nathan Gray, graduated May 2017 (ice models and statistical mechanics on the metaplectic group). First job: visiting instructor at Mount Holyoke College.
- William Grodzicki, graduated May 2017 (Bessel models for representations of p-adic groups). First job: Instructor in mathematics, St. Olaf College
- Ben Strasser, graduated June 2019 (Hecke algebra approach to p-adic functionals). First job: General Dynamics Mission Systems, Bloomington, MN
- Katherine Weber, graduated August 2021 (Schubert calculus and solvable lattice models). Here are several papers she completed during this period:
Current students at Minnesota:
- Andy Hardt, 6th year (character theory for Hecke algebras and monoid Hecke algebras)
- Claire Frechette, 6th year (metaplectic forms and quantum groups)
- Emily Tibor, 5th year (p-adic representation theory)
Graduate Students at MIT
While at MIT, I graduated four Ph.D. students. Their names, interests, and first jobs are listed below:
- Mario DeFranco, (visiting professor at MIT, 2014-2015), worked on projects in p-adic representation theory related to the Bessel functional. Defended thesis in April, 2014.
- Catherine Lennon, (software developer for RJMetrics, graduated May, 2011), wrote a thesis on finite field hypergeometric function identities for traces of Hecke operators and trace of Frobenius for elliptic curves. Here are the two papers that comprised her thesis:
- Sawyer Tabony, (visiting assistant professor at Boston College, graduated May, 2011) wrote a thesis on symmetric function theory from lattice models in statistical mechanics and Hecke algebra computations for the metaplectic group.
- Peter McNamara (Szego asst. professor at Stanford, graduated May, 2010), worked on Whittaker functions for metaplectic forms over local fields with connections to crystal bases, and symmetric function theory. Here are the two papers that comprised his thesis:
Undergradute Student Research at Stanford
Bob Hough, Stanford Class of '07, wrote a comprehensive Summer Research Journal on Gauss sums, culling from Ireland and Rosen; Berndt, Evans, and Williams; and papers of Weil, Yamamoto, etc. with some original synthesis.
Carl Erickson, Stanford Class of '07, has been working on behavior of the Riemann Zeta Function in the critical strip. The resulting article appeared in the Stanford Undergraduate Research Journal and can be found in the Spring 2005, Issue 4 of SURJ.
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Teaching at Minnesota ('12 - present)
For Spring '17, I am teaching Math 3283W, "Sequences, Series, and Foundations," which serves as both a first course in elementary analysis and an introduction to mathematical proofs.
During Fall '16, I taught Math 8300, a graduate topics course on Iwahori-Hecke algebras.
In Spring '16, I taught Math 3593H, integral calculus on manifolds using linear algebra, from the book of Hubbard and Hubbard.
In Fall '15, I am taught an undergraduate honors course, Math 3592H using the book of Hubbard and Hubbard. Lecture notes will be posted there throughout the semester. The course will continue in Spring 2016.
For Fall '14, I taught a graduate course in algebraic number theory, Math 8251, using Neukirch's book. Lecture notes, which just expand on and rework the book, are available at 8251 lecture notes.
In Spring '14, I taught a second-semester graduate course in complex analysis focusing on the Riemmann mapping theorem and Riemann surfaces, Math 8702. Scans of lecture notes I gave are linked there (culled from Ahlfors, Miranda, and Donaldson's books).
For Fall '13 and Fall '12, I've been teaching an introductory graduate course in complex analysis, Math 8701.
For Spring '13, I was on leave as an organizer of a special semester program at Brown's new NSF-sponsored institute ICERM.
Teaching at MIT ('06 - '12)
For Fall '11, I'm teaching 18.03 -- Differential Equations.
During the '10-'11 academic year, I taught a seminar on additive number theory from Nathanson's books and a graduate course in automorphic forms (1/2 classical, 1/2 analytic aspects from Borel's book). I tried to write detailed course notes -- some of which are little more than recasting of notes of Milne, or Shimura's book, or Borel's book. They are available here: 18.785 notes.
During the '09-'10 academic year, once again I taught 18.01, first-year calculus and was on Junior Leave in Spring '10.
For Spring '09, I taught 18.786: Topics in Algebraic Number Theory on "Tate's Thesis," closely following the books of Ramakrishnan-Valenza ("Fourier Analysis on Number Fields" -- the actual name for Tate's thesis) and the classic Weil's "Basic Number Theory." There was no course website.
During Fall '08, I taught 18.01, first-year calculus, using Simmons. (These course webpages are retired each semester, as their contents are used in future semesters.)
During Spring '08, I taught 18.784: The Mathematical Legacy of Ramanujan, a small seminar course. I hope to eventually put up students' final projects, which were outstanding and original.
For Fall, '07, I taught 18.781: Theory of Numbers, a first course in number theory using Niven, Zuckerman, and Montgomery, but also featuring special topics like cubic reciprocity based on the treatment in Ireland and Rosen's book.
During Spring '07, I taught 18.103: Fourier Analysis - Theory and Applications. This course covers Lebesgue measure and integration theory and Fourier analysis, using the book by Adams and Guillemin. We'll discuss applications to probability along the way, and if time permits, how both the probability and the Fourier analysis are used in modern analytic number theory.
For Fall '06, I taught 18.786: Topics in Algebraic Number Theory on "Reciprocity Laws."
Teaching at Stanford ('03 -'06, Skeletons of the course webpages remain at most links. Email me if you are interested in their contents.)
During the Spring quarter, 2006, I am teaching two courses:
- Math 152
- Math 249B - An Introduction to Langlands Program
During the Fall quarter, 2005, I taught Math 51 and Math 248, an introduction to automorphic forms, co-taught with Dan Bump. The course website for Math 51 can be found at:
For the Spring quarter, 2005, I taught just one course:
During Winter quarter, 2005, I taught two courses:
For the Fall quarter, 2004, I taught Math 52: Integral Calculus of Several Variables, a course on integration techniques culminating in the theorems of Stokes, Gauss, and Green.
During Spring quarter, 2004, I was on leave.
During Winter quarter, 2004, I taught two courses:
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Math 109: Applied Group Theory, an introduction to group theory focusing on groups as measures of symmetry.
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Math 248B: Algebraic Number Theory, covering basic introduction to automorphic L-functions.
During Fall quarter, 2003, I taught also Math 51: Linear Algebra and Differential Calculus of Several Variables. Click on the link for the web page.
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Other Seminars at MIT and in the Boston Area
MIT number theory seminar, organized by Bjorn Poonen. TUESDAYS 4:30-5:30 in 2-139 (when there's no BC-MIT seminar).
MIT STAGE Seminar, Seminar on Topics in Arithmetic, Geometry, Etc., run by Greg Minton, Abhinav Kumar, and Bjorn Poonen - FRIDAYS 11-12, 2-139
MIT Lie Groups Seminar, organized by David Vogan. WEDNESDAYS, 4:30-5:30, 2-143
MIT Combinatorics Seminar - WEDNESDAYS AND FRIDAYS, 4:15
BU Algebra Seminar, which is secretly always about number theory - MONDAYS, 4:15
Links
Collaborators' Websites:
- Dan Bump, Stanford
- Jeff Hoffstein, Brown
- Sol Friedberg, Boston College
- Andrew Schultz, Wellesley College
- Tony Licata, ANU
- Gautam Chinta, CUNY
- Paul Gunnells, UMass
- Jennifer Beineke, WNEC
- Sharon Frechette, Holy Cross
- Alina Bucur, UCSD
The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota.
Last change: September 4, 2015.
