Jiantong Liu

Jiantong Liu

Math PhD student at UIUC

Talks

Apr 2025Graduate Student Homotopy Theory SeminarUntitled
Apr 2025THH Learning SeminarComputing $TC(𝔽_p)$ (2 talks)
Apr 2025Motivic and K-theory Literature SeminarReinterpretating Universal Localizing Invariant
Oct 2024Motivic and K-theory Literature SeminarCategorifying Spectra
Ramzi, Sosnilo, and Winges's recent paper 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 by Antieau-Gepner-Heller. [Slides]
Oct 2024Étale Cohomology Learning SeminarLeray-Serre Spectral Sequence, Purity and Gysin Sequence
Sep 2024Graduate Algebraic Geometry and Commutative Algebra SeminarA Tropical Proof of the Brill-Noether Theorem (2 talks)
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.
Sep 2024Étale Cohomology Learning SeminarProper Base-change Theorem
Jun 2024Étale Cohomology Learning SeminarÉtale Cohomology
May 2024Étale Cohomology Learning SeminarDescent Theory
Feb 2024Equivariant Homotopy Theory Learning SeminarEquivariant Cohomology
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.
Feb 2024Algebraic K-theory Reading Seminar$+=Q$ Theorem (2 talks)
Nov-Dec 2023Algebraic K-theory Reading SeminarSuslin's Rigidity Theorem (2 talks)
Nov 2023Algebraic K-theory Reading SeminarQuillen's Devissage Theorem and Localization Theorem
Nov 2023Algebraic K-theory Reading SeminarQuillen's Detection Theorem
Oct 2023Algebraic K-theory Reading SeminarBrauer Lift
Sep 2023Graduate Algebraic Geometry and Commutative Algebra SeminarTriangulated Category's Christmas Wish List
We discuss 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.
Jan 2023AMS Special Session at Joint Mathematics MeetingsBounds in Simple Hexagonal Lattice and Classification of 11-stick Knots
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.
Mar 2022UCLA Directed Reading Program ColloquiumEnriched Categories and Applications
I gave a presentation at the end of the quarter for my directed reading project with Ben Spitz.