Professor, Mathematics
School of Physical Sciences

Riemann surfaces and theta functions; moduli spaces and modular curve generalizations; Diophantine equation

math.uci.edu/~mfried/

1967-69: Institute for Advanced Study, Princeton, New Jersey.
Associate Professor, SUNY at Stony Brook, New York, 1969--1974.
Visiting Associate Professor, M.I.T., Boston, Massachusetts, September 1971--January 1972.
Visiting Associate Professor, University of Michigan, Ann Arbor, January 1972--June 1972.
Member, Institute for Advanced Study, Princeton, January 1973--June 1973.
Professor, UC Irvine, July 1974---present.
Visiting Full Professor, Tel-Aviv University, Spring 1976.
Fulbright Research Fellow, Helsinki University, September 1982---January 1983.
Director: UCI Summer Mathematics Institute for K-12 teachers, 1982--87.
Lady Davis Research Professorship, Hebrew University in Jerusalem, September 1984--January 1985.
Visiting Full Professor (VFP), Hebrew University, Fall 1988.
Full Professor, University of Florida, July 1986--Fall 1989.
Member, Institute for Advanced Studies, Hebrew University, Jerusalem, Israel, September 1991--June 1992.
VFP, Erlangen University, Fall 1994, under the Alexander Humboldt program.
VFP, Institut of Experimental Mathematics, Essen, Germany, Spring-Summer 1995, under the Alexander Humboldt program.
VFP Hebrew University, Spring 1999.
Visiting Research Prof. at MSRI, Fall 1999.
VFP, University of

1967-69: Institute for Advanced Study, Princeton, New
Jersey. Associate Professor, SUNY at Stony Brook, New York,
1969--1974. Visiting Associate Professor, M.I.T., Boston,
Massachusetts, September 1971--January 1972. Visiting
Associate Professor, University of Michigan, Ann Arbor,
January 1972--June 1972. Member, Institute for Advanced
Study, Princeton, January 1973--June 1973. Professor, UC
Irvine, July 1974---present. Visiting Full Professor,
Tel-Aviv University, Spring 1976. Fulbright Research
Fellow, Helsinki University, September 1982---January
1983. Director: UCI Summer Mathematics Institute for K-12
teachers, 1982--87. Lady Davis Research Professorship,
Hebrew University in Jerusalem, September 1984--January
1985. Visiting Full Professor (VFP), Hebrew University,
Fall 1988. Full Professor, University of Florida, July
1986--Fall 1989. Member, Institute for Advanced Studies,
Hebrew University, Jerusalem, Israel, September 1991--June
1992. VFP, Erlangen University, Fall 1994, under the
Alexander Humboldt program. VFP, Institut of Experimental
Mathematics, Essen, Germany, Spring-Summer 1995, under the
Alexander Humboldt program. VFP Hebrew University, Spring
1999. Visiting Research Prof. at MSRI, Fall 1999. VFP,
University of

Famous difficulties arise in nonlinear equations. Though solutions may exist, we often find they have no presentation in known functions. Rather, however, than solutions, we want special properties of solutions. The monodromy method combines finite groups and algebraic geometry. We use it to translate the regular inverse Galois problem into studying rational points on moduli spaces. These are modular curve generalizations, and like modular curves their properties give solutions to classical problems. Avoiding direct solutions of nonlinear equations did the job. Here is a list of solved problems that motivate Fried's present research.


Schur's conjecture; Davenport's problem; The Galois stratification procedure for the theory of finite fields; Carlitz's conjecture and a near complete description of exceptional polynomials; An enhancement of Shafarevich's conjecture to present the absolute Galois group of the rationals as an extension of known groups; Diophantine reduction of the Inverse Galois problem.

Solutions of these classical problems feed into several general programs since the middle 90's. Pub. #4 is an example contribution to the Modular Tower approach for expanding the method's applications. Many related topics, and related writers, appear in Fried's synopsis of the 18 papers in Arithmetic fundamental groups and noncommutative algebra, Proceedings of Symposia in Pure Mathematics, 70 (2002) editors M. Fried and Y. Ihara, 1999 von Neumann Conference on Arithmetic Fundamental Groups and Noncommutative Algebra, August 16-27, 1999 MSRI. This prelude to the volume is at http://math.uci.edu/~mfried/psfiles/msrivol01.html.


Part I: GQ action on moduli spaces of covers
Part II: Curve covers in positive characteristic
Part III: Special groups for covers of the punctured sphere
Part IV: Fundamental

The Main Conjecture of Modular Towers and
its higher rank generalization}, in Groupes de Galois arithmetiques et differentiels
(Luminy 2004; eds. D.~Bertrand and P.~D\`ebes), Sem.~et Congres, Vol.~{\bff 13}, 2006.

The Main Conjecture of Modular Towers and
its higher rank generalization}, in {\sl Groupes de Galois arithmetiques et differentiels\/}
(Luminy 2004; eds. D.~Bertrand and P.~D\`ebes), Sem.~et Congres, Vol.~{\bff 13}, 2006.

The Main Conjecture of Modular Towers and its higher rank generalization, in Groupes de Galois arithmetiques et differentiels
(Luminy 2004; eds. D.~Bertrand and P.D\`ebes), Sem. et Congres, Vol. 13, 2006.

Solving diophantine problems over all residue class fields of a number field …, Annals Math. 104 (1976), 203--233.

Arithmetic of 3 and 4 branch point covers}: a bridge provided by noncongruence subgroups SL_2(Z), Progress in Math. Birkhauser 81 (1990), 77--11

with P.~Debes, Arithmetic variation of fibers in families: Hurwitz monodromy criteria for rational points on all members of the family, Crelles J. 409 (1990), 106--137

with H.Voelklein, The inverse Galois problem and rational points on moduli spaces, Math.~Annalen 290 (1991), 771--80

with H. Voelklein, The embedding problem over an Hilbertian-PAC field, Annals of Math 135 (1992), 469--481

with R. Guralnick and J. Saxl, Schur Covers and Carlitz's Conjecture, Israel J.~Thompson Volume 82 (1993), 157--225

with Y.~Kopeliovic, Applying Modular Towers to the inverse Galois problem, Geometric Galois Actions II Dessins d'Enfants, Mapping Class Groups and Moduli 243, London Mathematical Society Lecture Note series, (1997) 172--1

Variables Separated Polynomials and Moduli Spaces, No. Theory in Progress, eds. K.Gyory, H.Iwaniec, J.Urbanowicz, proceedings of the Schinzel Festschrift, Summer 1997 Zakopane, Walter de Gruyter, Berlin-New York (Feb. 1999), 169--

with P. Bailey, Hurwitz monodromy, spin separation and higher levels of a Modular Tower, in Proceedings of Symposia in Pure Mathematics 70 (2002) editors M.~Fried and Y.~Ihara, 1999 von Neumann Symposium on Arithmetic Fundamental Groups and Noncommutative Algebra, August 16-27, 1999 MSRI, 79--2

with Arianne Mezard, Configuration spaces for wildly ramified covers, in Proceedings of Symposia in Pure Mathematics 70 (2002) editors M. Fried and Y. Ihara, 1999 von Neumann Symposium on Arithmetic Fundamental Groups and Noncommutative Algebra, August 16-27, 1999 MSRI, 353--

NSF Panel chair for Mathematics Education awards at NSF: 1986--1989. Associate Editor for Research Announcements, Bull. Amer. Math. Society (1991-1994). CRDF (Civilian Research and Development Foundation panel) Mathematics panel chair, NSF affiliated granting agency in Washington for fSU joint projects with US researchers, Oct. 5, 2001. Editor for "Finite Fields and their Applications" Academic Press Journal (1994-2004). Member of Organizing Committee for MSRI semester, "The Inverse Galois Problem and Arithmetic Fundamental Groups" (spring 1999). Editor with Y. Ihara of the resulting von Neumann symposium volume (to appear in 2002). Chair, AMS Summer Institute in Finite Fields, Seattle conference (July 1997), sole editor of resulting volume (1999). Chair, Committee on Summer Institutes and Special Symposium, AMS (1996-1999). Editor of the 1999 von Neumann Symposium on Arithmetic Fundamental Groups and Noncommutative Algebra, Proceedings of Symposia in Pure Mathe

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Last updated
04/16/2006