# Research in Algebra

### Including number theory, algebraic geometry, and combinatorics

We have large groups of researchers active in number theory and algebraic geometry, as well as many individuals who work in other areas of algebra: groups, noncommutative rings, Lie algebras and Lie super-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.

## Courses

### Undergraduate upper division 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.

## 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*

## Senate Faculty

Name | Title | Research Interests |
---|---|---|

George M. Bergman | Professor Emeritus, Professor of the Graduate School | Associative rings, Universal algebra and category theory, Counterexamples |

Richard E. Borcherds | Professor | Lie algebras, Vertex algebras, Automorphic forms |

Sylvie Corteel | Professor | Combinatorics |

David Eisenbud | Professor | Algebraic geometry, Commutative algebra, Computation |

Edward Frenkel | Professor | Representation theory, Integrable systems, Mathematical physics |

Mark D. Haiman | Professor | Algebra, combinatorics, and algebraic geometry |

Robin C. Hartshorne | Professor Emeritus | Algebraic geometry, History of geometry |

Tsit-Yuen Lam (林節玄) | Professor Emeritus, Professor of the Graduate School | Algebra |

Hendrik W. Lenstra, Jr. | Professor Emeritus | Algebraic number theory, Algorithms |

David Nadler | Professor | Geometric representation theory, symplectic geometry |

Andrew P. Ogg | Professor Emeritus | Number theory, Elliptic curves, Modular forms |

Arthur E. Ogus | Professor Emeritus | Algebraic geometry |

Martin Olsson | Professor | Algebraic and arithmetic geometry |

Alexander Paulin | Teaching Professor | Number theory, p-adic automorphic forms |

Nicolai Reshetikhin | Professor Emeritus | Mathematical physics, Low-dimensional topology, Representation theory |

John L. Rhodes | Professor Emeritus | Algebra, Semigroups, Automata |

Kenneth A. Ribet | Professor | Algebraic number theory, Algebraic Geometry |

Vera Serganova | Professor | Super-representation theory |

Vivek Shende | Associate Professor | Geometry |

Sug Woo Shin | Professor | Number theory, automorphic forms |

Zvezdelina Stankova | Teaching Professor & Director, Berkeley Math Circle | Algebraic geometry, representation theory, combinatorics, Olympiad problem solving, Berkeley Math Circle |

Bernd Sturmfels | Professor | Algebraic Geometry, Combinatorics, Optimization and Statistics |

Constantin Teleman | Professor | Lie groups, Algebraic geometry, Topology, Quantum field theory |

Paul A. Vojta | Professor | Diophantine approximation, Nevanlinna theory (especially as related to diophantine approximation), Arakelov theory |

Mariusz Wodzicki | Professor | Non-commutative and algebraic geometry, Analysis, K-theory |

## Visiting Faculty

Name | Title | Research Interests |
---|---|---|

Hélène Barcelo | Research Associate | Algebraic combinatorics, discrete homotopy and homology theory |

Juan Antonio Carrasco López | Visiting Scholar | Computational Linear Algebra, Numerical Methods for Continuous-time Markov Chains |

Jesse Elliott | Visiting Professor | Ring theory, analytic number theory, foundations of mathematics, philosophy of mathematics and science |

Emiliano Gómez | Academic Coordinator/Instructor | Number theory, Galois representation |

Kangjin Han | Visiting Scholar | Projective geometry and syzygies, Geometry of tensors |

David Hernandez | Visiting Scholar | Quantum groups, infinite dimensional Lie algebras, cluster algebras, representation theory and its interactions with algebraic geometry and mathematical physics |

Arun Sharma | Unit 18 Lecturer | Combinatorics, Ramsey theory, error-correcting codes |

Kelli Talaska | Continuing Lecturer | Algebraic combinatorics |

## Postdocs

Name | Title | Research Interests |
---|---|---|

Owen F. Barrett | Morrey Visiting Assistant Professor | Algebraic and arithmetic geometry, ℓ-adic cohomology, local acyclicity, singular support |

Lea Beneish | Morrey Visiting Assistant Professor | Number theory, representation theory |

Daniel Bragg | RTG Postdoctoral Scholar & Unit 18 Lecturer | Algebraic geometry, positive characteristic |

Juliette Emmy Bruce | NSF Postdoctoral Scholar | Algebraic geometry, commutative algebra, arithmetic geometry |

Ivan Danilenko | Morrey Visiting Assistant Professor | Algebraic geometry, representation theory, mathematical physics. |

Gabriel Dorfsman-Hopkins | RTG Postdoctoral Scholar & Unit 18 Lecturer | Arithmetic geometry, p-adic geometry, mathematical illustration, 3D printing |

Sebastian Eterović | RTG Postdoctoral Scholar & Unit 18 Lecturer | Arithmetic geometry, model theory |

Peter Haine | NSF Postdoctoral Scholar and UC President's Postdoctoral Fellow | Homotopy theory, (derived) algebraic geometry, and related subjects |

Jackson S. Morrow | RTG Postdoctoral Scholar & Unit 18 Lecturer | Arithmetic geometry, non-Archimedean geometry, number theory |

Christopher Ryba | Morrey Visiting Assistant Professor | Representation theory, algebraic combinatorics. |

Koji Shimizu | Morrey Visiting Assistant Professor | Number theory and algebraic geometry |

Andrés R. Vindas Meléndez | NSF Postdoctoral Scholar | Algebraic and geometric combinatorics |

## Faculty with Related Research Interests

Name | Title | Research Interests |
---|---|---|

Olga Holtz | Professor | Numerical analysis, matrix theory, algebra and combinatorics, computational complexity |

Ralph McKenzie | Professor Emeritus | General algebra, Logic |

Calvin C. Moore | Professor Emeritus | Representations and actions of topological groups, Operator algebras |

Marc A. Rieffel | Professor | Non-commutative harmonic analysis, Operator algebras, Quantum geometry |

Thomas Scanlon | Professor | Model theory and applications to number theory |

Ruixiang Zhang | Assistant Professor | Harmonic analysis, analytic number theory, additive combinatorics |

## Graduate Students

## Recent Ph.D.s

Name | Dissertation Title | Dissertation Supervisor | Year |
---|---|---|---|

Alexander Bertoloni Meli | The Local Langlands Correspondence, Rapoport-Zink Spaces, and Shimura Varieties | 2020 | |

Madeline Brandt | Tropical Geometry of Curves | Bernd Sturmfels | 2020 |

Christopher Eur | The Geometry of Divisors on Matroids | David Eisenbud | 2020 |

Alexander Sherman | Spherical and Symmetric Supervarieties | 2020 | |

Ben Wormleighton | Numerics and stability for orbifolds with applications to symplectic embeddings | David Eisenbud | 2020 |

Mengyuan Zhang | Liaison of curves and bundles | David Eisenbud | 2020 |

Alexander Carney | The arithmetic Hodge-index theorem and rigidity of algebraic dynamical systems over function fields | Xinyi Yuan | 2019 |

Benjamin Gammage | Microlocal sheaves and mirror symmetry | David Nadler | 2019 |

Felix Gotti | Matroids and convex geometry in combinatorics and algebra | Lauren Williams | 2019 |

Christopher R Miller | On the k-Schur Positivity of k-Bandwidth LLT Polynomials | 2019 | |

Minseon Shin | Computations of the cohomological Brauer group of some algebraic stacks | Martin Olsson | 2019 |

Dylan Yott | Special Cycles on GSpin Shimura Varieties | Xinyi Yuan | 2019 |

Shishir Agrawal | Deformations of overconvergent isocrystals on the projective line | David Nadler | 2018 |

Harrison Chen | A Localization Theorem for Derived Loop Spaces and Periodic Cyclic Homology | David Nadler | 2018 |

Justin Chen | On Betti Tables, Monomial Ideals, and Unit Groups | David Eisenbud | 2018 |

Bryan Gillespie | The Generalized External Order, and Applications to Zonotopal Algebra | Olga Holtz | 2018 |

Watson Ladd | Algebraic Modular Forms on SO5(Q) and the Computation of Paramodular Forms | Ken Ribet | 2018 |

Daniel Lowengrub | Applications of the Intersection Theory of Singular Varieties | Vivek Shende | 2018 |

Shelly Manber | Asymptotics of the Tate-Shafarevich Group | Xinyi Yuan | 2018 |

Tao Su | A Hodge-theoretic study of augmentation varieties associated to Legendrian knots/tangles | Vivek Shende, Richard E.Borcherds | 2018 |

Brandon Williams | Computing modular forms for the Weil representation | Richard Borcherds | 2018 |

Anastasia Chavez | Posets, Polytopes, and Positroids | Lauren Williams, Federico Ardila | 2017 |

Joe Kileel | Algebraic Geometry for Computer Vision | Bernd Sturmfels | 2017 |

Bo Lin | Combinatorics and Computations in Tropical Mathematics | Bernd Sturmfels | 2017 |

Emmanuel Tsukerman | Combinatorial Analysis of Continuous Problems | Lauren Williams, Bernd Sturmfels | 2017 |

Qiao Zhou (Elaine) | Applications of Toric Geometry to Geometric Representation Theory | David Nadler | 2017 |