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.

- 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, - 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, - 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