Timothy Hodges

Timothy J Hodges

Professor Emeritus

Education

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

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

Research and Practice Interests

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

Positions and Work Experience

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

03-1995 -04-1995 Visiting Research Professor, University of Poitiers, Poitiers, France

1993 -2023 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

2023 -To Present Program Director, National Science Foundation, Alexandria, VA

08-2018 -10-2018 Acting Department Head, Department of Mathematical Sciences, University of Cincinnati,

Research Support

2011 C. P. Taft Research Center Summer Research Fellowship Role:PI Type:Fellowship Level:College

Grant: #MDA 904-01-1-0084 2001 -2003 National Security Agency Degeneration of R-Matrices and Structure of Non-Standard Quantum Groups Role:PI $36,565

Grant: #MDA 904-99-1-0026 1999 -2001 National Security Agency Algebraic Structure of Nonstandard Quantum Groups Role:PI $31,481

Grant: #DMS 9501484 1995 -1998 National Science Foundation Algebraic Structure of Quantum Groups Role:PI $50,000

Grant: #MDA 904-93-3042 1993 -1995 Algebraic Structure of Quantum Groups Role:PI $24,000

1993 Taft Foundation, University of Cincinnati Summer Faculty Research Grant Role:PI Type:Grant

Grant: #MDA 904-91-H0045 1991 -1993 National Security Agency Quantum and Classical Enveloping Algebras of Semisimple Lie Algebras Role:PI $58,459

Grant: #MDA 904-89-H2046 1989 -1991 National Security Agency and National Science Foundation Noncommutative Algebras Role:PI $41,811

1988 Taft Foundation, University of Cincinnati Summer Faculty Research Grant Role:PI Type:Grant

Grant: #DMS 8602291 1986 -1988 National Science Foundation Structure and K-Theory of Group Rings and Rings of Differential Operators Role:PI $40,329

1985 University Research Council, University of Cincinnati Summer Faculty Research Scholarship Role:PI Type:Grant

Abbreviated Publications

Peer Reviewed Publications

Timothy J Hodges, Sergio Molina and Jacob Schlather, On the existence of homogeneous semi-regular sequences, Journal of Algebra, 476, 519-547

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

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

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

A. Diene, J. Ding, J. E. Gower, T. J. Hodges, & Z. Yin (2006). 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.

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

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

Giaquinto, Anthony, & Hodges, Timothy J. (1998). Nonstandard solutions of the Yang-Baxter equation. Lett. Math. Phys., 44(1), 67 - 75.

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

Hodges, Timothy J., Levasseur, Thierry, & Toro, Margarita (1997). Algebraic structure of multiparameter quantum groups. Adv. Math., 126(1), 52 - 92.

Hodges, Timothy J., & Holland, Martin (1994). Chern characters, reduced ranks and D-modules on the flag variety. Proc. Edinburgh Math. Soc. (2), 37(3), 477 - 482.

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

Hodges, Timothy J., & Levasseur, Thierry (1993). Primitive ideals of C_q[SL(3)]. Comm. Math. Phys., 156(3), 581 - 605.

Hodges, Timothy J. (1992). 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.

Hodges, Timothy J. (1990). 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.

Jacques Alev, Timothy J.  Hodges, and Juan-Diego Velez, Fixed rings of the Weyl algebra A_1(C). J. Algebra, 130(1), 83 - 96 (1990)

Hodges, Timothy J. (1989). 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.

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

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

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

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

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

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

Hodges, Timothy J., & Smith, S. Paul (1985). 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.

Hodges, Timothy J., & Smith, S. Paul (1985). Sheaves of noncommutative algebras and the Bernstein-Beilinson equivalence of categories. Proc. Amer. Math. Soc., 93(3), 379-386.

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

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

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

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

Timothy J. Hodges and Steven B. Tesser,  Representing Clifford algebras as crossed products. J. Algebra, 123 (2), 500 - 505, 1989

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

Timothy J. Hodges and Kyunghee Kim, Behaviour of Krull dimension under extension of the base field. Comm. Algebra, 19(1), 143 - 156, 1991

Timothy J. Hodges and Thierry Levasseur, Primitive ideals of C_q[SL(n)]. J. Algebra, 168(2), 455 - 468, 1994

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

Timothy J. Hodges and Milen Yakimov, Triangular Poisson structures on Lie groups and symplectic reduction. Contemporary Mathematics, 391, 123-134,

Robin Endelman and Timothy J. Hodges, Degenerations and representations of twisted Shibukawa-Ueno R-operators. Lett. Math. Phys., 68 (3), 151-164,

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

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

Timothy J. Hodges and Milen Yakimov, The double and dual of a quasitriangular Lie bialgebra. Math. Res. Lett., 8 (1), pp 91-105,

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

Robin Endelman and Timothy J.  Hodges, Generalized Jordanian R-matrices of Cremmer-Gervais type. Lett. Math. Phys., 52 (3), pp 225-237, (2000)

Jintai Ding and Timothy J.  Hodges,  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

Timothy J. Hodges, Christophe Petit, and Jacob Schlather, First Fall Degree and Weil Descent, Finite Fields and their Applications, 30, 155-177, 2014

Timothy J. Hodges and Jacob Schlather, The degree of regularity of a quadratic polynomial, Journal of Pure and Applied Algebra, 217, 207-217, 2013

Timothy J. Hodges and Sergio Molina, Homological characterization of bounded F2-regularity, Journal of Algebra, 588, pp 148-165, 2021

Book

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

Review

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

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

Presentations

Invited Presentations

(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 Gr¨obner 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. . Level:International

(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,

Service

(A&S Faculty Senate ) Vice President Type:University/College Service Level:College

(Faculty Senate Budget and Priorities ) Committee Member Type:University/College Service Level:University

(Graduate Affairs Committee ) Committee Member Type:Departmental Service Level:Department

(Graduate Student Evaluation Committee ) Committee Chair Type:Departmental Service Level:Department

(A&S Graduate Council ) Committee Member Type:University/College Service Level:College

(A&S Graduate Council Executive Committee ) Committee Member Type:University/College Service Level:College

(A&S RPT Committee ) Committee Chair Type:University/College Service Level:College

Proceedings of the London Mathematical Society Reviewer Type:Editorial Service Level:International

(Graduate Program Director ) Type:Departmental Service Level:Department

(UC Hepcats ) Faculty Coordinator Type:Service to Student Groups Level:University

(Algebra Prelim ) Committee Member Type:Departmental Service Level:Department

Keywords

Algebra

Faculty Development Activities

06-13-2011 -06-17-2011 Instructional Technology Workshop University of Cincinnati Type:Workshop