Stephen D. Miller

Professor and Chair of Mathematics, Rutgers University

Research interests

Automorphic Forms
L-functions (See the Database of Automorphic L-functions)
Analytic Number Theory

Department of Mathematics
Hill Center-Busch Campus
Rutgers, The State University of New Jersey
110 Frelinghuysen Rd
Piscataway, NJ 08854-8019

E-mail: miller@math.rutgers.edu
Fax: (732)-445-5530

Some recent items: What's this background image?

Generating cryptographically-strong random lattice bases and recognizing rotations of Zⁿ, joint with Tamar Lichter Blanks, in PQCrypto 2021. Tamar's Video presentation.

Lecture notes on lattices from the automorphic point of view, from a Spring 2020 course. So far the notes cover a proof of Siegel's integration formula using Eisenstein series. Future updates will center on Borel and Harish-Chandra's reduction theory.

A template method for Fourier coefficients of Langlands Eisenstein series, with Dorian Goldfeld and Michael Woodbury, to appear in Rivista di Matematica della Universita di Parma, Proceedings of the "Second Symposium on Analytic Number Theory".

On the convergence of Kac-Moody Eisenstein series, with Lisa Carbone, Howard Garland, Kyu-Hwan Lee, and Dongwen Liu, preprint.

On Arthur's unitarity conjecture for split real groups, with Joseph Hundley, to appear in the American Journal of Mathematics.

Stopping time signatures for some algorithms in cryptography, with Percy Deift and Thomas Trogdon, preprint.

Universal optimality of the E₈ and Leech lattices and interpolation formulas, with Henry Cohn, Abhinav Kumar, Danylo Radchenko, and Maryna Viazovska, preprint.

On the nonexistence of automorphic eigenfunctions of exponential growth on SL(3,Z)\SL(3,R)/SO(3,R), with Tien D. Trinh, preprint. Video presentation.

Kissing numbers and transference theorems from generalized tail bounds, with Noah Stephens-Davidowitz, preprint.

Coppersmith's lattices and ``focus groups'': an attack on small-exponent RSA, with Bhargav Narayanan and Ramarathnam Venkatesan, Journal of Number Theory (to appear).

A Spectral reciprocity formula and non-vanishing for L-functions on GL(4)xGL(2) , with Valentin Blomer and Xiaoqing Li, preprint.

Weights, raising and lowering operators, and K-types for automorphic forms on SL(3,R), with Jack Buttcane, preprint.

The balanced Voronoi formulas for GL(n), with Fan Zhou, preprint.

Estimates on Eisenstein distributions for reciprocals of p-adic L-functions: the case of irregular primes, with Stephen Gelbart, Ralph Greenberg, and Freydoon Shahidi, preprint.

The sphere packing problem in dimension 24, with Henry Cohn, Abhinav Kumar, Danylo Radchenko, and Maryna Viazovska, preprint. Video of IAS lecture.

Some properties of optimal functions for sphere packing in dimensions 8 and 24, with Henry Cohn, preprint.

Cryptanalysis of the NFL schedule (expository piece).

Non-abelian analogs of lattice rounding, with Evgeni Begelfor and Ramarathnam Venkatesan, to appear in Groups Complexity Cryptology.

SL(2,Z)-invariance and D-instanton contributions to the D^6R^4 interaction, with Michael B. Green, and Pierre Vanhove, Communications in Number Theory and Physics 9 (2015), pp.307-344.

Entirety of cuspidal Eisenstein series on loop groups, with Howard Garland and Manish Patnaik, to appear in the American Journal of Mathematics.

What mathematics is really behind the distributional Γ-factors? (for the Schmid 70th conference).

A p-adic integral for the reciprocal of L-functions, with Stephen Gelbart, Alexei Pantchichkine, and Freydoon Shahidi, to appear in a Contemporary Mathematics volume in memory of Ilya Piatetski-Shapiro.

Residual automorphic forms and spherical unitary representations of exceptional groups , to appear in Annals of Mathematics. Click here for the programs and data files.

Small representations, string instantons, and Fourier modes of Eisenstein series, with Michael Green and Pierre Vanhove; includes appendix "Special Unipotent Representations" by Dan Ciubotaru and Peter E. Trapa.

Fourier coefficients of automorphic forms, character variety orbits, and small representations, with Siddhartha Sahi, to appear in Journal of Number Theory.

The archimedean theory of the Exterior Square L-functions over Q, with Wilfried Schmid,Journal of the American Mathematical Society, 25 (2012), pp. 465-506.

Adelization of automorphic distributions and mirabolic Eisenstein series, with Wilfried Schmid, to appear in Contemporary Mathematics, volume in honor of Gregg Zuckerman's 60th birthday.

Pairings of automorphic distributions, with Wilfried Schmid, to appear in Mathematische Annalen.

On the rapid decay of cuspidal automorphic forms, with Wilfried Schmid, to appear in Advances in Mathematics.

Eisenstein series for higher-rank groups and string theory amplitudes, with Michael B. Green, Jorge Russo, and Pierre Vanhove, Communcations in Number Theory and Physics 4 (2010), 551-596.

Non-degeneracy of Pollard Rho Collisions, with Ramarathnam Venkatesan, International Mathematics Research Notices.

A method for computing general automorphic forms on general groups, Forum Mathematicum.

Conference on Analytic Number Theory and Higher Rank Groups, NYU, May 19-23, 2008. Power point for lectures 1 and 3, scanned notes for lecture 2.
(Click here for older powerpoint format.)

Slides from the Technion conference talk (12/20/2005) on "Applications of automorphic distributions to analytic number theory": powerpoint or html.

Other Powerpoint presentations (can be viewed with a browser, e.g. Internet Explorer):

Chemistry and Graph Theory slides (10/14/2004)

Slides from a general talk (click here for the powerpoint file).