Spectral Mackey functors and equivariant algebraic K-theory (I) by Barwick, Clark Edward

“…Finally, we employ this technology to lay the foundations of equivariant stable homotopy theory for profinite groups; the lack of such foundations has been a serious impediment to progress on the conjectures of Gunnar Carlsson. …”
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Compatible actions for finite cyclic groups of p-power order by Mohamad, Mohd Sham, Mohd Aziz, Mohd Khairul Bazli, Adam, Samsudin, Adli Aminuddin, Adam Shariff, Yusof, Yuhani, Nor Haniza, Sarmin

“…The concept of the nonabelian tensor product of groups has its origins in the algebraic K-theory and the homotopy theory. This concept is defined on the actions which are compatible to each other. …”
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Compatible actions for finite cyclic groups of p-power order by Mohamad, Mohd Sham, Mohd Aziz, Mohd Khairul Bazli, Adam, Samsudin, Adli Aminuddin, Adam Shariff, Yusof, Yuhani, Nor Haniza, Sarmin

“…The concept of the nonabelian tensor product of groups has its origins in the algebraic Ktheory and the homotopy theory. This concept is defined on the actions which are compatible to each other. …”
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Compatibility conditions and nonabelian tensor products of finite cyclic groups of ρ-power order by Mohd Sham, Mohamad

“…The nonab elian tensor pro ducts of groups originated from a generalized Van Kamp en Theorem and its construction has its origins in algebraic K-theory and in homotopy theory.In this research,cyclic groups of p-p ower order where p is a prime numb er are considered.The aim of this research is to prove that the nonabelian tensor pro ducts of some finite cyclic groups of p-p ower order are cyclic.This research starts with the characterization of automorphisms of cyclic groupsof p-p ower order using numb er theoretical results where the order of the actions are considered.Then,the necessary and sufcient conditions for the actions to b ecompatible are determined for a pair of finite cyclic groups.Finally,by using a general expansion formula,the nonab elian tensor pro ducts of some cyclic groups of p-p ower order are proven to b e cyclic.The results of this research show that the nonabelian tensor pro duct of cyclic groups of p-p ower order where p is an odd prime with two-sided actions are cyclic.Furthermore,the nonab elian tensor product of cyclic groups of 2-power order with two-sided actions and b oth actions have order greater than two have been proven to b e also cyclic…”
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New Hopf Structures on Binary Trees by Stefan Forcey, Aaron Lauve, Frank Sottile

“…The multiplihedra $\mathcal{M}_{\bullet} = (\mathcal{M}_n)_{n \geq 1}$ form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic combinatorics community, it is nestled between two families of polytopes that certainly are not: the permutahedra $\mathfrak{S}_{\bullet}$ and associahedra $\mathcal{Y}_{\bullet}$. …”
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Topological aspects of brane fields: Solitons and higher-form symmetries by Salvatore D. Pace, Yu Leon Liu

“…The solitons are singularities of the n-brane field, and we classify them using the homotopy theory of $E_n$-algebras. We find that the classification of codimension ${k+1}$ topological solitons with ${k\geq n}$ can be understood using homotopy groups of $\mathcal{E}_n$. …”
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Symmetry-Based Approach to Superconducting Nodes: Unification of Compatibility Conditions and Gapless Point Classifications by Seishiro Ono, Ken Shiozaki

“…While most previous studies are based on the homotopy theory, our theory is on the basis of the symmetry-based analysis of band topology, which enables systematic diagnoses of nodes in all nonmagnetic and magnetic space groups. …”
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The Gysin triangle via localization and A[superscript 1]-homotopy invariance by Van den Bergh, Michel, Trigo Neri Tabuada, Goncalo Jorge

“…Finally, as a third application, we construct explicit bridges relating motivic homotopy theory and mixed motives on the one side with noncommutative mixed motives on the other side. …”
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The algebraic K-theory of the chromatic filtration and the telescope conjecture by Levy, Ishan

“…We develop tools for understanding the algebraic K-theory of categories such as those coming from the chromatic filtration of the stable homotopy category, and apply these tools to improve our understanding of the large scale structure of stable homotopy theory and understand Ravenel's telescope conjecture. …”
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Exact categories, Koszul duality, and derived analytic algebra by Kelly, J

“…</p> <p>We then develop the homotopy theory of algebras in <em>Ch</em>(<em>E</em>). …”