We provide a brief description of the mathematics that led to Daniel Quillen's introduction of model categories, a summary of his seminal work “Homotopical algebra”, and a brief description of some of the developments in the field since.

We prove the injective stability theorem for unitary K1 under the usual stable range condition on the ground ring. This improves the stability theorem of A. Bak, V. Petrov and G. Tang where a stronger Λ-stable range condition was used.

Let X be a smooth projective variety over a finite field . We discuss the unramified cohomology group H3nr(X, ℚ/ℤ(2)). Several conjectures put together imply that this group is finite. For certain classes of threefolds, H3nr(X, ℚ/ℤ(2)) actually vanishes. It is an open question whether this holds for arbitrary threefolds. For a threefold X equipped with a fibration onto a curve C, the generic fibre of which is a smooth projective surface V over the global field (C), the vanishing of H3nr(X, ℚ/ℤ(2)) together with the Tate conjecture for divisors on X implies a local-global principle of Brauer–Manin type for the Chow group of zero-cycles on V. This sheds new light on work started thirty years ago.

Let U/ℂ be a smooth quasi-projective variety of dimension d, CHr (U,m) Bloch's higher Chow group, and

clr,m: CHr (U,m) ⊗ ℚ → homMHS (ℚ(0), H2r−m (U, ℚ(r)))

the cycle class map. Beilinson once conjectured clr,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of clr,m in more detail (as well as at the “generic point” of U) in terms of kernels of Abel-Jacobi mappings. When r = m, we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of clm,m at the generic point is the same for integral or rational coefficients.

Let k be a field with resolution of singularities, and X a separated k-scheme of finite type with structure map g. We show that the slice filtration in the motivic stable homotopy category commutes with pullback along g. Restricting the field further to the case of characteristic zero, we are able to compute the slices of Weibel's homotopy invariant K-theory [24] extending the result of Levine [10], and also the zero slice of the sphere spectrum extending the result of Levine [10] and Voevodsky [23]. We also show that the zero slice of the sphere spectrum is a strict cofibrant ring spectrum HZXsf which is stable under pullback and that all the slices have a canonical structure of strict modules over HZXsf. If we consider rational coefficients and assume that X is geometrically unibranch then relying on the work of Cisinski and Déglise [4], we deduce that the zero slice of the sphere spectrum is given by Voevodsky's rational motivic cohomology spectrum HZX ⊗ ℚ and that the slices have transfers. This proves several conjectures of Voevodsky [22, conjectures 1, 7, 10, 11] in characteristic zero.

Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence.

We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological K0-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory.

We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordism theory.

In this short paper we try to describe the fundamental contribution of Quillen in the development of abstract homotopy theory and we explain how he uses this theory to lay the foundations of rational homotopy theory.

We use Godeaux–Serre varieties of finite groups, projective representation theory, the twisted Atiyah–Segal completion theorem, and our previous work on the topological period-index problem to compute the étale index of Brauer classes α ∈ Brét(X) in some specific examples. In particular, these computations show that the étale index of α differs from the period of α in general. As an application, we compute the index of unramified classes in the function fields of high-dimensional Godeaux–Serre varieties in terms of projective representation theory.

Let p be a prime and F a field of characteristic different from p. Suppose all p-primary roots of unity are contained in F. Let α ∈ pBr(F) which has a cyclic splitting field. We prove that γi(α) = 0 for all i ≥ 2, where γi : pBr(F) → K2i(F)/pK2i(F) are the divided power operations of degree p. We also show that if char F ≠ 2, √−1 ∈ F*. D ∈2 Br(F), indD = 8 and a ∈ F* such that ind DF(√a) = 4, then γ3(D) = {a,s}γ2(D) for some s ∈ F*. Consequently, we prove that if D, considered as a division algebra, has a subfield of degree 4 of certain type, then γ3(D) = 0. At the end of the paper we pose a few open questions.

We develop a suitable version of the stable module category of a finite group G over an arbitrary commutative ring k. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the finitely presented kG-modules. The main idea is to form a localisation of the usual version of the stable module category with respect to the filtered colimits of weakly injective modules. There is also an analogous version of the homotopy category of weakly injective kG-modules and a recollement relating the stable category, the homotopy category, and the derived category of kG-modules.

In 1969 Quillen discovered a deep connection between complex cobordism and formal group laws which he announced in [Qui69]. Algebraic topology has never been the same since. We will describe the content of [Qui69] and then discuss its impact on the field. This paper is a writeup of a talk on the same topic given at the Quillen Conference at MIT in October 2012. Slides for that talk are available on the author's home page.

We show that for a field k, algebraic over ℚ, and F a field of characteristic zero, the triangular spectra of mixed Tate motives (resp. mixed Artin-Tate motives), denoted by DMT(k)F (resp. DMAT(k)F), are both isomorphic to spec(F). We further compute the Picard groups of DMT(k)F and DMAT(k)ℚ.

We give a brief introduction to two fundamental papers by Daniel Quillen appearing in the Annals, 1971. These papers established the foundations of equivariant cohomology and gave a qualitative description of the cohomology of an arbitrary finite group. We briefly describe some of the influence of these seminal papers in the study of cohomology and representations of finite groups, restricted Lie algebras, and related structures.

Let B = A[[t;σ,δ]] be a skew power series ring such that σ is given by an inner automorphism of B. We show that a certain Waldhausen localisation sequence involving the K-theory of B splits into short split exact sequences. In the case that A is noetherian we show that this sequence is given by the localisation sequence for a left denominator set S in B. If B = ℤp[[G]] happens to be the Iwasawa algebra of a p-adic Lie group G ≅ H ⋊ ℤp, this set S is Venjakob's canonical Ore set. In particular, our result implies that

is split exact for each n ≥ 0. We also prove the corresponding result for the localisation of ℤp[[G]][] with respect to the Ore set S*. Both sequences play a major role in non-commutative Iwasawa theory.

We establish a presentation for K1 of any small exact category P, based on the notion of “mirror image sequence,” originally introduced by Grayson in 1979; as part of the proof, we show that every element of K1(P) arises from a mirror image sequence. This provides an alternative to Nenashev's presentation in terms of “double short exact sequences.”

We address the question of lifting the étale unipotent fundamental group of curves to the level of algebraic cycles and show that a sequence of algebraic cycles whose sum satisfies the Maurer-Cartan equation would do the job. For any elliptic curve with the origin removed and the curve , we construct such a sequence of algebraic cycles whose image under the cycle map gives rise to the étale unipotent fundamental group of the curve.

In the 1960's and 1970's, the Adams Conjecture figured prominently both in homotopy theory and in geometric topology. Quillen sketched one way to attack the conjecture and then proved it with an entirely different line of argument. Both of his approaches led to spectacular and beautiful new mathematics.

We study self-duality of Grothendieck's blended extensions in the context of a tannakian category. The set of equivalence classes of symmetric, resp. antisymmetric, blended extensions is naturally endowed with a torsor structure, which enables us to compute the unipotent radical of the associated monodromy groups in various situations.

We survey the genesis and development of higher algebraic K-theory by Daniel Quillen.