Given a finite commutative monoid M, we show that submonoids of M× [n] - where [n] = {0,1,…,n} is equipped with the max operation ∨ - may be enumerated via the transfer matrix method. When M is also idempotent, we show that there are finitely many integers λ and rational numbers b_λ (only depending on M) such that the number of submonoids of M× [n] is ∑_λb_λλ^n. This answers a question of Knuth regarding ternary (and higher order) max-closed relations, and has applications to the enumeration of saturated transfer systems in equivariant infinite loop space theory.

We prove that Hill's characteristic function $\chi$ for transfer systems on a lattice $P$ surjects onto interior operators for $P$. Moreover, the fibers of $\chi$ have unique maxima which are exactly the saturated transfer systems. In order to apply this theorem in examples relevant to equivariant homotopy theory, we develop the theory of saturated transfer systems on modular lattices, ultimately producing a ``matchstick game'' that puts saturated transfer systems in bijection with certain structured subsets of covering relations. After an interlude developing a recursion for transfer systems on certain combinations of bounded posets, we apply these results to determine the full lattice of transfer systems for rank two elementary Abelian groups.

We identify the motivic $KGL/2$-local sphere as the fiber of $\psi^3-1$ on $(2,\eta)$-completed Hermitian $K$-theory, over any base scheme containing $1/2$. This is a motivic analogue of the classical resolution of the $K(1)$-local sphere, and extends to a description of the $KGL/2$-localization of an arbitrary motivic spectrum. Our proof relies on a novel conservativity argument that should be of broad utility in stable motivic homotopy theory.

We provide a general recursive method for constructing transfer systems on finite lattices. Using this we calculate the number of homotopically distinct $N_\infty$ operads for dihedral groups $D_{p^n}$, $p > 2$ prime, and cyclic groups $C_{qp^n}$, $p \neq q$ prime. We then further display some of the beautiful combinatorics obtained by restricting to certain homotopically meaningful $N_\infty$ operads for these groups.

We perform Hochschild homology calculations in the algebro-geometric setting of motives over algebraically closed fields. The homotopy ring of motivic Hochschild homology contains torsion classes that arise from the mod- p p motivic Steenrod algebra and generating functions defined on the natural numbers with finite non-empty support. Under Betti realization, we recover Bökstedt’s calculation of the topological Hochschild homology of finite prime fields.

Communities summer conference Homotopical Combinatorics, one of four topical research conferences offered this year that are focused on collaborative research and professional development for early-career mathematicians.Additional information can be found at https://www.

We provide a general recursive method for constructing transfer systems on finite lattices. Using this, we calculate the number of homotopically distinct N-infinity operads for dihedral groups D(p)n , p > 2 prime, and cyclic groups C(qp)n , p not equal q prime. We then further display some of the beautiful combinatorics obtained by restricting to certain homotopically meaningful N-infinity operads for these groups.

We investigate the rich combinatorial structure of premodel structures on finite lattices whose weak equivalences are closed under composition. We prove that there is a natural refinement of the inclusion order of weak factorization systems so that the intervals detect these composition closed premodel structures. In the case that the lattice in question is a finite total order, this natural order retrieves the Kreweras lattice of noncrossing partitions as a refinement of the Tamari lattice, and model structures can be identified with certain tricolored trees.

We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics . In the case of a finite total order [ n ], we enumerate all model structures, exhibiting a rich combinatorial structure encoded by Shapiro’s Catalan triangle. This is an application of previous work of the authors on the theory of N_∞ -operads for cyclic groups of prime power order, along with new structural insights concerning extending choices of certain model structures on subcategories of [ n ].

We isolate a class of groups -- called lossless groups -- for which homotopy classes of $G$-$N_\infty$ operads are in bijection with certain restricted transfer systems on the poset of conjugacy classes $\operatorname{Sub}(G)/G$.

We provide a general recursive method for constructing transfer systems on finite lattices. Using this we calculate the number of homotopically distinct N_∞ operads for dihedral groups D_p^n, p > 2 prime, and cyclic groups C_qp^n, p ≠ q prime. We then further display some of the beautiful combinatorics obtained by restricting to certain homotopically meaningful N_∞ operads for these groups.

Transfer systems are combinatorial objects which classify $N_\infty$ operads up to homotopy. By results of A. Blumberg and M. Hill, every transfer system associated to a linear isometries operad is also saturated (closed under a particular two-out-of-three property). We investigate saturated and linear isometric transfer systems with equivariance group $C_{p^mq^n}$, the cyclic group of order $p^mq^n$ for $p,q$ distinct primes and $m,n\ge 0$. We give a complete enumeration of saturated transfer systems for $C_{p^mq^n}$. We also prove J. Rubin's saturation conjecture for $C_{pq^n}$; this says that every saturated transfer system is realized by a linear isometries operad for $p,q$ sufficiently large (greater than $3$ in this case).

For a finite group G, we introduce the complete suboperad $Q_G$ of the categorical G-Barratt-Eccles operad $P_G$. We prove that $P_G$ is not finitely generated, but $Q_G$ is finitely generated and is a genuine $E_\infty$ G-operad (i.e., it is $N_\infty$ and includes all norms). For G cyclic of order 2 or 3, we determine presentations of the object operad of $Q_G$ and conclude with a discussion of algebras over $Q_G$, which we call biased permutative equivariant categories.

We compute the homotopy groups of the η-periodic motivic sphere spectrum over a finite-dimensional field k with characteristic not 2 and in which -1 a sum of four squares. We also study the general characteristic 0 case and show that the η-periodic slice spectral sequence over Q determines the η-periodic slice spectral sequence over all extensions of Q. This leads to a speculation on the role of a "connective Witt-theoretic J-spectrum" in η-periodic motivic homotopy theory.

1. Amuse-gueule 2 2. Some linear algebra 3 3. The quadratic square: quadratic forms, symmetric matrices, quadratic spaces, and symmetric bilinear forms 5 4. Equivalence, congruence, and isometry 7 5. Regular forms 8 6. Diagonalization of forms 10 7. Hyperbolic spaces 16 8. Witt decomposition and cancellation 18 9. Chain equivalence 21 10. Tensor product of vector spaces and quadratic forms 22 11. Group completion 25 12. The Witt and Grothendieck-Witt rings 29 13. The I-adic filtration of the Witt ring and the “dimterminant” homomorphism 31 14. First computations of Witt and Grothendieck–Witt rings 33 15. Presentations of the Witt and Grothendieck–Witt rings 35 16. Orderings and signatures 37 17. Total signature and Pfister’s local-global principle 40 18. Pfister forms 41 19. Multiplicative forms 43 20. A glimpse into function fields and the Hauptsatz 45 21. Quaternion algebras 47 22. Local fields 53 23. Hilbert reciprocity 58 References 61

We analyze the ring tmf_*tmf of cooperations for the connective spectrum of topological modular forms (at the prime 2) through a variety of perspectives: (1) the E_2-term of the Adams spectral sequence for tmf ^ tmf admits a decomposition in terms of Ext groups for bo-Brown-Gitler modules, (2) the image of tmf_*tmf in the rationalization of TMF_*TMF admits a description in terms of 2-variable modular forms, and (3) modulo v_2-torsion, tmf_*tmf injects into a certain product of copies of TMF_0(N)_*, for various values of N. We explain how these different perspectives are related, and leverage these relationships to give complete information on tmf_*tmf in low degrees. We reprove a result of Davis-Mahowald-Rezk, that a piece of tmf ^ tmf gives a connective cover of TMF_0(3), and show that another piece gives a connective cover of TMF_0(5). To help motivate our methods, we also review the existing work on bo_*bo, the ring of cooperations for (2-primary) connective K-theory, and in the process give some new perspectives on this classical subject matter.

Let F be a field of characteristic different than 2. We establish surjectivity of Balmer's comparison map rho^* from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor-Witt K-theory. We also comment on the tensor triangular geometry of compact cellular motivic spectra, producing in particular novel field spectra in this category. We conclude with a list of questions about the structure of the tensor triangular spectrum of the stable motivic homotopy category.

We determine systematic vanishing regions for the bigraded homotopy sheaves of the motivic sphere spectrum over a field of characteristic different from two.

In previous work, the authors constructed and studied a lift of the Galois correspondence to stable homotopy categories. In particular, if $L/k$ is a finite Galois extension of fields with Galois group $G$, there is a functor $c_{L/k}^*$ from the $G$-equivariant stable homotopy category to the stable motivic homotopy category over $k$ such that $c_{L/k}^*(G/H_+) = Spec(L^H)_+$. We proved that when $k$ is a real closed field and $L=k[i]$, the restriction of $c_{L/k}^*$ to the $\eta$-complete subcategory is full and faithful. Here we "uncomplete" this theorem so that it applies to $c_{L/k}^*$ itself. Our main tools are Bachmann's theorem on the $(2,\eta)$-periodic stable motivic homotopy category and an isomorphism range for the map on bigraded stable stems induced by $C_2$-equivariant Betti realization.

For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and eta (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full and faithful embedding after eta-completion if a motivic version of Serre's finiteness theorem is valid. We produce strong necessary conditions on the field extension L/k for this functor to be full and faithful. Along the way, we produce several results on the stable C_2-equivariant Betti realization functor and prove convergence theorems for the p-primary C_2-equivariant Adams spectral sequence.