Talks and Presentations

Thomason-Trobaugh K-theory has served as a foundational definition of algebraic K-theory, further characterized by a universal property in the language of $\infty$-categories by Blumberg-Gepner-Tabuada. Thomason and Trobaugh also extended their construction to a non-connective spectrum via Bass delooping, resulting in a K-theory that satisfies certain representability conditions, as well as Zariski and Nisnevich descent. This naturally raises the question of finding a notion of homotopy-invariant K-theory that supports the desired representability and satisfies suitable descent properties. We will provide an overview of Cisinski's approach to this problem, demonstrating that several such notions coincide, possess the expected representability, and, in fact, satisfy even stronger descent properties.

Recall that Blumberg, Gepner, and Tabuada constructed a universal (finitary) localizing invariant into some $\infty$-category $\mathcal M_\text{loc}$ corepresenting non-connective K-theory. In particular, that means an equivalence on the corresponding functor categories. However, it is hard to find out anything useful about $\mathcal M_\text{loc}$ a priori because the construction. It turns out that, once restricted to $\operatorname{Cat}_\infty^\text{perf}$, the category $\mathcal M_\text{loc}$ can be thought of as the Dwyer-Kan localization of the category at a class of exact functors, called the motivic equivalences. This was recently proven by Ramzi, Sosnilo, and Winges. The proof is anchored upon the fact that the category $\operatorname{Cat}_\infty^\text{perf}$ along with this class of functors can be made into a model of cofibration categories, which has nice properties whenever the cofibration category is small. Due to cardinality issues, however, to prove this for $\mathcal M_\text{loc}$, one has to filter the category by a family of small full subcategories given by $\kappa$-compact objects, where each $\kappa$ is countably closed. The choice of countably closed cardinals is nice because 1) there are enough supply of them for a filtration, i.e., given any cardinal, there is a countably closed cardinal bigger than it, and 2) it preserves enough constructions for $\kappa$-compact objects. In particular, once restricting $\operatorname{Cat}_\infty^\text{perf}$ to $\operatorname{Cat}_\infty^{\text{perf}, \kappa}$ for countably closed $\kappa$, we retrieve a small cofibration category by taking corresponding wide subcategories. Therefore it suffices to localize each small subcategory of the filtration, prove that it satisfies the universal property, and then taking a colimit across all such cardinals. One can also be motivated to study $\aleph_1$-finitary localizing invariants, i.e., ones preserving $\aleph_1$-filtered colimits, for instance from the fact that constructions like $\operatorname{TR}$ and $\operatorname{TC}$ are only $\aleph_1$-finitary but not finitary. The proof was done by studying properties of $\aleph_1$-compactness, then extending the properties proved for countably closed cardinals to general cardinals, and inverting a class called "simple motivic equivalences" on the corresponding functor categories. It turns out that the universal $\aleph_1$-finitary localizing invariant coincides with the universal finitary localizing invariant given by $\mathcal M_\text{loc}$.

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.

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.

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.

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.

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.

I gave a presentation about my directed reading project mentored by Ben Spitz when I was an undergraduate student at UCLA.