Mathematics Building, Room 422
Columbia University
New York, NY 10027 
(212) 854-4304
       Fax: (212) 854-8962


Research Interests:  Number Theory curriculum vitae
Some Preprints: (pdf files)
A standard zero free region for Rankin-Selberg L-functions (with X. Li)
On the determination of the Plancherel measure for Lebedev-Whittaker transforms on GL(n)
(with A. Kontorovich)
Fourier expansions of GL(2) newforms at various cusps
                                        (with J. Hundley and M. Lee)
Natural boundaries and the correct notion of integral moments of L-functions
                                                                   (with A. Diaconu and P. Garrett)
Moments for L-functions for GL(r) x GL(r-1) (with A. Diaconu and P. Garrett)
A converse theorem for double Dirichlet series (with N. Diamantis)
Voronoi Formulas on GL(n) (with X. Li)
Second Moments of GL(2) L-Functions (with A. Diaconu)
Counting Congruence Subgroups (with A. Lubotzky and L. Pyber)
Multiple Dirichlet Series and Moments of Zeta and L-Functions
                                                 (with A. Diaconu and J. Hoffstein)
Modular Forms, Elliptic Curves, and the ABC Conjecture
The Gauss Class Number problem for Imaginary Quadratic Fields
The Elementary proof of the Prime Number Theorem,  An Historical Perspective 


Automorphic Representations and L-Functions for the General Linear Group, Volume 1
Automorphic Representations and L-Functions for the General Linear Group, Volume 2
Automorphic Forms and L-Functions for the Group GL(n,R) 
Blog with corrections of Errors in the above books
                    (Goldfeld Conference photos)

                    Joint COLUMBIA-CUNY-NYU Number Theory Seminar
                    Bretton Woods Workshop on Multiple Dirichlet Series(2005)    (Conference photos)
                    AAG key exchange
                    Edinburgh Workshop on Multiple Dirichlet Series(2008)
                    Decision Regarding World Record Musky Challenge
                    My Daughter's art work (including my portrait above)
                    Some Interesting Chemistry
                    Very accurate clock


Mathematics V3027,  MAKING, BREAKING CODES
 TR 2:40am-3:55pm Location: Room 312 Mathematics Building

Textbook: Introduction to cryptography with coding theory (second edition)
by Wade Trappe and Lawrence C. Washington, Prentice Hall
Joćo Guerreiro
Help Room Hours:  Tuesdays: 10am -11:30am,     Thursdays: 1pm - 2:30pm

TA: Paul Lewis
Help Room Hours:  Wednesdays
: 11am - 2pm

EXAM 1: Thursday, October 8  (20% of total grade for this course)
EXAM 2:  Thursday, November 12   (20% of total grade for this course)
Projected FINAL EXAM: 
Thursday, December 17:  1:10-4pm (35% of total grade for this course)

HOMEWORK:   (25% of total grade for this course)
Please put all homework in the box just opposite Room 417 Math Building.
Deadline is Thursday 11:59pm. All homework is from the Trappe-Washington book.

HW#1 due Thursday, September 17:   Page 55, Problems 2, 3, 4, 5, 8, 10, 11
HW#2 due Thursday, September 24:   Page 104, Problems 1, 2, 3, 5, 12, 13, 18, 19, 26
HW#3 due  Thursday, October 1:   Page 192, Problems 1, 3, 7, 9, 11  Page 214, Problems 1, 2, 11
HW#4 due Thursday, October 8:   Page 105, Problems 9, 10  Page 195, Problem 21  Page 215, Problems 3, 4, 6, 10
HW#5 due Thursday, October 15:   Page 108, Problems 25, 26,  Page 239, Problems 1, 2, 3, 4
HW#6 due Thursday, October 22:   Page 252, Problems 1, 2, 4, 5, 8,  Page 286, Problem 4,   Page 315, Problems 2, 3
HW#7 due Thursday, October 29:  
Page 321, Problems 1, 2, 3, 4, 5, 6
 HW#8 due Thursday, November 5:  
Page 370, Problems 1, 2, 3, 4, 5, 10   
HW#9 due Thursday, November 12:   Page 110, Problems 33, 34,  Page 447,  Problem 12(a),  Construct a finite field of 8 elements.
HW#10 due Thursday, November 19:   Page 445, Problems 1,  2,  3(a),(b),(c)
HW#11 due Thursday, December 3:   Page 445, Problems  4,  5(a),(b),  6, 8
HW#12 due Thursday, December 10:   Page 445, Problems  9, 10, 11, 12, 13

Mathematics G9905
Location: 507 Mathematics Building
TUESDAYS  11:40a.m. - 12:55p.m.

September 22
Wei Zhang Super-positivity of L-functions
September 29
David Soudry
On local descent of supercuspidal representations
 from GL(n) to classical groups
October 6
Ioan Filip A local relative trace formula for spherical varieties and distinguished representations
October 13 Tian An Wong
Explicit formulas via trace formulas
October 20 Daniel Litt
Hyperbolicity of moduli spaces and function field arithmetic
October 27 Ali Altug
Trace formula, analytic number theory, and related problems
November 3
Election Day  - - University Holiday
November 10 Rahul Krishna
A new relative trace formula approach to Waldspurger's formula for toric periods
November 17 Joćo Guerreiro
The classical and adelic versions of the GL(3) Kuznetsov trace formula
November 24 Bingrong Huang
Hybrid sup-norm bounds for Eisenstein series
December 1 Vivek Pal
Mazur-Tate conjecture