Jiantong Liu

Jiantong Liu

Math PhD student at UIUC

Talks and Presentations

Étale Cohomology

June 14, 2024

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

May 20, 2024

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

February 09, 2024

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

February 06, 2024

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

February 02, 2024

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

December 05, 2023

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]

Quillen’s Detection Theorem

November 07, 2023

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

October 10, 2023

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

September 27, 2023

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

January 06, 2023

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.