Algebra Research - UC Berkeley Department of Mathematics

We have a large active group of researchers in number theory and algebraic geometry, as well as many individuals who work in other areas of algebra: groups, noncommutative rings, Lie algebras and super-Lie algebras, representation theory, combinatorics, game theory, and coding. A number of members of the algebra group belong to the Research Training Group in Representation Theory, Geometry and Combinatorics, which runs activities and supports grad students and postdocs in its areas of interest.

Faculty

Tenured and tenure-track:

George M. Bergman, (Emeritus), Associative rings, universal algebra and category theory, counterexamples.
Richard E. Borcherds, Lie algebras, vertex algebras, automorphic forms.
Robert F. Coleman, Arithmetic geometry.
David Eisenbud, Algebraic geometry, commutative algebra, computation.
Edward Frenkel, Representation theory, integrable systems, mathematical physics.
Mark D. Haiman, Algebra, combinatorics, and algebraic geometry.
Robin C. Hartshorne (Emeritus), Algebraic geometry.
Gerhard P. Hochschild (Emeritus), Lie groups, Algebraic groups, Homological algebra.
Tsit-Yuen Lam, (Emeritus), Groups, rings, fields, classical algebraic K-theory, quadratic forms.
Hendrik W. Lenstra, Jr. (Emeritus, not in residence), Algebraic number theory, algorithms.
Andrew P. Ogg (Emeritus), Number theory, Elliptic curves, Modular forms.
Arthur E. Ogus, Algebraic geometry.
Martin Olsson, Algebraic and arithmetic geometry.
Bjorn Poonen, Number theory and algebraic geometry.
John L. Rhodes (Emeritus, not in residence), Algebra, semigroups, automata.
Kenneth A. Ribet, Algebraic number theory, algebraic geometry.
Vera Serganova, Super-representation theory.
Bernd Sturmfels, Combinatorics and algebraic geometry, with emphasis on applied and computational aspects.
Constantin Teleman,Lie groups, algebraic geometry, topology, quantum field theory.
Paul A. Vojta, Diophantine approximation, Nevanlinna theory - especially as related to diophantine approximation, Arakelov theory.
Lauren Williams, Combinatorics, and connections to representation theory, geometry, and statistical physics.
Mariusz Wodzicki, Non-commutative and algebraic geometry, analysis, K-theory.

Current Visitors and Postdoctoral Fellows:

Marco Aldi, Algebraic and geometric applications of string theory.
David Hill, Representation theory and Lie theory, using techniques also from combinatorics and the theory of symmetric functions.
Reimundo Heluani, Geometric representation, Mathematical physics.
Valerie Hower, Topological data analysis, computational biology, geometric combinatorics.
Scott Morrison, Topological quantum field theories, Khovanov homology, blob homology; fusion categories; diagrammatic representation theory.
Chung Pang Mok, Special values of L-functions, and the applications of theta-lifting in arithmetic geometry.
Alexander George Marshall Paulin, Number theory, p-adic aspects of the Langlands program.
Michael Rose, Algebraic geometry, number theory.
David Rydh, Algebraic geometry, algebraic stacks.
Raman Sanyal, Discrete geometry, topological combinatorics, and combinatorial commutative algebra
Noam Shomron, Representation theory of Lie super-algebras.
Zvezdelina Stankova, Algebraic geometry.
Joshua Sussan, Representation theory, low dimensional topology, and categorification.
Luis Tabera, Tropical geometry.
Mauricio Velasco, Commutative algebra, algebraic geometry, combinatorics, and algebraic geometry.

Faculty listed under other sections who are also interested in algebra:

Elwyn R. Berlekamp (Emeritus) (Applied Mathematics), Combinatorial game theory, algebraic coding theory, electrical engineering, computer science.
Ralph McKenzie (Emeritus, not in residence) (Foundations), General algebra, logic.
Calvin C. Moore (Emeritus) (Modern Analysis), Representations and actions of topological groups, operator algebras.
Lior Pachter (Applied Mathematics), Mathematical and computational biology, combinatorics.
Marc A. Rieffel (Modern Analysis), Non-commutative harmonic analysis, operator algebras, quantum geometry.
Thomas Scanlon (Foundations), Model theory and applications to number theory.

Courses

Undergraduate upper divison courses

Math 110 (and honors version, Math H110). Linear algebra.
Math 113 (and honors version, Math H113). Introduction to abstract algebra.
Math 114. Second course in abstract algebra. (Instructor's choice; usually Galois Theory)
Math 115. Introduction to number theory.
Math 116. Cryptography.
Math 143. Elementary algebraic geometry.
Math 172. Combinatorics.

The Mathematics Department also offers, at the undergraduate level, courses which may include algebraic topics along with others: Problem Solving (H90), Experimental Courses (191), a Special Topics course (195), and several courses of directed and independent individual and group work (196-199).

Graduate courses

Math 245A. General theory of algebraic structures.
Math 249. Algebraic combinatorics.
Math 250A. Groups, rings and fields.
Math 250B. Multilinear algebra and further topics.
Math 251. Ring theory.
Math 252. Representation theory.
Math 253. Homological algebra.
Math 254A-254B. Number theory.
Math 255. Algebraic curves.
Math 256A-256B. Algebraic geometry.
Math 257. Group theory.

Topics courses and Seminars

2008-2009

Math 290. Seminar - Commutative algebra and algebraic geometry, David Eisenbud
Math 290. Seminar - Number theory, Kenneth Ribet

Spring 2009

Math 274. Topics in Algebra - Tropical geometry, Bernd Sturmfels
Math 274. Topics in Algebra - Infinitesimal geometry, Mariusz Wodzicki

Fall 2008

Math 290. Seminar - Algebraic geometry, David Eisenbud and Daniel Erman
Math 290. Seminar - Discrete mathematics, Bernd Sturmfels
Math 290. Seminar - Representation theory, geometry and combinatorics, Mark Haiman and Nicolai Reshetikhin
Math 290. Seminar - Student arithmetic geometry seminar, Martin Olsson
Student Seminar. Student algebraic and arithmetic geometry seminar, David Brown, Daniel Erman and Anthony Varilly

2007-2008

Math 270. Hot Topics - Derived algebraic geometry and topology, Peter Teichner
Math 290. Seminar - Commutative algebra and algebraic geometry, David Eisenbud
Math 290. Seminar - Representation theory, geometry and combinatorics, Mark Haiman and Nicolai Reshetikhin

Spring 2008

Math 274. Topics in Algebra - Real p-adic analysis, Robert Coleman

Fall 2007

Math 274. Topics in Algebra - Locally finite lie algebras and their representations with a view toward open problems, Ivan Penkov
Math 290. Seminar - Student representation theory, geometry and combinatorics, Vera Serganova and Peter Tingley
Math 290. Seminar - Number theory, Kenneth Ribet
Math 290. Seminar - Perverse sheaves, Joel Kamnitzer and Xinwen Zhu

Earlier years, from 2006-2007

Current Thesis Students: Name (Adviser)

Eric Antokoletz (Wodzicki)
David Berlekamp (Eisenbud)
David Brown (Poonen)
Morgan Brown (Eisenbud)
Edward Carter (Eisenbud)
Dustin Cartwright (Sturmfels)
Li-Chung Chen (Haiman)
Charles Crissman (Eisenbud)
Maria Cueto (Sturmfels)
Ishai Dan-Cohen (Ogus + Olsson)
Jeffrey Doker (Pachter + M.Beck at SFSU)
Daniel Erman (Eisenbud)
Alexander Fink (Sturmfels)
Seth Foster (Bergman)
Anton Geraschenko (Serganova)
Jaime Haletky (Poonen)
An Huang (Borcherds)
Yuhao Huang (Olsson)
Sarah Iveson (Haiman)
Chul Lee (Borcherds)
Shaowei Lin (Sturmfels)
Jae-Young Park (Olsson)
Claudiu Raicu (Eisenbud)
Manuel Reyes (Lam)
Matthew Satriano (Olsson)
Arun Sharma (Haiman + R.Graham at UCSD)
Shenghao Sun (Olsson)
Anne Shiu (Sturmfels + Pachter)
David Spivak (Graber)
Bianca Viray (Poonen)

Recent Ph.D.s: Name (Adviser) Thesis title

2009

Jonah Blasiak (Haiman) Cyclage, catabolism, and the Affine Hecke Algebra
Seth Clayton Dutter (Vojta) Logarithmic Jet Spaces and Intersection Multiplicities
Sevak Mkrtchyan (Reshetikhin) Scaling Limits of Random Skew Plane Partitions
Brian David Rothbach (Haiman) Borel Orbits of x² = 0 in gl_n
Andrew James Tolland (Frenkel) Gromov-Witten Gauge Theory
Anthony Varilly (Poonen) Arithmetic of del Pezzo Surfaces of Degree 1
Xinwen Zhu (Frenkel) Gerbal Representations of Double Loop Groups

2008

Jameel Al-Aidroos (Eisenbud + Graber at Cal Tech) Perfect Pairings in the Tautological Rings of the Moduli Spaces of Stable Curves
Jennifer Berg (Serganova) On the Center of the Lie Superalgebra q(n)⁽²⁾
Elizabeth Dan-Cohen (Serganova + Wolf) Structure of Root-Reductive Lie Algebras
Jesse Fern (Serganova + Whaley) Calculations of Quantum Error Correction and Fault Tolerance Thresholds
David Freeman (Ribet + E.Schaefer at Santa Clara U.) Constructing Abelian Varieties for Pairing-Based Cryptography
Peter Huggins (Sturmfels + Pachter) Polytopes in Computational Biology
Dimitar Jetchev (Ribet) CM Points, Selmer Groups, Component Groups, and Euler Systems
Jason Morton (Sturmfels) Geometry of Conditional Independence
David Zywina (Poonen) The Large Sieve and Galois Representations

2007

Sami Assaf (Haiman) Dual Equivalence Graphs, Ribbon Tableaux and MacDonald Polynomials
Tathagata Basak (Borcherds) Complex Reflection Groups and Dynkin Diagrams
Ioan Berbec (Coleman) Group Schemes over Artinian Rings and Applications
Scott Carnahan (Borcherds) Monstrous Lie Algebras and Generalized Moonshine
Aaron Gray (Ogus) Functoriality of the Logarithmic Riemann-Hilbert Correspondence
Aaron Greicius (Poonen) Elliptic Curves with Surjective Global Galois Representation
Crystal Hoyt (Serganova) Kac-Moody Superalgebras of Finite Growth
James Kelley (Wodzicki) Homotopical Syzygies in K-theory
Grace Lyo (Carlsson & Ribet) Semilinear Actions of Galois Groups and the Algebraic K-theory of Fields
Carl Miller (Ogus) Cohomology of p-Torsion Sheaves on Characteristic p-Curves
Jared Weinstein (Ribet) Automorphic Representations with Local Constraints
Soroosh Yazdani (Ribet) Modular Forms with Odd Congruence Numbers
Josephine Yu (Sturmfels) Combinatorial Aspects of Tropical Geometry