Timothy J Hodges , PHD

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PhD: University of Leeds Leeds, England, 1981 (Mathematics)

BSc: University of Leeds Leeds, England, 1977 (Mathematics)

Abstract algebra: cryptography, quantum groups, Poisson groups, Lie algebras, Lie bialgebras, noncommutative algebra, noetherian rings, K-theory

2023 -2024 Program Director, Algebra and Number Theory, National Science Foundation, Alexandria, VA

2020 -2021 Editor in Chief, Journal of Mathematical Cryptology,

2018 - Acting Department Head, Department of Mathematical Sciences, University of Cincinnati,

2015 -2018 Program Director, Algebra and Number Theory, Secure and Trustworthy Cyberspace, National Science Foundation, Alexandria, Virginia

2011 -2015 Graduate Program Director, Department of Mathematical Sciences, University of Cincinnati,

2001 -2010 Head, Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio

1995 - Visiting Research Professor, University of Poitiers, Poitiers, France

1993 -2022 Professor, University of Cincnnati, Cincinnati, Ohio

1990 -1991 Visiting Associate Professor, Universite de Bretagne Occidentale, Brest, France

1987 -1993 Associate Professor, University of Cincinnati, Cincinnati, Ohio

1984 -1987 Assistant Professor, University of Cincinnati, Cincinnati, Ohio

1981 -1984 Instructor, University of Utah, Salt Lake City, Utah

1980 -1981 Teaching Associate, University of California at Santa Barbara, Santa Barbara, California

2011 C. P. Taft Research Center Summer Research Fellowship

2001 -2003 National Security Agency Degeneration of R-Matrices and Structure of Non-Standard Quantum Groups

1999 -2001 National Security Agency Algebraic Structure of Nonstandard Quantum Groups

1995 -1998 National Science Foundation Algebraic Structure of Quantum Groups

1993 -1995 Algebraic Structure of Quantum Groups

1993 Charles P. Taft Foundation, University of Cincinnati Summer Faculty Research Grant

1991 -1993 National Security Agency Quantum and Classical Enveloping Algebras of Semisimple Lie Algebras

1989 -1991 National Security Agency and National Science Foundation Noncommutative Algebras

1988 Charles P. Taft Foundation, University of Cincinnati Summer Faculty Research Grant

1986 -1988 National Science Foundation Structure and K-Theory of Group Rings and Rings of Differential Operators

1985 University Research Council, University of Cincinnati Summer Faculty Research Scholarship

Peer Reviewed Publications

Introduction to Belavin-Drinfeld Quantum Groups. In Recent Advances in Noncommutative Algebra and Geometry 801 (2024): pp. 111-124.

Semi-regularity of pairs of Boolean polynomials, Finite Fields and their Applications, 79 (2022), 101992 (with H. Iyer)

Homological characterization of bounded F2-regularity, Journal of Algebra, 588, pp 148-165, 2021 (with S. Molina)

On the existence of homogeneous semi-regular sequences, Journal of Algebra, 476, 519-547, 2017 (with S. Molina and J. Schlather)

First fall degree and Weil descent, Finite Fields and their Applications, 30, 155-177, 2014 (with C. Petit and J. Schlather)

The degree of regularity of a quadratic polynomial, Journal of Pure and Applied Algebra, 217, 207-217, 2013 (with Jacob Schlather)

Growth of the ideal generated by a quadratic multivariate function over GF(3). Journal of Algebra & its Applications, 12 (2013), 1250219 (23 pages) (with J. Ding, V. Kruglov, D. Schmidt and S. Tohaneanu)

Apolipoprotein A-I structural organization in high density lipoproteins isolated from human plasma, Nature Structural and Molecular Biology 18, 416-422 (2011) (with R. Huang, R. A. G. D. Silva, W. G. Jerome, A. Kontush, M.J. Chapman, L. K. Curtiss, and W. S. Davidson)

Inverting HFE is Quasi-Polynomial for all Fields. In: Advances in Cryptology -- CRYPTO 2011, Rogaway, Phillip (Ed.) Lecture Notes in Computer Science, vol. 6841 pp. 724-742, 2011 (with J. Ding)

Growth of the ideal generated by a quadratic Boolean function, PQCrypto 2010, Lecture Notes in Computer Science 6061, Springer, 13-27 (2010) (with J. Ding and V. Kruglov)

Dimension of the linearization equations of the Matsumoto-Imai cryptosystems. In Oyvind Etrehus (Eds.), Coding and Cryptography (pp. 242-251). Lecture Notes in Computer Science 3969, Springer-Verlag (2006)  (with A. Diene, J. Ding, J. E. Gower, and Z. Yin)

Triangular Poisson structures on Lie groups and symplectic reduction. Contemporary Mathematics, 391, 123-134 (2005) (with M. Yakimov)

Degenerations and representations of twisted Shibukawa-Ueno R-operators. Lett. Math. Phys., 68 (3), 151-164, (2004) (with R. Endelman)

Cryptanalysis of an implementation scheme of the tamed transformation method cryptosystem. J. Algebra & Appl., 3 (3), 273-282, (2004) (with J. Ding)

The Yang-Baxter equation for operators on function fields. In A. Pressley (Eds.), Quantum Groups and Lie theory, 168-174. Cambridge University Press (2002) (with J. Ding)

The double and dual of a quasitriangular Lie bialgebra. Math. Res. Lett., 8 (1), pp 91-105, (2001)  (with M. Yakimov)

Generating functions for Cremmer-Gervais quantum groups. J. Pure Appl. Algebra, 157(1), pp 69-79, (2001)

Generalized Jordanian R-matrices of Cremmer-Gervais type. Lett. Math. Phys., 52 (3), pp 225-237, (2000)  (with R. Endelman)

The Cremmer-Gervais solution of the Yang-Baxter equation. Proc. Amer. Math. Soc., 127(6), 1819 - 1826 (1999)

Non-standard quantum groups associated to Belavin-Drinfeld triples. Contemporary Mathematics, 214, 63 - 70 (1998)

Nonstandard solutions of the Yang-Baxter equation. Lett. Math. Phys., 44(1), 67 - 75 (1998)  (with A. Giaquinto)

Double quantum groups and Iwasawa decomposition. J. Algebra, 192(1), 303 - 325 (1997)

Algebraic structure of multiparameter quantum groups. Adv. Math., 126 (1), 52 - 92 (1997)  (with T. Levasseur and M. Toro)

On the Cremmer-Gervais quantizations of SL(n). Internat. Math. Res. Notices, 10, 465 - 481, (1995)

Chern characters, reduced ranks and D-modules on the flag variety. Proc. Edinburgh Math. Soc. (2), 37 (3), 477 - 482 (1994)  (with M. Holland)

Primitive ideals of C_q[SL(n)]. J. Algebra, 168(2), 455 - 468, (1994)  (with T. Levasseur)

Noncommutative deformations of type-A Kleinian singularities. J. Algebra, 161(2), 271 - 290 (1993)

Primitive ideals of C_q[SL(3)]. Comm. Math. Phys., 156(3), 581 - 605 (1993)  (with T. Levasseur)

Morita equivalence of primitive factors of U(sl(2)). In Vinay Deodhar (Eds.), Kazhdan-Lusztig theory and related topics (pp. 175, 179). Providence, RI: Contemporary Mathematics 139, Amer. Math. Soc.(1992)

Behaviour of Krull dimension under extension of the base field. Comm. Algebra, 19 (1), 143 - 156, (1991) (with K. Kim and R. Resco)

Fixed rings of the Weyl algebra A_1(C). J. Algebra, 130 (1), 83 - 96 (1990) (with J. Alev and J.-D. Velez)

Ring-theoretical aspects of the Bernstein-Beilinson theorem. In S. K. Jain and S. R. Lopez-Permouth (Eds.), Noncommutative ring theory (pp. 155-163). Berlin: Lecture Notes in Math. 1448, Springer (1990)

K-Theory of Noetherian rings. In M.-P. Malliavin (Eds.), Seminaire d'Algebre Paul Dubreil et Marie-Paul Malliavin, 39eme Annee (pp. 246, 268). Berlin: Lecture Notes in Math. 1404, Springer (1989)

Equivariant K-theory for noetherian rings. J. London Math. Soc. (2), 39 (3), 414 - 426, (1989)

Representing Clifford algebras as crossed products. J. Algebra, 123 (2), 500 - 505, (1989)  (with S. Tesser)

Noetherian rings with big indecomposable modules. Bull. London Math. Soc., 21(3), 249 - 254 (1989)  (with J. T. Stafford)

K-theory of D-modules and primitive factors of enveloping algebras of semisimple Lie algebras. Bull. Sci. Math. (2), 113(1), 85-88 (1989)

A noncommutative Bass-Tate sequence. Bull. Soc. Math. Belg. Ser. B, 39(3), 255-257, (1987)

K-theory and right ideal class groups for HNP rings. Trans. Amer. Math. Soc., 302(2), 751-767 (1987)

A rank two indecomposable projective module over a Noetherian domain of Krull dimension one. Bull. London Math. Soc., 19 (2), 139-144 (1987)  (with J. Osterburg)

On the primitivity of polynomial rings with nonprimitive coefficient rings. Proc. Amer. Math. Soc., 98 (4), 553-558 (1986)

On the global dimension of certain primitive factors of the enveloping algebra of a semisimple Lie algebra. J. London Math. Soc. (2), 32(3), 411-418 (1985)  (with S. P. Smith)

Sheaves of noncommutative algebras and the Bernstein-Beilinson equivalence of categories. Proc. Amer. Math. Soc., 93 (3), 379-386 (1985)  (with S. P. Smith)

Krull and global dimensions of Weyl algebras over division rings. J. Algebra, 91(2), 334-359 (1984)  (with K. R. Goodearl and T. H. Lenagan)

An example of a primitive polynomial ring. J. Algebra, 90 (1), 217-219, (1984)

The Krull dimension of skew Laurent extensions of commutative rings. Comm. Algebra, 12(11), 1301-1310 (1984)

On Ore and skew Laurent extensions of noetherian rings. J. Algebra, 73(1), 56-64 (1981)  (with J. C. McConnell)

Conference/Workshop Proceedings

Recent Advances in Noncommutative Algebra and Geometry 801 (2024)  (with K. A. Brown, M. Vancliff, and J. J. Zhang, editors)

Quantum Theory and Symmetries, Proceedings of the Third International Symposium. World Scientific (2004) (with P. Argyres, F. Mansouri, J. Scanio, P. Suranyi, & R. Wijewardhana, editors)
 

Review

A guide to quantum groups. Bull. London Math. Soc., 506-508, 1997

Torsion Theories. Bull. Amer. Math. Soc., 148-151, 1989

Invited Presentations

(08-2023. ) Belavin-Drinfeld quantum groups .Canada–Mexico–US Conference in Representation Theory, Noncommutative Algebra, and Categorification, Montreal, Canada.

(07-2022. ) Belavin-Drinfeld quantum groups .Recent Advances in Noncommutative Algebra and Geometry, University of Washington, Seattle, WA.

(04-2018. ) Structure of Belavin-Drinfeld Quantum Groups .American Mathematics Society Sectional Meeting, OSU, Columbus, OH.

(10-2015. ) Mathematical Problems in Post-Quantum Cryptography .United States Naval Academy, Annapolis, MD,

(08-2015. ) Existence of Semi-Regular Sequences .SIAM Conference on Applied Algebraic Geometry (AG15), National Institute for Mathematical Sciences, Daejeon, South Korea.

(07-2015. ) The operational degree of Grobner basis algorithms for systems of equations over finite fields .The 12th International Conference on Finite Fields and Their Applications, Skidmore College, NY.

(01-2015. ) Mathematical problems in multivariate cryptography .DIMACS Workshop on the Mathemtics of Post Quantum cryptgraphy, Rutgers University.

(10-2013. ) Semi-Regular Sequences over F2 .AMS Special Session on Algebraic Cryptography, University of Louisville.

(08-2013. ) Degree of Regularity of Generalized HFE Cryptosystems .SIAM Conference on Applied Algebraic Geometry, Fort Collins, Colorado.

(10-2012. ) First Fall Degree and Weil Descent .Universite Catholique de Louvain, Louvain-la-Neuve, Belgium.

(09-2012. ) First Fall and Degree and Weil Descent .YACC 2012, Porquerolles, France.

(08-2011. ) Inverting HFE Systems is Quasi-polynomial for all Fields .CRYPTO 2011, Santa Barbara, CA, USA. Conference. .

(05-2010. ) Growth of ideals in the algebra of Boolean functions .PQ Crypto 2010, Darmstadt, Germany.

(01-2010. ) Growth of ideals in Boolean Algebras .Special session on algebraic cryptography, AMS annual meeting, San Francisco.

(10-2004. ) Fermat's Conjecture .Undergraduate Colloquium talk, Loyola University, Chicago.

(07-2004. ) Algebraic structure of triangular quantum groups .International Conference on Quantum Groups, Technion, Haifa, Israel.

(06-2003. ) Degeneration of R-matrices .12-th International Colloquium on Quantum Groups and Integrable Systems, Prague, Czech Republic.

(05-2001. ) Symplectic foliations of Poisson groups .Midwest Conference on Lie Algebras and Related Topics, University of Wisconsin-Milwaukee.

(01-2001. ) Symplectic foliations of triangular Poisson groups .Workshop on quantum and semi-classical structures, Fields Institute, Toronto.

(04-2000. ) Recent developments in quantum groups .IU/UC Algebra Day, Indiana University.

(03-2000. ) Generalized Jordanian R-matrices of Cremmer-Gervais type .Workshop on Quantum Groups, Morelia, Mexico.

(02-2000. ) Degeneration of R-matrices .Workshop on Interactions between noncommutative algebra and geometry, Mathematical Sciences Research Institute, Berkeley, CA.

(07-1999. ) R-matrices on rational function fields .LMS Symposium on Quantum Groups, Durham University, England.

(10-1998. ) Algebraic structure of the quantum Lorentz group .Special session on noncommutative algebra, AMS meeting, Wake Forest University.

(03-1998. ) Quantization of Belavin-Drinfeld triples .Special session on quantum groups, AMS meeting, Kansas State University.

(10-1997. ) The Cremmer-Gervais solution of the Yang-Baxter equation .Special session on enveloping algebras and quantum groups, AMS meeting, Milwaukee.

(08-1997. ) Nonstandard quantum groups .Ring theory and Representation Theory, Oberwolfach, Germany.

(05-1997. ) Nonstandard quantum groups .Special session on Noncommutative Ring Theory, Wayne State University,Detroit.

(03-1997. ) Quantum groups - symmetry for the 21st century .Colloquium talk, University of Louisville.

(06-1996. ) Non-standard quantum groups associated to Belavin-Drinfeld triples .Joint Summer Research Conference on Quantization, Mount Holyoke College.

(03-1995. ) The algebraic structure of Cremmer-Gervais quantum groups .Seminaire d'algebre, Universite de Paris VII.

(03-1995. ) Seminar talk, Universite de Poitiers, France.

(01-1995. ) Algebraic structure of Cremmer Gervais quantum groups .Special Session on noncommutative algebra, AMS meeting, San Francisco.

(08-1993. ) Ring theory and Representation Theory .Oberwolfach, Germany.

(07-1993. ) Workshop on Ring Theory, University of Washington.

(01-1993. ) Primitive Spectra of Quantum Groups .Special session on Quantum Groups, AMS meeting, San Antonio, Texas.

(09-1991. ) Primitive ideals of C[SLq(3)] and symplectic leaves of SL(3) .Mid-Atlantic Algebra Conference, Develoments in Algebra related to Quantum Groups, Wake Forest University.

(02-1991. ) A Bernstein-Beilinson theorem for Uq(sl(2)) .Seminar talk, Universite de Strasbourg.

(11-1990. ) Seminaire sur les algebres enveloppantes .Universite de Paris VI,

(10-1990. ) Seminaire d'algebre .Universite de Paris VI,

(06-1990. ) Analog of the Bernstein-Beilinson theorem for Uq(sl(2)) .Joint Summer Research Conference: Deformation and quantization of algebras, University of Massachusetts, Amherst.

(02-1990. ) A Bernstein-Beilinson theorem for Uq(sl(2)) .Special session on quantum groups, AMS Meeting, Penn State.

(12-1989. ) D-module methods for quantum groups .IU/UC Algebra Day, University of Cincinnati.

(10-1989. ) Morita equivalence of primitive factors of the enveloping algebra U(sl(2)) .Special session on Noncommutative algebra, AMS meeting , Muncie, IN.

(09-1989. ) Quantum D-modules .Midwest Noncommutative Ring Theory Conference, Ohio University.

(09-1989. ) Colloquium talk .Miami University.

(07-1989. ) K-theory of D-modules .Microprogram on Noncommutative Rings, Mathematical Sciences Research Institute, Berkeley.

(06-1989. ) Quantum Kleinian singularities .Ringe und Moduln, Oberwolfach, Germany.

(05-1989. ) Quantum Kleinian singularities .Special session on Noncommutative Ring Theory, AMS Meeting , Chicago.

(05-1988. ) Noncommutative Ring Theory .N. Illinois U.,

(06-1987. ) Fixed rings of Weyl algebras .Special session on Ring Theory and Invariant Theory, AMS meeting, Salt Lake City.

(06-1987. ) K-theory of Noetherian rings .Seminaire d'algebre,, University of Paris VII.

(07-1986. ) Seminar talk, University of Warwick, England.

(07-1986. ) Equivariant K-theory of Noetherian rings and schemes .Ring Theory Symposium, Leeds, England.

(04-1986. ) Equivariant K-theory of Noetherian rings .Midwest Ring Theory Conference, Milwaukee.

(04-1984. ) Sheaves of twisted differential operators .Colloquium talk, University of Milwuakee.

(08-1983. ) Differential operators on projective space .NATO Advanced Study Institute, Antwerp, Belgium.

(04-1983. ) Global dimension of simple factors of the enveloping algebra of semi-simple Lie algebras .Special session on Noncommutative Ring Theory, AMS Meeting , Salt Lake City.

(03-1983. ) Sheaves of differential operators on projective space .Special session on Noncommutative Ring Theory, AMS Meeting , Norman, Oklahoma.

(09-1982. ) Seminar talk, University of Southern California.

(06-1981. ) Seminar talk, University of Leeds.

(05-1981. ) Primitive polynomial rings .Seminar talk, University of Southern California.

(04-1980. ) Krull dimension of skew-Laurent polynomial rings .British Mathematical Colloquium,