Untitled Talk
Talk, UIUC Graduate Student Homotopy Theory Seminar, Urbana, IL
Talks and Presentations
Untitled Talk
Talk, UIUC Motivic and K-theory Literature Seminar, Urbana, IL
Untitled Talk
Talk, UIUC THH Learning Seminar, Urbana, IL
Categorifying Spectra
Talk, UIUC Motivic and K-theory Literature Seminar, Urbana, IL
Ramzi, Sosnilo, and Winges’s recent paper Every Spectrum is the K-theory of a Stable Infinity Category constructs a functorial assignment that sends each spectrum to a small idempotent complete stable $\infty$-category, known as a categorification of the spectrum. An interesting consequence of this result is a disproof of a conjecture regarding the theorem of heart for non-connective spectra, proposed in K-theoretic Obstructions to Bounded t-structures. [Slides]
Leray-Serre Spectral Sequence, Purity and Gysin Sequence
Talk, Étale Cohomology Learning Seminar, Urbana, IL
Using the machinery of Leray-Serre spectral sequence, I will present a proof of the purity theorem and its consequences.
A Tropical Proof of the Brill-Noether Theorem
Talk, UIUC Graduate Algebraic Geometry and Commutative Algebra Seminar, Urbana, IL
This is a continuation of my talk from last week. Recall that we observed how the degeneration technique provides limits of line bundles on the stable curve, but these limits are not unique, which prevents us from classifying them effectively. This week, we will see that this “obstacle” also presents an opportunity to explore the twisting of the limit line bundles through the chip-firing game, ultimately leading to a proof of the Brill-Noether non-existence theorem.
Proper Base-change Theorem
Talk, Étale Cohomology Learning Seminar, Urbana, IL
We will give a proof of the proper base-change theorem, which requires reducing the given claim to a much simpler case and then applying consequences of Artin approximation.
How to Overcomplicate the Brill-Noether Theorem
Talk, UIUC Graduate Algebraic Geometry and Commutative Algebra Seminar, Urbana, IL
The Brill-Noether theorem, as a purely algebro-geometric result, is concerned with the mappings of algebraic curves into projective spaces. Classic proofs, such as those by Griffiths-Harris and Lazarsfeld, rely solely on degeneration arguments and properties of vector bundles. However, modern developments have revealed surprising connections between the theorem and combinatorics. In this talk (and the subsequent one), I will present an argument that involves a mixture of degeneration techniques with a combinatorial argument of the chip-firing game from Cools-Draisma-Payne-Robeva.
Étale Cohomology
Talk, Étale Cohomology Learning Seminar, on Zoom
Roughly speaking, étale cohomology is an algebraic analogue of singular cohomology that extends sheaf cohomology. As we study topological spaces over Zariski topology (therefore associated to the small Zariski site) by considering sheaves and sheaf cohomology, having a stronger topology admits more open subsets, which makes studying small étale site over étale topology using schemes and étale cohomology the “correct” analogue. It turns out that étale cohomology also has strong connections with many other cohomologies, e.g., Galois cohomology and Čech cohomology. We will justify the said analogue by studying flasque sheaves and the higher direct image functor. Finally, we will show that étale cohomology has properties similar to those in the Eilenberg-Steenrod axioms. [Notes]
Descent Theory
Talk, Étale Cohomology Learning Seminar, on Zoom
Before starting any discussion about étale topology or étale cohomology, we discuss the notion of a descent. Roughly speaking, a descent asks for uniqueness/recoverability of gluing along the pullback (as a notion of restriction), which generalizes the gluing condition on a topology. We introduce descent theory over Zariski topology, with respect to fpqc and/or fppf morphisms, which includes descent of quasi-coherent sheaves, descent of properties of morphisms, descent of schemes, as well as an argument that allows “passage to limit” through a directed system. With the categorical formulation of descent data, we allow a generalization of descent theory over étale topology and/or with respect to a covering, or over general fibered categories over sites. [Notes]
$+=Q$ Theorem, Part 2
Presentation, Algebraic K-theory Reading Seminar, UIUC
Given an exact category $\mathcal A$, there are two natural notions of K-groups on $\mathcal A$, one is given by the K-group of the Quillen exact structure, and the other is given by the K-group of the symmetric monoidal structure on its subcategory $S = i\mathcal A$, the isomorphism category of $\mathcal A$. It was observed by Quillen and later proven by Grayson using the $S^{-1}S$-construction in his 1976 paper that the two notions agree under mild assumptions, and in particular the K-groups of a ring $R$ agree with the K-groups of $\mathcal P(R)$, the category of finitely-generated projective $R$-modules. [Notes]
Equivariant Cohomology
Talk, Equivariant Homotopy Theory Learning Seminar, UIUC
We first introduce two notions of cohomology in the equivariant setting, namely Bredon cohomology and Borel cohomology, and we will see how they come up in the proof of Smith Theory. We will also introduce an equivariant (stable) version of Brown Representability, which roughly produces an equivalence between equivariant cohomology theories and G-spectra. [Seminar] [Notes]
$+=Q$ Theorem, Part 1
Presentation, Algebraic K-theory Reading Seminar, UIUC
We study the properties of the $S^{-1}S$-construction for a given symmetric monoidal category $S$ from Grayson’s 1976 paper. [Notes]
Suslin’s Rigidity Theorem, Part 2
Presentation, Algebraic K-theory Reading Seminar, UIUC
We continued our discussion of Suslin’s K-theory of algebraically closed fields and proved his rigidity theorem. [Notes]
Suslin’s Rigidity Theorem, Part 1
Presentation, Algebraic K-theory Reading Seminar, UIUC
We gave an overview of Suslin’s K-theory of algebraically closed fields, which can be found in his 1983 paper, 1984 paper, and his 1986 ICM talk. Using Quillen’s theorems, we introduced crucial properties developed in his 1983 paper, and constructed specialization map for closed points on a curve. [Notes]
Quillen’s Devissage Theorem and Localization Theorem
Presentation, Algebraic K-theory Reading Seminar, UIUC
We used Quillen’s theorem A and B to prove Devissage theorem and Localization theorem, along with certain important consequences. [Notes]
Quillen’s Detection Theorem
Presentation, Algebraic K-theory Reading Seminar, UIUC
This is part 6 of a series of presentations about Quillen’s 1972 paper. We proved Quillen’s detection theorem using tools like localization at fixed point set and homology of extended powers.
Brauer Lift
Presentation, Algebraic K-theory Reading Seminar, UIUC
This is part 2 of a series of presentations about Quillen’s 1972 paper. We discussed Atiyah–Segal completion theorem, Green’s theorem, Brauer character of representation, and constructed the Brauer lift as a $H$-map.
Triangulated Category’s Christmas Wish List
Talk, UIUC Graduate Algebraic Geometry and Commutative Algebra Seminar, Urbana, IL
We will discuss Paul Balmer’s work and introduce tensor triangulated category and Balmer spectrum, where we study geometric information over these particular triangulated categories. We will also look into motivations for studying tensor triangulated geometry, including its connection with commutative algebra and algebraic geometry.
Bounds in Simple Hexagonal Lattice and Classification of 11-stick Knots
Talk, AMS Special Session at Joint Mathematics Meetings, Boston, Massachusetts
The stick number and the edge length of a knot type in simple hexagonal lattice (sh-lattice) are the minimal number of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Moreover, we find lower bounds for any given knot’s stick number and edge length in sh-lattice using these properties in the cubic lattice. Finally, we show that the only non-trivial 11-stick knots in the sh-lattice are the trefoil knot ($3_1$) and the figure-eight knot ($4_1$). This is based on our work here.
Enriched Categories and Applications
Presentation, Directed Reading Program Colloquium, UCLA
I gave a presentation at the end of the quarter for my directed reading project with Ben Spitz.