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Timothy J Hodges

Title: Professor
Office: 4428D French Hall
Tel: 513-556-4040
Email: timothy.hodges@uc.edu
Web: https://math.uc.edu/~hodgestj

Education

  • PhD, University of Leeds, Leeds, England, 1981 (Mathematics).
  • BSc, University of Leeds, Leeds, England, 1977 (Mathematics).

Research Information

Research Interests

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

Research Support

  • (PI), NSF IPA - Algebra Program Director, National Science Foundation. (NSF IPA 2015-2016), $201,919.00. 08/24/2015 to 08/23/2016.
  • (PI), Summer Research Fellowship, C. P. Taft Research Center. 2011.
  • (PI), Degeneration of R-Matrices and Structure of Nonstandard Quantum Groups, National Security Agency. (MDA904-01-1-0084), $36,565.00. 07/13/2001 to 07/12/2003.
  • (PI), Degeneration of R-Matrices and Structure of Non-Standard Quantum Groups, National Security Agency. (MDA 904-01-1-0084), $36,565. 2001 to 2003.
  • (PI), Algebraic Structure of Nonstandard Quantum Groups, National Security Agency. (MDA 904-99-1-0026), $31,481. 1999 to 2001.
  • (PI), Algebraic Structure of Non-Standard Quantum Groups, National Security Agency. (MDA9049910026/P00002), $31,481.00. 11/25/1998 to 02/24/2001.
  • (PI), Algebraic Structure of Quantum Groups, National Science Foundation. (DMS 9501484), $50,000. 1995 to 1998.
  • (PI), Algebraic Structure of Quantum Groups. (MDA 904-93-3042), $24,000. 1993 to 1995.
  • (PI), Summer Faculty Research Grant, Taft Foundation, University of Cincinnati. 1993.
  • (PI), Quantum and Classical Enveloping Algebras of Semisimple Lie Algebras, National Security Agency. (MDA 904-91-H0045), $58,459. 1991 to 1993.
  • (PI), Noncommutative Algebras, National Security Agency and National Science Foundation. (MDA 904-89-H2046), $41,811. 1989 to 1991.
  • (PI), Summer Faculty Research Grant, Taft Foundation, University of Cincinnati. 1988.
  • (PI), Structure and K-Theory of Group Rings and Rings of Differential Operators, National Science Foundation. (DMS 8602291), $40,329. 1986 to 1988.
  • (PI), Summer Faculty Research Scholarship, University Research Council, University of Cincinnati. 1985.

Publications

Peer Reviewed Publications

  • Timothy J Hodges, Sergio Molina and Jacob Schlather, On the existence of homogeneous semi-regular sequences, Journal of Algebra, 476 (2017) 519-547
  • Hodges, Timothy J., Petit, Christophe and Schlather, Jacob, First Fall Degree and Weil Descent, Finite Fields and their Applications, 30 (2014), 155-177
  • 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)[Link]
  • Hodges Timothy J. and Schlather, Jacob, The degree of regularity of a quadratic polynomial, Journal of Pure and Applied Algebra, 217 (2013) 207-217[Link]
  • 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, Jintai & Hodges, Timothy J., (2011)  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
  • 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., & Yakimov, Milen (2005). Triangular Poisson structures on Lie groups and symplectic reduction. Contemporary Mathematics, 391, 123.
  • Endelman, Robin, & Hodges, Timothy J. (2004). Degenerations and representations of twisted Shibukawa-Ueno R-operators. Lett. Math. Phys., 68(3), 151.
  • Ding, Jintai, &  Hodges, Timothy J. (2004). Cryptanalysis of an implementation scheme of the tamed transformation method cryptosystem. J. Algebra & Appl., 3(3), 273.
  • Ding, Jintai, & Hodges, Timothy J. (2002). The Yang-Baxter equation for operators on function fields. In A. Pressley (Eds.), Quantum Groups and Lie theory (pp. 168, 174). Cambridge University Press.
  • Hodges, Timothy J., & Yakimov, Milen (2001). The double and dual of a quasitriangular Lie bialgebra. Math. Res. Lett., 8(1), 91, 105.
  • Hodges, Timothy J. (2001). Generating functions for Cremmer-Gervais quantum groups. J. Pure Appl. Algebra, 157(1), 69.
  • Endelman, R., & Hodges, Timothy J. (2000). Generalized Jordanian R-matrices of Cremmer-Gervais type. Lett. Math. Phys., 52(3), 225-237.
  • 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. (1995). On the Cremmer-Gervais quantizations of SL(n). Internat. Math. Res. Notices, 10, 465 - 481.
  • Hodges, Timothy J., & Levasseur, Thierry (1994). Primitive ideals of C_q[SL(n)]. J. Algebra, 168(2), 455 - 468.
  • 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., & Kim, Kyunghee (1991). Behaviour of Krull dimension under extension of the base field. Comm. Algebra, 19(1), 143 - 156.
  • Alev, Jacques, Hodges, Timothy J., & Velez, Juan-Diego (1990). Fixed rings of the Weyl algebra A_1(C). J. Algebra, 130(1), 83 - 96.
  • 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.
  • 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. (1989). Equivariant K-theory for noetherian rings. J. London Math. Soc. (2), 39(3), 414 - 426.
  • Hodges, Timothy J., & Tesser, Steven B. (1989). Representing Clifford algebras as crossed products. J. Algebra, 123(2), 500 - 505.
  • 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. (1987). 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. (1984). An example of a primitive polynomial ring. J. Algebra, 90(1), 217-219.
  • Hodges, Timothy J. (1984). 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.

Books

  • 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

  • (1997). A guide to quantum groups. Bull. London Math. Soc., 506-508.
  • (1989). Torsion Theories. Bull. Amer. Math. Soc., 148-151.

Work in Progress

  • Hodges, Timothy J and Schlather Jacob, Periodic sums of generalized binomial coefficients, in preparation

Presentations & Lectures

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. [Link]
  • (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.

Experience & Service

Work Experience

  • 2001 to 2010, Head, University of Cincinnati, Cincinnati, Ohio.
  • 03-1995 to 04-1995, Visiting Research Professor, University of Poitiers, Poitiers, France.
  • 1993 to Present, Professor, University of Cincnnati, Cincinnati, Ohio.
  • 1990 to 1991, Visiting Associate Professor, Universite de Bretagne Occidentale, Brest, France.
  • 1987 to 1993, Associate Professor, University of Cincinnati, Cincinnati, Ohio.
  • 1984 to 1987, Assistant Professor, University of Cincinnati, Cincinnati, Ohio.
  • 1981 to 1984, Instructor, University of Utah, Salt Lake City, Utah.
  • 1980 to 1981, Teaching Associate, University of California at Santa Barbara, Santa Barbara, California.

Service

  • Vice President, A&S Faculty Senate,
  • Committee Member, Faculty Senate Budget and Priorities,
  • Committee Member, Graduate Affairs Committee,
  • Committee Chair, Graduate Student Evaluation Committee,
  • Committee Member, A&S Graduate Council,
  • Committee Member, A&S Graduate Council Executive Committee,
  • Committee Chair, A&S RPT Committee,
  • Reviewer, Proceedings of the London Mathematical Society
  • Graduate Program Director,
  • Faculty Coordinator, UC Hepcats,
  • Committee Member, Algebra Prelim,
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