# Graduate Colloquium

The graduate students of the Mathematics Department coordinate a quarterly colloquium. The talks are generally every Thursday from 4-5 PM, with an informal tea (get-together) beforehand.

**Spring 2021 Grad Colloquium Thursdays**

**"tea" @ ~3:30pm, talk @ 4pm, California Time**

**Schedule ***(click dates for abstract)*

4/8 | Andrew Kobin | What is a stack ? (part 2) |

4/15 | Sandra Nair | So what is a Shimura Curve? |

4/22 | Herman Yu | Localizations of Triangulated Categories |

4/29 | Philip Barron | An Introduction to Supercuspidal Representations |

5/6 | Justin Lake | Singular Plane Curves, Zariski's Moduli Problem, and the Monster Tower |

5/13 | Yufei Shan | The Heat Kernel Estimate on the Asymptotically Hyperbolic Manifold and the Stability of the Asymptotically Hyperbolic Einstein Manifold |

5/20 | Malachi Alexander | An Introduction to Building Theory |

5/27 | John McHugh | Perfect Isometries and Broué's Abelian Defect Group Conjecture |

**Special Time: "pre-talk" review of part 1 from 3:30 - 4 PM; talk starts at 4 PM**

**Andrew Kobin, UCSC**

*What is a stack? (part 2)*

Many classification problems in math are made harder by nontrivial automorphisms of the objects one is trying to classify. Groupoids, and ultimately stacks, are the right language to solve these problems in a satisfying way. In this talk, I will gently ease the audience into the waters of algebraic geometry while sprinkling in the motivating principles behind stacks. Then we will look at some important examples of algebraic stacks, including quotient stacks and the moduli stack of elliptic curves. The talk will be preceded by a "pre-talk" where key facts and ideas from the first part of this talk, that took place last quarter, will be recalled.

**Sandra Nair, University of Michigan**

*So what is a Shimura Curve?*

Shimura varieties are one of the objects needed to realize the ambitious Langlands’ correspondence. They are infamous for being rather technical. Historically, their one-dimensional avatar, called Shimura curves, were studied first, in the context of class field theory. In this talk, we will build up to a working definition of what a Shimura curve is, using classical modular curves as our Ariadne’s Thread.

**Herman Yu, UCSC**

*Localizations of Triangulated Categories*

Much of the theory of triangulated categories was developed in parallel in two different settings: homological algebra/algebraic geometry and algebraic topology. As such, the concept of "localization" in triangulated categories also come in two flavors: that of a Verdier quotient (Verdier localization) and that of a "localization" functor (Bousfield localization). Our talk will delve into both of these ideas and look at some basic properties and results that arise. We then proceed to look at smashing localizations and their connection the world of tensor-triangular geometry, specifically to the Balmer spectrum, via the use of tensor-idempotents.

**Philip Barron, UC Santa Cruz**

*An Introduction to Supercuspidal Representations*

Representations of the finite group \(GL_n(\mathbb{F}_p)\) are either "cuspidal" or they arise by induction from cuspidal representations of \(GL_m(\mathbb{F}_p)\) for \(m < n\). In a similar way, representations of \(GL_n(\mathbb{Q}_p)\) - and all other reductive \(p\)-adic groups - arise from "supercuspidal" representations of smaller groups. In this way, supercuspidal representations play a key role in the representation theory of \(p\)-adic groups. In this talk, we will develop the necessary theory to define supercuspidal representations for general \(p\)-adic reductive groups. We will then explore methods of constructing certain classes of supercuspidal representations using the Bruhat-Tits building. Then we will apply these methods to construct supercuspidal representations of the group \(SL_2(\mathbb{Q}_p)\).

**Justin Lake, UC Santa Cruz**

**Singular Plane Curves, Zariski's Moduli Problem, and the Monster Tower**

We will take a look at irreducible curve germs in the plane \(k^2\), where \(k\) is a field of characteristic \(0\). To them, we will associate a numerical semigroup using the theorem of Puiseux. We will then explore the moduli space of Zariski, and the problem of classification of plane branches. Finally, we will take a look at the monster tower construction for the plane, and discuss how it may be used to classify plane branches.

**Yufei Shan, UC Santa Cruz**

**joint offering with our G&A Seminar**

**The Heat Kernel Estimate on the Asymptotically Hyperbolic Manifold and the Stability of the Asymptotically Hyperbolic Einstein Manifold**

In this talk, first I am going to review the heat kernel estimate on the asymptotically hyperbolic manifold which is non-trapping and no resonance at the bottom of the spectrum (the result of Xi Chen, Andrew Hassell). Basically, their idea is to relate the Schwartz kernel of the resolvent of the Laplacian operator to the heat kernel. Then, by the result of R. Mazzeo and R. Melrose [1] which can give us the behavior of the Schwartz kernel of the resolvent as the spectrum parameter is bounded, and the result of R. Melrose, A. Sa Barreto and A. Vasy [2] which can give us the behavior of the Schwartz kernel of the resolvent for unbounded spectrum parameter, we can get the heat kernel estimate by the estimate of the corresponding Schwartz kernel.

Next, I am trying to give some ideas about how to generalize the above result into the symmetric two-tensor case. Finally, I will give a stability result of the asymptotically hyperbolic Einstein manifold based on the heat kernel estimate for the symmetric two-tensor on the asymptotically hyperbolic manifold.

[1] Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature

[2] Analytic continuation and semiclassical resolvent estimates on asymptotically hyperbolic spaces

**Malachi Alexander, UC Santa Cruz**

**An Introduction to Building Theory**

Jacques Tits invented buildings as a framework to unify the study of complex semisimple Lie groups and algebraic groups over an arbitrary field. We will give a brief introduction to the theory of buildings, working towards giving a proper understanding of the definition. As part of this introduction, we will define Coxeter groups through their geometric interpretation, reflection groups, construct the Coxeter complex and give simpler examples of buildings. If time permits, we will cover how one constructs a building associated with a group.

**John McHugh, UC Santa Cruz**

*Perfect Isometries and Broué's Abelian Defect Group Conjecture*

Broué's abelian defect group conjecture is a famous unsolved problem in the modular representation theory of finite groups. It can be roughly stated as follows: let \(R\) be a complete discrete valuation ring with residue field \(F\) of characteristic \(p\). Let \(G\) be a finite group and assume that \(R\) contains a primitive \(|G|\)-th root of unity. Finally, let \(B\) be a block algebra of \(RG\) with defect group \(D\). If \(D\) is abelian, then there is an "equivalence" between \(B\) and a "corresponding" block algebra of \(RN_{G}(D)\). What "corresponding" here means can be made precise, but there are different formulations of the conjecture based on what "equivalence" means. We will discuss the different versions of the conjecture after introducing some of the main players: perfect isometries, isotypes, and bounded derived categories.