# 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 |

Tony Feng | Assistant Professor | Number Theory, Arithmetic Geometry, Langlands Program, Algebraic Topology, Representation Theory |

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 |

Hannah K. Larson | Assistant Professor, starting Summer 2023 | Algebraic geometry, intersection theory. |

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 |

Andrei Okounkov | Professor | Representation theory (of combinatorial flavor) and its application to algebraic geometry, mathematical physics, probability, special functions, and other fields. |

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 |

Yunqing Tang | Assistant Professor | Arithmetic geometry, number theory |

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 |

Arash Farahmand | Unit 18 Lecturer | Nonlinear Programming, Frobenius Numbers |

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

Marcus Nilsson | Visiting Professor | Number theory, Cryptography, Dynamical systems |

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

Florian Ian Sprung | Visiting Professor | Elliptic Curves, Iwasawa Theory, Modular Forms, decidability, arithmetic statistics |

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 |

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

Colleen Delaney | Postdoctoral Scholar | Mathematical physics, quantum topology, applied category theory, quantum computation |

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

Nicolle Gonzalez | Morrey Visiting Assistant Professor | Diagrammatic categorification, representation theory, and algebraic combinatorics arising in connection to Macdonald polynomials, Demazure modules, and homological knot invariants |

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

Borislav Mladenov | Morrey Visiting Assistant Professor | Algebraic geometry, mathematical physics |

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. |

Alexandra Utiralova | Morrey Visiting Assistant Professor | Algebra, representation theory, tensor categories |

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

Yegor Zenkevich | Postdoctoral Fellow | Mathematical physics, representation theory |

## Faculty with Related Research Interests

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

Vadim Gorin | Associate Professor | Integrable probability, random matrices, asymptotic representation theory |

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

Name | Dissertation Supervisor |
---|---|

Ahmad Abassi | |

Yulia Alexandr | Bernd Sturmfels, Serkan Hosten |

Chan Bae | |

Ansuman Bardalai | |

Aaron Brookner | |

Thomas Browning | |

Yifan Chen | |

Dennis Chen | David Nadler |

Daniel Chupin | |

Franny Dean | |

Adam Dhillon | |

CJ Dowd | |

Sean Gonzales | |

Aubrey Gross | |

Connor Halleck-Dube | |

Mitsuki Hanada | |

Robin Huang | |

Daigo Ito | |

Eric Jankowski | |

Kabir Kapoor | |

Luhang Lai | |

Catherine Lee | |

John Lentfer | |

Feiyang Lin | |

Rose Lopez | |

Anya Michaelsen | |

Claire Mirocha | |

Xianglong Ni | |

John Nolan | |

Ronan O'Gorman | |

Yuji Okitani | |

Yanshuai Qin | Xinyi Yuan |

Joseph Stahl | |

Jiazhen Tan | |

Zhongkai Tao | |

Jeremy Taylor | |

Pranav Trivedi | |

Rebecca Whitman |

## Recent Ph.D.s

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

Ravi Fernando | Saturated de Rham-Witt complexes with unit-root coefficients | Martin Olsson | 2023 |

Nicolas Brody | Groups Acting on Products of Trees | Ian Agol | 2022 |

Christopher Kuo | Symplectic geometric methods in microlocal sheaf theory | Vivek Shende | 2022 |

Kyle Miller | Singularity Theory for Extended Cobordism Categories and an Application to Graph Theory | Ian Agol | 2022 |

Ritvik Ramkumar | The Geometry of Hilbert Schemes on Projective Space | David Eisenbud | 2022 |

Weitong Wang | Distribution of Class Groups | Melanie Wood | 2022 |

Michael Christianson | Combinatorial Invariants on Smooth Projective Spherical Varieties | Martin Olsson | 2021 |

Anningzhe Gao | Some results on arithmetic aspects of K3 surfaces and abelian varieties | Martin Olsson | 2021 |

Ian Gleason | Specialization maps for Scholze’s category of diamonds | Sug Woo Shin | 2021 |

Calvin McPhail-Snyder | SL_2(C)-holonomy invariants of links | Nicolai Reshetikhin | 2021 |

Isabelle Shankar | Sums of Squares and Symmetric Polynomials | Serkan Hosten, Jon Wilkening | 2021 |

Melissa Sherman-Bennett | On the combinatorics of cluster structures on positroid varieties | Lauren Williams, Sylvie Corteel | 2021 |

German Stefanich | Higher Quasicoherent Sheaves | David Nadler | 2021 |

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 |