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, 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, 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.
John R. Stallings (Emeritus), Topology, group 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.
Mariusz Wodzicki, Non-commutative and algebraic geometry, analysis, K-theory.

Visitors and postdoctoral fellows (2007-2008):

Marco Aldi, Algebraic and geometric applications of string theory.
Michel Broue, General conjectures on representation theory of finite groups, complex reflection groups and related topics, Hecke algebras and Braid groups Spetses.
Trevor Bruen, Discrete mathematics, mathematical and computational biology, biostatistics, applied algebraic geometry and probability.
Shamgar Gurevitch, Applications of representation theory, algebraic geometry to quantum mechanics, harmonic analysis.
David Hill, Representation theory and Lie theory, using techniques also from combinatorics and the theory of symmetric functions.
Joel Kamnitzer, Geometric and combinatorial representation theory.
Dagan Karp, Algebraic geometry, Gromov-Witten theory.
Chandrashekhar Khare (visiting from U.Utah), Galois representations and modular forms.
Sarah Mason, Algebraic and enumerative combinatorics.
Noam Shomron, Representation theory of Lie super-algebras.
Chung Pang Mok, Special values of L-functions, and the applications of theta-lifting in arithmetic geometry.
Ivan Penkov, Algebra, Lie theory, geometry.
Zvezdelina Stankova, Algebraic geometry.
Joshua Sussan, Representation theory, low dimensional topology, and categorification.
Mauricio Velasco, Commutative algebra, combinatorics, and algebraic geometry.
Julius Zelmanowitz (Emeritus, UCSB; Deputy Director, MSRI), Rings and representation theory.

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 172. Combinatorics.

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

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

2006-2007

Math 290. Seminar - Commutative algebra and algebraic geometry, David Eisenbud and Brian Osserman
Math 290. Seminar - Number theory, Bjorn Poonen

Spring 2007

Math 274. Topics in algebra - Stacks, Martin Olsson
Math 290. Seminar - Elliptic curves, Bjorn Poonen
Math 290. Seminar - Noncommutative geometry, Wodzicki
Math 290. Student seminar in algebraic statistics, Sturmfels
Math 290. Seminar - Topics in the geometric Langlands Program, Frenkel
Student seminar. Number theory, Varilly

Fall 2006

Math 274. Topics in algebra - Commutative rings, Tsit-Yuen Lam
Math 290. Arithmetic geometry, Bjorn Poonen

Current Thesis Students: Name (Adviser)

Jameel Al-Aidroos (Eisenbud + Graber at Cal Tech)
Eric Antokoletz (Wodzicki)
Jennifer Berg (Serganova)
Jonah Blasiak (Haiman)
David Brown (Poonen)
Edward Carter (Eisenbud)
Dustin Cartwright (Sturmfels)
Li-Chung Chen (Haiman)
Ishai Dan-Cohen (Ogus)
Elizabeth Dan-Cohen (Serganova + Wolf)
Seth Dutter (Vojta)
Daniel Erman (Eisenbud)
Jesse Fern (Serganova + Whaley)
Seth Foster (Bergman)
David Freeman (Ribet + E.Schaefer at Santa Clara U.)
Anton Geraschenko (Serganova)
Jaime Haletky (Poonen)
Peter Huggins (Sturmfels + Pachter)
Sarah Iveson (Haiman)
Dimitar Jetchev (Ribet)
Jason Morton (Sturmfels)
Brian Rothbach (Haiman)
Matthew Satriano (Olsson)
Arun Sharma (Haiman + R.Graham at UCSD)
Shenghao Sun (Olsson)
Anne Shiu (Sturmfels + Pachter)
David Spivak (Graber)
Andrew Tolland (Frenkel)
Anthony Varilly (Poonen)
Bianca Viray (Poonen)
Xinwen Zhu (Frenkel)
David Zywina (Poonen)

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

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

2006

Alexander James Diesl (Lam) Classes of Strongly Clean Rings
Thomas J. Dorsey (Bergman) Cleanness and Strong Cleanness of Rings of Matrices
Alexander S. Dugas (Lam) Stable Equivalences of Finite Dimensional Algebras: Relative Homology and Lifting Problems
Nicholas Eriksson (Sturmfels) Algebraic Combinatorics for Computational Biology
Ron A. Fertig (Ribet) Almost Rational Torsion Points on Abelian Varieties
Walter Kim (Coleman) Ramification Points on the Eigencurve and the Two Variable Symmetric Square p-adic L-function
Zachary Mesyan (Bergman) On Endomorphism Rings of Infinite Direct Sums of Modules
Pace Peterson Nielsen (Lam) The Exchange Property for Modules and Rings
Jiawang Nie (Demmel and Sturmfels) Global Optimization of Polynomial Functions and Applications
Shahed Sharif (Poonen) Construction of Curves with Prescribed Period and Index

2005

Gautam Borooah (Lam) Clean Conditions and Incidence Rings
Patrick Corn (Poonen) Del Pezzo Surfaces and the Brauer-Manin Obstruction
Dragos Ghioca (Poonen and Scanlon) The Arithmetic of Drinfeld Modules
Chris Hillar (Sturmfels) Solving Polynomial Systems with Special Structure
Bonnie Huggins (Poonen) Fields of Moduli and Fields of Definition of Curves
Aaron Levin (Vojta) Generalizations of Siegel's and Picard's theorems
Chu-Wee Lim (Ribet) Decomposition of Spaces of Cusp Forms over Q, and Variants of Partial Nim
Nghi Nguyen (Poonen) Whitney Theorems and Lefschetz Pencils over Finite Fields
Mark Rinker (Stallings) Height Functions, Maps to the Integers, and the Whitehead Graph of a Finite Presentation
Daniel Schepler (Ogus) Logarithmic Nonabelian Hodge Theory in Characteristic p
David Speyer (Sturmfels) Tropical Geometry
Seth Sullivant (Sturmfels) Toric Ideals in Algebraic Statistics
Ronald van Luijk (Lenstra) Rational points on K3 surfaces
Oscar Villareal (Poonen) Countable Unions of Subvarieties of Semiabelian Varieties
John Voight (Lenstra) Quadratic forms and quaternion algebras: algorithms and arithmetic
Alexander Woo (Haiman) Ideals of the Polynomial Ring Generated by Irreducible Symmetric Group Representations and Ellingsrud-Stromme Cells on the Hilbert Scheme

2004

Samit Dasgupta (Ribet) Gross-Stark Units, Stark-Heegner Points, and Class Fields of Real Quadratic Fields
Daniel Giaimo (Eisenbud) On the Castelnuovo-Mumford Regularity of Curves and Reduced Schemes
George Kirkup (Eisenbud) Random Variables with Completely Independent Subcollections
Peter Pribik (Frenkel) Integrable Soliton Hierarchies for so(2n) via Intertwining Operators
Nicholas Proudfoot (Knutson, now at UCSD) Hyperkähler Analogues of Kähler Quotients
Kevin Purbhoo (Knutson, now at UCSD) Vanishing and Non-Vanishing Criteria for Branching Schubert Calculus
Nirit Sandman (Vojta) Tamari Lattices and a Geometric Problem in Antitrust
Sankaran Viswanath (Borcherds) Stabilization of Tensor Products in Kac-Moody Algebras

2003

Donald Barkauskas (Stallings) Centralizers in Fundamental Groups of Graphs of Groups
Alexander Barnard (Borcherds) The Singular Theta Correspondence, Lorentzian Lattices and Borcherds-Kac-Moody Algebras
Ruchira Datta (Sturmfels) Algebraic Methods in Game Theory
Mark Davis (Lam) Split Extensions and Generalizations of Injectivity
Mike Develin (Sturmfels) Topics in Discrete Geometry
Elena Drozd (Hartshorne) Curves on a Nonsingular Quadric Hypersurface in P4: Existence and Liaison Theory
Kirsten Eisenträger (Poonen) Hilbert's Tenth Problem and Arithmetic Geometry
Jesse Elliott (Lenstra) Witt-Burnside Rings
Megumi Harada (Knutson, now at UCSD) The Sympletic Geometry of the Gel'fand-Cetlin-Molev Basis for Representation of Sp(2n,C)
David Helm (Ribet) Jacobians of Shimura Curves and Jacquet-Langlands Correspondences
Amit Khetan (Sturmfels) Formulas for Resultants
Nina Nemyrovska (Serganova) On Atypical Representations of Classical Lie Superalgebras of Defect One
Sinan Unver (Vojta) Ramification on Arithmetic Surfaces and p-adic Multi-zeta Values

2002

Scott Annin (Lam) Associated and Attached Primes over Noncommutative Rings
Francesco Calegari (Ribet) Ramification and Semistable Abelian Varieties.
Joseph Israel (Bergman) Amalgamation and Unimodality.
Laura Matusevich (Sturmfels) Combinatorial aspects of Hypergeometric Functions.
Matthew Szczesny (Frenkel) Twisted Vertex Operators, Algebraic Curves, and Prym Varieties.
Howard Thompson (Ogus) On Toric Log Schemes
Wayne Whitney (Lenstra) Functorial Cohen Rings

2001

John Flynn (Lenstra) Near-Exceptionality Over Finite Fields
Oliver King (Borcherds) A Mass Formula for Unimodular Lattices with No Roots
Kenneth McMurdy (Coleman) A Splitting Criterion for Galois Representations Associated to Exceptional Modular Forms
William Murray (Lam) Frobenius Algebras, Independence of Field, and Quadratic Forms
Mircea Mustata (Eisenbud) Singularities and Jet Schemes
Martin Olsson (Ogus) Log Algebraic Stacks and Moduli of Log Schemes
Noam Shomron (Serganova) Representations of Cartan-Type Lie Superalgebras
Gregory Smith (Eisenbud) Computational Methods for Studying Sheaves
David Wasserman (Bergman) Epimorphisms and Dominions in Varieties of Lattices

2000

Amod Agashe (Ribet) The Birch and Swinnerton-Dyer Formula for Modular Abelian Varieties of Analytic Rank Zero
James Borger (Lenstra) On Conductors over Discrete Valuation Rings with General Residue Fields
Emiliano Gomez (Coleman) Deformations of Pseudorepresentations
Christine Heitsch (Rhodes) Computational Complexity of Generalized Pattern Matching
Diane Maclagan (Sturmfels) Structures on Sets of Monomial Ideals
Gregory Marks (Lam) 2-Primal Rings
Ezra Miller (Sturmfels) Resolutions and Duality for Monomial Ideals
David Romano (Lenstra) Galois Groups of Strongly Eisenstein Polynomials
Lawren Smithline (Coleman) Slopes of p-adic Modular Forms
William Stein (Lenstra) Explicit Approaches to Modular Abelian Varieties
Harrison Tsai (Sturmfels) Algorithms for Algebraic Analysis