BAGELS is a weekly grad-student-led seminar on Algebraic Geometry, where bagels and mathematical ideas are shared. It takes place on Wednesdays from 3:15-4:15pm in JWB 208. If you have any questions about the seminar (other than what BAGELS stands for), feel free email one of the organizers: Daniel Apsley and Rahul Ajit. (apsley(at)math.utah.edu, rahulajit(at)math.utah.edu)
| January 15 | Rahul Ajit | A survey on Vanishing theorems in Birational Geometry. | After giving motivation for why should one care about Vanishing theorems of cohomology groups, I'll survey major vanishing results appearing ubiquitously in birational geometry over complex numbers. As I want to make this talk very accessible, I won't give any detailed proof. |
| January 22 | Daniel Apsley | Quotients in Algebraic Geometry | Quotients, or understanding the geometry of orbits of a given group action, is an important concept in geometry. In this talk, I want to discuss the problem of constructing quotients in algebraic geometry by reviewing which quotients can be constructed as schemes via GIT, and concluding with some generalities on algebraic spaces and stacks, to show that these abstract notions provide a more convenient framework for this problem. |
| Febuary 5 | Zach Mere | Differentials on arc spaces | In this talk, we'll describe the sheaves of Kähler differentials of the arc space and jet schemes. The resulting formulas can be applied to derive new results and simplify the proofs of some theorems in the literature. We'll start by briefly reviewing some of the basic definitions and facts in the theory of arc spaces, then we'll compute several examples. |
| Febuary 12 | Jack Cook | Algebraic Geometry in Representation Theory Part 1: Real reductive groups, the Borel-Weil-Bott theorem, and Localization Theory. | Representation theory has taken a rather geometric turn in the past 30 years. This started however with Borel and Weil in the 1950s. They first gave a geometric realization of irreducible representations of a compact Lie group. Over the years since then, other geometric techniques have been developed. We will discuss the history of this geometric advancement and see how one uses them to study representations. |
| Febuary 19 | Jack Cook | Algebraic Geometry in Representation Theory Part 2: The Nilpotent Cone, Perverse Coherent Sheaves, and the Lusztig-Vogan Bijection. | Given the geometric nature of the previous talk, we now want to see how this can be abused to obtain novel results about representations. Lusztig and Vogan independently conjectured the existence of a certain bijection between representations of a maximal compact subgroup and certain geometric data. The first proof was done for complex groups by Bezrukavnikov using an idea of Deligne. We will go over the structure theory of the nilpotent cone and how Bezrukavnikov proved his result. Time permitting we will talk about generalizations of his result and techniques being used to compute this abstract bijection. |
| Febuary 26 | Jonathon Fleck | Rationality, Cremona Groups, and Singularities of Fano Varieties | A variety is said to be rational if it is birational to some projective space. A common strategy showing a variety is not rational is naturally to show one of its birational invariants differ from that of projective space. In this talk, I will focus on a theorem of Iskovskikh and Manin which disproves rationality of smooth quartic threefolds in P4. I will start with its inspiration, a proof of Noether's theorem on the structure of the Cremona group of P2, discuss a low dimensional case which is similar in spirit to the case of quartic threefolds, and mention how these ideas generalize to higher dimensions using singularities. |
| March 19 | Gari Chua | Continued Fractions and Resolutions of Singularities of Affine Toric Surfaces | An affine toric surface is an irreducible surface containing a 2-dimensional complex torus, and such that the natural action of the torus on itself extends to an action on the entire surface. Affine toric varieties can be described combinatorially in terms of polyhedral cones and fans. In this talk, we will explore the link between certain continued fraction expansions and resolutions of singularities of toric surfaces. In particular, we will see that for toric surfaces obtained from cones in certain forms, that computing a continued fraction expansion results in a series of blowups on the toric surface that constitute a resolution of singularities. We focus on explicit computations. |
| March 26 | Qingyuan Xue | Quasi-Log Schemes in Birational Geometry | Quasi-log schemes, which generalize the concept of pairs, play a crucial role in birational geometry, particularly in the study of the Minimal Model Program (MMP) for pairs that may not be log canonical. In this talk, I will begin by reviewing fundamental definitions in birational geometry. I will then introduce quasi-log schemes, highlighting their key properties and similarities to usual pairs. If time permits, I will discuss the MMP in the context of quasi-log schemes. |
| April 2 | Joseph Sullivan | Title TBD | Abstract TBD |
| April 9 | Will Legg | Title TBD | Abstract TBD |
| April 16 | Yu-Ting Huang | Title TBD | Abstract TBD |