# 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 |

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 | Algebraic geometry |

Martin Olsson | Professor | Algebraic and arithmetic geometry |

Nicolai Reshetikhin | Professor | 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 | Assistant Professor | Geometry |

Sug Woo Shin | Associate 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 |

Lauren Williams | Professor | Combinatorics, and connections to representation theory, geometry, and statistical physics |

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

Xinyi Yuan | Assistant Professor | Number theory |

## Visiting Faculty

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

Alexander Alldridge | Visiting Scholar | Supergeometry, Lie supergroups |

Hélène Barcelo | Visiting Scholar | Algebraic combinatorics, discrete homotopy and homology theory |

Sylvie Corteel | Visiting Miller Professorship | Combinatorics |

Emiliano Gómez | Lecturer | Number theory, Galois representation |

Jeremy Lovejoy | Lecturer | Number theory and combinatorics |

Alexander Paulin | Lecturer | Number theory, p-adic automorphic forms |

Kelli Talaska | Lecturer | Algebraic combinatorics |

## Postdocs

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

Carolyn R. Abbott | Morrey Visiting Assistant Professor | Geometric group theory, low-dimensional topology |

David Dynerman | Morrey Visiting Assistant Professor | Applied algebra, cryo-electron microscopy, implementation of algorithms |

Khrystyna Serhiyenko | Postdoc | Algebra, representation theory, cluster algebras |

Andrey Smirnov | Morrey Visiting Assistant Professor | Representation theory, mathematical physics |

## Faculty with Related Research Interests

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

Denis Auroux | Professor | Symplectic topology and mirror symmetry |

Elwyn R. Berlekamp | Professor Emeritus | Combinatorial game theory, Algebraic coding theory, Electrical engineering, Computer science |

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 |

Lior Pachter | Professor | Mathematical and computational biology |

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

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

## Graduate Students

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

Shishir Agrawal | |

Alexander Bertoloni Meli | |

Madeline Brandt | Bernd Sturmfels |

Nicolas Brody | |

Harrison Chen | David Nadler |

Justin Chen | David Eisenbud |

Yael Degany | Katrin Wehrheim |

Christopher Eur | |

Ravi Fernando | Martin Olsson |

Benjamin Gammage | David Nadler |

Anningzhe Gao | |

Bryan Gillespie | Olga Holtz |

Felix Gotti | |

Christopher Kuo | |

Watson Ladd | Ken Ribet |

Luhang Lai | |

Daniel Lowengrub | |

Shelly Manber | Xinyi Yuan |

Christopher R Miller | |

Kyle Miller | |

Ritvik Ramkumar | David Eisenbud |

Melissa Sherman-Bennett | |

Minseon Shin | |

Joseph Stahl | |

German Stefanich | |

Tao Su | Vivek Shende, Richard E.Borcherds |

Charles Wang | |

Brandon Williams | Richard Borcherds |

Ben Wormleighton | David Nadler |

Dylan Yott | |

Mengyuan Zhang | David Eisenbud |

## Recent Ph.D.s

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

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 |

Chang-Yeon Cho | Topological types of Algebraic stacks | 2016 | |

Maria Monks Gillespie | A combinatorial approach to the q,t-symmetry in Macdonald polynomials | Mark Haiman | 2016 |

Will Johnson | Fun with Fields | 2016 | |

Olya Mandelshtam | Combinatorics of the Asymmetric Simple Exclusion Process | Lauren Williams | 2016 |

Alexander Shapiro | Grothendieck resolution, affine Grassmannian, and Yangian | 2016 | |

Piotr Achinger | Arthur E. Ogus | 2015 | |

Natth Bejraburnin | Lior Pachter | 2015 | |

Katrina Honigs | Martin Olsson | 2015 | |

Adam Kalman | Lauren Williams | 2015 | |

Eric C. Peterson | Constantin Teleman | 2015 | |

Zvi Rosen | 2015 | ||

Lasse Peter Mannisto | Albanese and Picard 1-Motives in Positive Characteristic | Martin Olsson | 2014 |

Khoa Nguyen | Arithmetic Dynamics of Diagonally Split Polynomial Maps | Paul A. Vojta, Thomas Scanlon | 2014 |

Andrew Niles | Moduli of Elliptic Curves via Twisted Stable Maps | Martin Olsson | 2014 |

Qingchun Ren | Computations and Moduli Spaces for Non-archimedean Varieties | Bernd Sturmfels | 2014 |

Pablo Solis | Antimicrobial Applications of Ambient-Air Plasmas | Constantin Teleman | 2014 |

Thanh Quang Vu | Combinatorial patterns in syzygies | David Eisenbud | 2014 |

Harold M. Williams | Cluster Algebras and Integrable Systems | Nicolai Reshetikhin | 2014 |

Adam L. Boocher | Superflatness | David Eisenbud | 2013 |

Andrew J. Critch | Algebraic Geometry of Hidden Markov and Related Models | Bernd Sturmfels | 2013 |

Michael W. Daub | Complex and p-adic Computations of Chow-Heegner Points | Kenneth A. Ribet | 2013 |

Daniel Halpern-Leistner | Geometric Invariant Theory and Derived Categories of Coherent Sheaves | Constantin Teleman | 2013 |

Theodore Johnson-Freyd | Perturbative Methods in Path Integration | Nicolai Reshetikhin | 2013 |

Lilit Martirosyan | The Representation Theory of the Exceptional Lie Superalgebras F(4) and G(3) | Vera Serganova | 2013 |

Matthew Tucker-Simmons | Marc A. Rieffel, Nicolai Reshetikhin | 2013 | |

Hwajong Yoo | Modularity of residually reducible Galois representations and Eisenstein ideals | Kenneth A. Ribet | 2013 |

Morgan Brown | Cox Rings and Partial Amplitude | David Eisenbud | 2012 |

Melody Chan | Tropical curves and metric graphs | Bernd Sturmfels | 2012 |

Sarah Iveson | Mark D. Haiman | 2012 | |

Chul Hee Lee | Algebraic Structures in Modular q-hypergeometric Series | Richard E. Borcherds | 2012 |

Edgard Rincon | Tropical Linear Spaces and Applications | Olga Holtz, Bernd Sturmfels | 2012 |

George Schaeffer | 2012 | ||

George W. Melvin | Constantin Teleman |