My primary area of research is harmonic analysis and its applications to elliptic partial differential equations with non-smooth coefficients. I also have a research interest in cryptography, in particular lattice based cryptography and the NTRU public key cryptosystem.

My primary area of research is harmonic analysis and its applications to elliptic partial differential equations with non-smooth coefficients.
In harmonic analysis, I have specific interests in multiparameter Fourier analysis: differentiation and covering lemmas, multi-linear operators, and singular integrals. My research in elliptic pde is primarily in the area of boundary value problems associated to divergence and non-divergence second and higher order operators whose coefficients lack smoothness.
I also have a research interest in cryptography, in particular lattice based cryptography.

9/16/89 - 9/15/93: "Alfred P. Sloan Research Fellowship" - $23,384

7/1/90 - 12/31/95: NSF - "Presidential Young Investigator" -
DMS-9058463 - $125,000

7/1/92 - 12/31/94: NSF - "Harmonic Analysis and Elliptic Partial
Differential Equations" - $54,474

7/1/97 - 6/30/00: NSF - "Harmonic Analysis and Elliptic PDE" - $121,008

7/1/00 - 6/30/03: NSF - "Harmonic Analysis and Linear Elliptic Equations" - $82,062

2006-2009: NSF grant DMS0600389 – "Topics in Harmonic Analysis and Elliptic Partial Differential Equations"

2009-2012 NSF DMS-0901139 Research Grant, 2009-2013, $435,307

2010 - 2014 Australia Research Council Grant, Discovery Project DP120100399, with X. Duong (MacQuarie University, M. Lacey (Georgia Tech), and L. Ward (University of South Australia), 2012-2014, $270,00

 

Multiparameter Riesz Commutators, with M. Lacey, S. Petermichl and B. Wick, to appear Amer. J. Math.

An Introduction to Mathematical Cryptography, with J. Hoffstein, J. Silverman, Series: Undergraduate Textbooks in Mathematics, July 2008, Springer.

BMO from dyadic BMO on the bidisc, with Lesley Ward ,J. London Math. Soc. Vol 77 (2), April 2008, p. 524-544.

The L^p Dirichlet problem for second order elliptic equations and a p-adapted square function, with M. Dindos and S. Petermichl, J. Funct. Anal., Vol. 249, issue 2, 2007, p. 372-392.

Multiparameter paraproducts, with C. Muscalu, T. Tao, C. Thiele, Revista M. Ibero. Vol. 22, Number 3, 2006, p. 963-976.

Variations on the theme of Journe's lemma, with C. Cabrillo, M. Lacey, U. Molter, JourneLemma.pdf to appear, Houston J. Math.

On Estimating the Lattice Security of NTRU, with N. Howgrave-Graham, J. Hoffstein, W. Whyte, NTRULattice-2005-1.pdf

A covering lemma for rectangles in $\R^n$, with R. Fefferman, Proc. AMS.133 No. 11, p.3235-3241

A short proof of the Coifman-Meyer paraproduct theorem, , with C. Muscalu, T. Tao, C. Thiele, C-M.pdf

Biparameter paraproducts, with C. Muscalu, T. Tao, C. Thiele, biparam.pdf , Acta Mathematica 193 (2004) , p.269-296.

NTRUSign: Digital signatures using the NTRU lattice, with J. Hoffstein, N. Howgrave-Graham, J.Silverman, W.Whyte, Proceedings of CT-RSA 2003, NTRUSign.pdf

Lectures at the L'Institut Fourier 2002 Summer School in Cryptology on NTRU encryption and digital signatures, grenoble.pdf

NTRUSign: Digital signatures using the NTRU lattice, extended version, with J. Hoffstein, N. Howgrave-Graham, J.Silverman, W.Whyte,2002 , NTRUSignextended.pdf

The Dirichlet problem for elliptic equations with drift terms, with C. Kenig, Publicaciones Matematiques 45 (2001), 199-217 Dir.dvi

NSS: An NTRU lattice-based signature scheme, with J. Hoffstein and J. Silverman, Proceedings of Eurocrypt 2001.

A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations. Adv. Math. 153 (2000), no. 2, 231--298. w/Kenig, C.; Koch, H.; Toro, T. AbsoluteContinuity.pdf

The absolute continuity of elliptic measure revisited. J. Fourier Anal. Appl. 4 (1998), no. 4-5, 463--468. w/Kenig, Carlos E.

NTRU: a ring-based public key cryptosystem. Algorithmic number theory (Portland, OR, 1998), 267--288, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. w/Hoffstein, Jeffrey; Silverman, Joseph H.

The inhomogeneous Dirichlet problem for $\Delta\sp 2$ in Lipschitz domains. J. Funct. Anal. 159 (1998), no. 1, 137--190. w/Adolfsson, Vilhelm

Multiparameter operators and sharp weighted inequalities. Amer. J. Math. 119 (1997), no. 2, 337--369. w/Fefferman, R.

Area integral estimates for higher order elliptic equations and systems. Ann. Inst. Fourier (Grenoble) 47 (1997), no. 5, 1425--1461. w/Dahlberg, B. E. J.; Kenig, C. E.; Verchota, G. C.

Vector potential theory on nonsmooth domains in ${R}\sp 3$ and applications to electromagnetic scattering. J. Fourier Anal. Appl. 3 (1997), no. 2, 131--192. w/Mitrea, Dorina; Mitrea, Marius;

Review of: Harmonic Analysis Techniques in Second Order Elliptic PDE, by Carlos Kenig, Bulletin of the AMS 33 (2) (1996), p. 229-236, review.pdf

Maximum principles for the polyharmonic equation on Lipschitz domains. Potential Anal. 4 (1995), no. 6, 615--636. w/Verchota, G. C.

The Neumann problem for elliptic equations with nonsmooth coefficients. II. A celebration of John F. Nash, Jr. Duke Math. J. 81 (1995), no. 1, 227--250 (1996). w/Kenig, Carlos E.

Littlewood-Paley estimates: some applications to elliptic boundary value problems. Partial differential equations and their applications (Toronto, ON, 1995), 221--238, CRM Proc. Lecture Notes, 12, Amer. Math. Soc., Providence, RI, 1997. fieldsinst.dvi

Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators. Ann. of Math. (2) 142 (1995), no. 1, 1--38. w/Verchota, Gregory C.

A maximum principle for biharmonic functions in Lipschitz and $C\sp 1$ domains. Comment. Math. Helv. 68 (1993), no. 3, 385--414. w/Verchota, G.

The Neumann problem for elliptic equations with nonsmooth coefficients. Invent. Math. 113 (1993), no. 3, 447--509. w/Kenig, Carlos E.

A martingale inequality related to exponential square integrability. Proc. Amer. Math. Soc. 118 (1993), no. 2, 541--546.

The Dirichlet problem in $L\sp p$ for the biharmonic equation on Lipschitz domains. Amer. J. Math. 114 (1992), no. 5, 923--972. w/Verchota, Gregory

Boundary value problems for higher order operators in Lipschitz and $C\sp 1$ domains. Fourier analysis and partial differential equations (Miraflores de la Sierra, 1992), 297--309, Stud. Adv. Math., CRC, Boca Raton, FL, 1995.

Area integral estimates for the biharmonic operator in Lipschitz domains. Trans. Amer. Math. Soc. 327 (1991), no. 2, 903--917. w/Verchota, Gregory

The theory of weights and the Dirichlet problem for elliptic equations. Ann. of Math. (2) 134 (1991), no. 1, 65--124. w/Fefferman, R. A.; Kenig, C. E.

Partial differential equations with minimal smoothness and applications. Proceedings of the IMA Participating Institutions Conference held at the University of Chicago, Chicago, Illinois, March 21--25, 1990. Edited by B. Dahlberg, E. Fabes, R. Fefferman, D. Jerison, C. Kenig and J. Pipher. The IMA Volumes in Mathematics and its Applications, 42. Springer-Verlag, New York, 1992. xii+220 pp. ISBN: 0-387-97774-0

The $h$-path distribution of the lifetime of conditioned Brownian motion for nonsmooth domains. Probab. Theory Related Fields 82 (1989), no. 4, 615--623. w/Kenig, Carlos E.

The oblique derivative problem on Lipschitz domains with $L\sp p$ data. Amer. J. Math. 110 (1988), no. 4, 715--737. w/Kenig, Carlos E.

Oblique derivative problems for the Laplacian in Lipschitz domains. Rev. Mat. Iberoamericana 3 (1987), no. 3-4, 455--472.

Hardy spaces and the Dirichlet problem on Lipschitz domains. Rev. Mat. Iberoamericana 3 (1987), no. 2, 191--247. w/Kenig, Carlos E.

Journé's covering lemma and its extension to higher dimensions. Duke Math. J. 53 (1986), no. 3, 683--690.

Bounded double square functions. Ann. Inst. Fourier (Grenoble) 36 (1986), no. 2, 69--82.

NSF Postdoctoral Fellowship, 1987-90

Sloan Foundation Fellowship, 1989-93

Presidential Young Investigator Award, 1990-95

Mathematical Sciences Research Institute Professor, 1997

NSF Mathematics Institute Award, ICERM, 2010 and 2015

Inaugural Fellow of the American Mathematical Society, 2012

Elected Member of the American Academy of Arts and Sciences, 2015

Fellow of the Society for Industrial and Applied Mathematicians, 2019

Fellow of American Association for the Advancement of Science,  2022

American Mathematical Society (AMS)

Association for Women in Mathematics (AWM)

Mathematical Association of America (MAA)

Society for Industrial and Applied Mathematics (SIAM)

I have taught a wide range of courses at the undergraduate level. The upper division courses are usually analysis, partial differential equations, probability, statistics, functional analysis and cryptography. At the graduate level, I typically teach real and complex analysis, PDE, probability, and topics courses in these areas.