Ben Brubaker Cecil and Ida B. Green Career Development Chair Associate Professor Department of Mathematics MIT Room 2-267 Cambridge, MA 02139 Phone: Fax: (617) 253-4079 (617) 253-4358 Email: brubaker@math.mit.edu somewhat current CV (PDF) (last updated Spring, 2010)

Research

My primary research interests are analytic number theory and representation theory. More specifically, I work on problems in automorphic forms and their generalizations on arithmetic covering groups.

This work is supported by an NSF Career Grant (DMS-0844185), the James Ferry, Jr. Fund for Innovation in Research Education, and a Cecil and Ida B. Green Career Development Chair.

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. He does a better job of keeping current, so you could try there first.

• Iwahori Whittaker functions and Demazure operators (with D. Bump and A. Licata), Submitted for publication.
• Whittaker coefficients of metaplectic, parabolic Eisenstein series (with S. Friedberg), In preparation.
• Eisenstein Series, Crystals, and Ice (expository paper with D. Bump and S. Friedberg), To appear in Notices of AMS (December 2011).
• Metaplectic Ice (with Bump, Chinta, Friedberg, and Gunnells), To appear in Birkhauser Progress in Math.
• Crystals of Type B and metaplectic Whittaker functions (with Bump, Chinta, and Gunnells), To appear in Birkhauser Progress in Math.
• Coefficients of the n-fold Theta function and Weyl group multiple Dirichlet series (with Bump, Friedberg, and Hoffstein), To appear in S.J. Patterson Festschrift.
• Schur polynomials and the Yang-Baxter equation (with D. Bump and S. Friedberg), To appear in Comm. Math. Phys.
• A crystal definition for symplectic multiple Dirichlet series (with J. Beineke and S. Frechette), To appear in Birkhauser Progress in Math.
• Weyl group multiple Dirichlet series of Type C (with J. Beineke and S. Frechette), To appear in Pacific Journal of Math.
• 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 MIT

I currently have one graduate student working with me, and three that graduated with a Ph.D. in the last two years. Their names, interests, and first jobs are listed below:

• Mario DeFranco, 3rd year graduate student, working on projects in p-adic Whittaker functions.
• Catherine Lennon, (post-doc at UPenn, 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:
  • Gaussian Hypergeometric Evaluations of Traces of Frobenius for Elliptic Curves
  • A Trace Formula for Certain Hecke Operators and Gaussian Hypergeometric Functions
• 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:
  • Metaplectic Whittaker Functions and Crystal Bases, Peter McNamara, Duke Math J. (2011)
  • Principal series representations of metaplectic groups over local fields, a nice presentation of results for general metaplectic covers.

Undergradute Student Research at Stanford

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Teaching at MIT ('06 - present)

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.)

• 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:

• Math 51: Linear Algebra and Differential Calculus of Several Variables

• Math 110: Applied Number Theory, a course on cryptography using number theory, field theory, and elliptic curves.

During Winter quarter, 2005, I taught two courses:

• Math 109: Applied Group Theory, an introduction to group theory focusing on groups as measures of symmetry.
• Math 263A: Lie Groups (course announcement here), an introductory course for graduate students.
  • 263A Homework 1 (pdf format)
  • 263A Homework 2 (pdf format)

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:

• Math 109: Applied Group Theory, an introduction to group theory focusing on groups as measures of symmetry.
• 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

Old Seminar Links

Stanford Representation Theory Seminar 2005-2006

Representation Theory Seminar,'05-'06

This year, the number theory seminar is jointly run between Stanford and AIM (American Institute of Mathematics). More information about the seminar can be foound at the following link:

AIM/Stanford Number Theory Seminar, Fall '04

Nat Thiem and I co-chair the Stanford Representation Theory Seminar for the academic year 2004-2005. A preliminary webpage is here:

Representation Theory Seminar,'04-'05

Number Theory Seminar, Spring '04

Links

• MathSciNet.
• Stanford Math Department.
• Brown Math Department.

• Dan Bump, Stanford
• Jeff Hoffstein, Brown
• Sol Friedberg, Boston College
• Tony Licata, Stanford
• Gautam Chinta, CUNY
• Paul Gunnells, UMass
• Jennifer Beineke, WNEC
• Sharon Frechette, Holy Cross
• Alina Bucur, UCSD

Last change: October 21, 2011.