Yuqing Shi

Research Interests

My research interests lie in algebraic topology, with special interests in periodic and chromatic homotopy theory, Koszul duality, Goodwillie calculus and stable infinity categories.

In my PhD thesis I considered unstable periodic homotopy theory. Here, "unstable" refers to the homotopy theory of homotopy types (instead of spectra). One shall view unstable periodic homotopy theory as an extension of rational homotopy theory of homotopy types: In rational homotopy theory we study the localisatin of the infinity category of simply connected homotopy types at the set of degree p self-maps of spheres for every prime number p. Fix a prime number p. For every natural number h, the v_h-periodic homomotopy theory is about the localisation of the infinity category of (simply-connected) p-local homotopy types at the set of certain "degree v_h self-maps" of certain finite complexes. The set of v₀ self-maps is the set of the degree p self-maps of spheres. Thus, the v₀-periodic homotopy theory recovers by construction the rational homotopy theory. See the introduction of my PhD thesis for more details.

In my master's thesis, I studied the application of manifold calculus to the theory of Vassiliev knot invariants. I still find the application of homotopy theory to geometric topology intriguing and would like to explore it further.

Article

• Universal property of the Bousfield--Kuhn functor. arXiv 2410.01116.

  We present a universal property of the Bousfield--Kuhn functor Φ_h of height h, for every positive natural number h. This result is achieved by proving that the costabilisation of the ∞-category of v_h-periodic homotopy types is equivalent to the ∞-category of T(h)-local spectra. A key component in our proofs is the spectral Lie algebra model for v_h-periodic homotopy types [Heu21]: We relate the costabilisation of the ∞-category of spectral Lie algebras with the costabilisations of the ∞-category of non-unital E_h-algebras, via our construction of higher enveloping algebras of spectral Lie algebras.

• Goodwillie's cosimplicial model for the space of long knots and its applications. Journal of Homotopy and Related Structures (2023). DOI. arXiv 2012.04036.

  We work out the details of a correspondence observed by Goodwillie between cosimplicial spaces and good functors from a category of open subsets of the interval to the category of spaces. Using this, we compute the first page of a integral Bousfield--Kan homotopy spectral sequence associated to the space of long knots arising from manifold calculus. Based on the methods in [Con08], we give a combinatorial interpretation of the differentials mapping into the diagonal terms, by introducing the notion of (i,n)-marked unitrivalent graphs.

Thesis

• Unstable Periodic Homotopy Theory. PhD thesis, Mathematical Institute of University of Utrecht. October 2023.

  In this thesis we study unstable v_h-periodic homotopy theory, where h is a natural number; here "unstable" refers to the homotopy theory of topological spaces. The work consists of two parts. In Part I we give a detailed exposition of the foundations of unstable v_h-periodic homotopy theory, sharpen existing results about v_h-periodic equivalences of H-spaces, and pose concrete questions and conjectures for future studies. The expository part follows papers by Bousfield, Dror Farjoun and Heuts and aims to assemble in one place the central notions and theorems of unstable localisations with a focus on unstable periodic homotopy theory. The goal of Part II is to understand unstable v_h-periodic phenomena from the point of view of Lie algebras in the stable infinity category of v_h-periodic spectra. We analyse the costabilisation of the infinity category of v_h-periodic homotopy types and obtain a universal property of the Bousfield--Kuhn functor.

• Vassiliev Invariants via Manifold Calculus [PDF]. Master thesis, Mathematical Institute of University of Bonn. September 2018.

  In the thesis, we give an alternative proof of a theorem by Budney–Conant–Koycheff–Sinha, which says that the manifold calculus tower of the space of long knots induces Vassiliev invariants. We calculate and interpret the first differentials ending at the diagonal of the integral homotopy spectral sequence associated to the manifold calculus tower. In an expository part, we give a detailed explaination of the correspondence between cosimplical spaces and "good" functors on the category of open subsets of the unit interval.

• The Alexander Polynomial [PDF]. Bachelor thesis, Mathematical Institute of University of Bonn. June 2016.

  The aim of the thesis is to understand basic concepts of knot theory and various constructions of the Alexander polynomial of a knot. We provided a detailed calculation of the Alexaner polynomial of torus knots and twists knots.