Tongtong Liang
I am a PhD student at University of Carlifornia, San Diego, advised by Zhouli Xu.
Research Interests
I am interested in algebraic topology and algebraic geometry, especially motivic homotopy theory and algebraic K-theory.
Writing
This article is an exposition on the homotopy-theoretic tools of cohomology operations applied to algebraic geometry and their inner workings. We first survey on the construction and mechanism of cohomology operations in classical homotopy theory. Particularly, we aim to explain how these operations detect elements and relations in homotopy classes along the Adams spectral sequences. Next, in order to explore analogous mechanism of cohomology operations in algebraic geometry, we introduce the framework of motivic homotopy theory as constructed by Morel and Voevodsky. Based on this framework, we study the constructions and properties of motivic power operations and related spectral sequences in motivic stable homotopy theory. Specifically, we would like to understand how motivic power operations exhibit the coherence encoded by norms in motivic homotopy theory and how motivic extended powers emerge in the motivic Adams spectral sequences.
This is a survey on Quillen's elementary proofs of some results of cobordism theory using power operations. We optimize the system of notations and clarify some vague arguments in Quillen's paper. Furthermore, we emphasize the relations among cobordism power operations, Landweber-Novikov operations and the formal group law associated to the complex cobordism theory. Particularly, we present a stable-homotopy-theoric construction of cobordism power operations in order to demonstrate the relations. Based on this, we give a different proof of Quillen's technical lemma by promoting a lemma in Rudyak's book from mod-2 case to mod-$p$ cases for all primes $p$.
This is a survey on Thom's solution to the Steenrod problem that is asking whether each homology class of a finite complex can be realized as a manifold. In particular, we clarify some vague arguments and calculations in Thom's paper. Following Thom's method, We first show how the problem is translated into a homotopy lifting problem by Thom's construction, then we calculate the obstructions of the corresponding lifting problems in terms of Steenrod operations. This survey aims to understand this method essentially, which is expected to enlighten us to think about how to generalize it to algebraic-geometric setting.
In algebraic topology, one often encounters diagrams of spaces that are commutative up to homotopy, rather than strictly commutative. However, by passing to the homotopy category, one loses the information of higher homotopies. This makes the corresponding algebraic invariants less effective to distinguish spaces. To give a more faithful algebraic picture for a geometric problem, it is desirable to devise machineries that capture higher homotopies. In this thesis, I show how the cup-i products and the Steenrod squares encode the data of higher homotopy types. From this perspective, I explain why the Steenrod squares and, more generally, cohomology operations for generalized cohomology theories work effectively as algebraic invariants for spaces in an attempt to understand the raison d’être of infinity-categorical algebra. This is based on investigating the literature and reorganizing theoretical and computational aspects of important tools in algebraic topology into an organic entirety through the theme of homotopy coherence. These include cohomology operations, simplicial sets, classifying spaces, and spectral sequences.
This is a survey on the Grothendieck-Riemann-Roch theorem. The idea of this article is to show that K- theory is the universal cohomology theory with multiplicative law $x+y-xy$, then Grothendieck-Riemann-Roch theorem follows the result.
Class groups and Riemann-Roch theorem are basic notion and theorems in arithmetic and algebra. I will introduce them in a geometric or topological way by using the language of schemes. First, we may turn arithmetical and algebraic objects (rings and modules) into geometric and topological objects (schemes and quasicoherent sheaves), then we use the methods of algebraic topology ((co)homology) to study them. We will see some theories in algebraic topology have an algebraic geometry version, which are very powerful.
This is a note on some commutative algebra, including going-up, going-down, dimension theory, regular ring and etc.