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Caltech
Division of Physics, Mathematics and Astronomy

Matthias Flach

Matthias Flach

Professor of Mathematics
Dipl., Johann-Wolfgang von Goethe University (Frankfurt), 1986; Ph.D., St. John's College, 1991. Associate Professor, Caltech, 1995-99; Professor, 1999-. Executive Officer for Mathematics, 2006-09.

Research Interests: Arithmetic algebraic geometry, special values of L-functions (conjectures of Bloch, Beilinson, Deligne, Bloch-Kato), Galois module theory, motivic cohomology

Overview

Matthias' work is in Arithmetic Algebraic Geometry. In particular his research focuses on special values of L-functions and the conjectures of Beilinson, Deligne, Bloch/Kato and Fontaine/Perrin-Riou. His recent results include a complete proof of the equivariant Tamagawa number conjecture for Dirichlet L-functions and new cases of the local Tamagawa number conjecture over tamely rami ed elds. In joint work with B.Morin he has been developing an idea originally due to Lichtenbaum (Weil-etale cohomology) which focuses on Zeta functions of arithmetic schemes rather than motivic L-functions and draws clear connections to Arakelov theory and Deningers ideas on motivic L-functions. The near term goal in this area is to nd precise special value conjectures independent of p-adic Hodge theory.

Matthias has also published about cohomology of topological groups. His mathematical interests include Galois module theory, algebraic K-theory and motivic cohomology, categorical algebra, topos theory and algebraic topology.

Selected Awards

  • Masterclass "Topological Hochschild homology and Zeta values", University of Copenhagen, Denmark, Jan 30-Feb 3, 2023
  • Alfred P. Sloan Research Fellow, 1996-2000
  • Heinz Maier Leibnitz Award, German Ministry of Education, Science and Technology, 1995

Selected Publications

For a complete publications list, see feeds.library.caltech.edu

Related Courses

Ma 1 abc. Calculus of One and Several Variables and Linear Algebra. 9 units (4-0-5); first, second, third terms, 2025-26. Prerequisites: high-school algebra, trigonometry, and calculus. Special section of Ma 1 a, 12 units (5-0-7). Review of calculus. Complex numbers, Taylor polynomials, infinite series. Comprehensive presentation of linear algebra. Derivatives of vector functions, multiple integrals, line and path integrals, theorems of Green and Stokes. Ma 1 b, c is divided into two tracks: analytic and practical. Students will be given information helping them to choose a track at the end of the fall term. There will be a special section or sections of Ma 1 a for those students who, because of their background, require more calculus than is provided in the regular Ma 1 a sequence. These students will not learn series in Ma 1 a and will be required to take Ma 1 d.
Instructors: Conlon, Gherman, Flach, T. Yu
Ma 160 abc. Number Theory. 9 units (3-0-6); first, second, third terms, 2025-26. Prerequisites: Ma 5. In this course, the basic structures and results of algebraic number theory will be systematically introduced. Topics covered will include the theory of ideals/divisors in Dedekind domains, Dirichlet unit theorem and the class group, p-adic fields, ramification, Abelian extensions of local and global fields.
Instructors: Fu, Yang, Flach
Ma 191 abc. Selected Topics in Mathematics. 9 units (3-0-6); first, second, third terms, 2025-26. Each term we expect to give between 0 and 6 (most often 2-3) topics courses in advanced mathematics covering an area of current research interest. These courses will be given as sections of 191. Students may register for this course multiple times even for multiple sections in a single term. The topics and instructors for each term and course descriptions will be listed on the math option website each term prior to the start of registration for that term.
Instructors: T. Yu, Pham, Zhang, Yang, Heinaevaara, Looi, Svoboda, Marcolli, El-Maazouz, Flach, Dong