Homepage of Gabriel Angelini-Knoll
I am a Ph.D. candidate at Wayne State University working under Andrew Salch. My research primarily focuses on how periodicity in the stable homotopy groups of spheres interacts with algebraic K-theory. Specifically, I am interested in exploring if periodic families from the homotopy groups of spheres of height n+1 can be detected in algebraic K-theory of the connective cover of the E(n)-local sphere. In my thesis, I focus on the case n=1 where I can show that the beta family, a periodic family of height two in the homotopy groups of spheres, is detected in algebraic K-theory of the p-completion of the connective cover of the E(1)-local sphere. I am also passionate about teaching and I have enjoyed teaching courses as a Graduate Teaching Assistant at WSU. For more information about my research interests and my teaching philosophy see my application materials below.
- Office: 1118 Faculty / Administration Building
- Email: gabriel dot angelini-knoll at wayne dot edu
- G. Angelini-Knoll. On topological Hochschild homology of the K(1)-local sphere. Submitted. pdf arXiv
I compute topological Hochschild homology of the connective cover of the K(1)-local sphere modulo p and v_1 using the THH-May spectral sequence.
- G. Angelini-Knoll, A. Salch. A May-type spectral sequence for topological Hochschild homology. Submitted. pdf arXiv
We construct a spectral sequence analogous to the May spectral sequence for higher order topological Hochschild homology. The spectral sequence takes a multiplicative filtered commutative monoid as input and it is constructed in any sufficiently nice model category. As an application, we give a upper bound on the topological Hochschild homology of a connective commutative ring spectrum in terms of the topological Hochschild homology of a generalized Eilenbegerg-MacLane spectrum.
- G. Angelini-Knoll, A. Salch. Maps of simplicial spectra whose realizations are cofibrations. Submitted. pdf arXiv
It is often useful to know when a map of simplicial symmetric spectra induces a cofibration on geometric realization. Reedy provided an answer to this question in his thesis, but the conditions of a Reedy cofibration between reedy cofibrant simplicial spectra are not usually easy to check. We provide conditions that are easy to check for a map of simplicial spectra to induce a cofibration on geometric realization. These conditions are modeled off of Segal's conditions for a map of simplicial topological spaces to induce a cofibration after geometric realization.
Unpublished papers and notes:
- G. Angelini-Knoll. K(n)-local homotopy groups via Lie algebra cohomology. pdf
We compute the homotopy groups of the v_2 telescope of the Smith-Toda complex V(1) using an Adams-Novikov spectral sequence. The input of this spectral sequence is computed using Hochschild-Serre spectral sequences, the Lie-May spectral sequence and then another May spectral sequence as outlined in Ravenel's "Complex Cobordism and the Stable Homotopy Groups of Spheres."
- G. Angelini-Knoll. Auslander-Reiten quiver of the category of unstable modules over a sub-Hopf algebra of the Steenrod algebra. pdf
Structure theorems for the category of unstable E(n)-modules are proven using Auslander-Reiten theory and Waldhausen algebraic K-theory. Here E(n) indicates the exterior subHopf algebra of the Steenrod algebra generated by the first n+1 Milnor primitives.
- G. Angelini-Knoll. Galois cohomology and algebraic K-theory of finite fields. pdf
A Master's Essay completed under the direction of Prof. Andrew Salch at Wayne State University. We compute algebraic K-theory of finite fields using the "motivic to algebraic K-theory" spectral sequence. In this case, we reduce motivic cohomology to etale cohomology and then reduce etale cohomology to galois cohomology of a profinite galois group.
- Young Topologist's Meeting 2015, EPFL Lausanne, Switzerland;
Topological Hochschild Homology of the connective cover of the K(1)-local sphere, Beamer Slides
- European Talbot Workshop 2015, Klosters, Switzerland; Fundamental Theorems of Algebraic K-theory Notes
- Graduate Student Topology and Geometry Conference 2015, UIUC Urbana-Champaign, Illinois;
Topological Hochschild Homology of the algebraic K-theory spectrum of a finite field, Beamer Slides
- MSRI Summer School: Algebraic Topology, Guanajuato, Mexico; On the paper Nilpotence and Stable Homotopy Theory II, Notes
- Kalamazoo College Undergraduate Seminar, Kalamazoo, Michigan; The Topologist's Snowflake and the Rose, Abstract