1.
Deformation bicomplex of module algebras - Yau, Donald
The deformation bicomplex of a module algebra over a bialgebra is constructed. It
is then applied to study algebraic deformations in which both the module
structure and the algebra structure are deformed. The cases of module
coalgebras, comodule (co)algebras, and (co)module bialgebras are also
considered.

2.
On affine morphisms of Hopf algebroids - Powell, Geoffrey M.L.
The paper considers the FPQC stacks which are associated to affine groupoid
schemes. Using a formulation of a descent datum in terms of morphisms of affine
groupoid schemes, explicit arguments are given which avoid appeal to the general
principle of faithfully flat descent. This theory is applied to consider the
notion of affine morphism.

3.
The Euler characteristic of a category as the sum of a divergent series - Berger, Clemens; Leinster, Tom
The Euler characteristic of a cell complex is often thought of as the alternating
sum of the number of cells of each dimension. When the complex is infinite, the
sum diverges. Nevertheless, it can sometimes be evaluated; in particular, this
is possible when the complex is the nerve of a finite category. This provides an
alternative definition of the Euler characteristic of a category, which is in
many cases equivalent to the original one.

4.
Excision for $K$-theory of connective ring spectra - Dundas, Bjørn Ian; Kittang, Harald Øyen
We extend Geisser and Hesselholt’s result on “bi-relative $K$-theory”
from discrete rings to connective ring spectra. That is,
if $\mathcal{A}$ is a homotopy cartesian n-cube of ring spectra (satisfying
connectivity hypotheses), then the $(n + 1)$-cube induced by the
cyclotomic trace
$$K(\mathcal{A}) \to TC(\mathcal{A})$$
is homotopy cartesian after profinite completion. In other words,
the fiber of the profinitely completed cyclotomic trace satisfies
excision.

5.
Splittings in the Burnside ring and in $SF_G$ - French, Christopher P.
Let $G$ be a finite $p$-group, $p \neq 2$. We construct a map from the space
$J_G$, defined as the fiber of $\psi^k-1: B_G O \to B_G Spin$, to the space
$(SF_G)_p$, defined as the 1-component of the zeroth space of the equivariant
$p$-complete sphere spectrum. Our map produces the same splitting of the
$G$-connected cover of $(SF_G)_p$ as we have described in previous work, but it
also induces a natural splitting of the $p$-completions of the component groups
of fixed point subspaces.

6.
Hopf-Hochschild (co)homology of module algebras - Kaygun, Atabey
We define a version of Hochschild homology and cohomology suitable for a class of
algebras admitting compatible actions of bialgebras, called module algebras. We
show that this (co)homology, called Hopf-Hochschild (co)homology, can also be
defined as a derived functor on the category of representations of an
equivariant analogue of the enveloping algebra of a crossed product algebra. We
investigate the relationship of our theory with Hopf cyclic cohomology and also
prove Morita invariance of the Hopf-Hochschild (co)homology.

7.
Homotopy types of truncated projective resolutions - Mannan, W. H.
We work over an arbitrary ring $R$. Given two truncated projective
resolutions of equal length for the same module, we consider their underlying
chain complexes. We show they may be stabilized by projective modules to obtain
a pair of complexes of the same homotopy type.

8.
Cofibrations in the category of Frölicher spaces: Part
I - Dugmore, Brett; Ntumba, Patrice Pungu
Cofibrations are defined in the category of Frölicher spaces
by weakening the analog of the classical definition to enable
smooth homotopy extensions to be more easily constructed,
using flattened unit intervals. We later relate smooth cofibrations
to smooth neighborhood deformation retracts. The
notion of smooth neighborhood deformation retract gives rise
to an analogous result that a closed Frölicher subspace A of
the Frölicher space $X$ is a smooth neighborhood deformation
retract of $X$ if and only if the inclusion $i : A \hookrightarrow X$ comes from a
certain subclass of cofibrations. As an application we construct
the right Puppe sequence.

9.
Model structure on operads in orthogonal spectra - Kro, Tore August
We generalize Berger and Moerdijk's results on axiomatic homotopy theory for
operads to the setting of enriched symmetric monoidal model categories, and show
how this theory applies to orthogonal spectra. In particular, we provide a
symmetric fibrant replacement functor for the positive stable model
structure.

10.
Beyond the hit problem: Minimal presentations of
odd-primary Steenrod modules, with application to CP($\infty) and BU - Pengelley, David J.; Williams, Frank
We describe a minimal unstable module presentation over the Steenrod algebra for
the odd-primary cohomology of infinite-dimensional complex projective space and
apply it to obtain a minimal algebra presentation for the cohomology of the
classifying space of the infinite unitary group. We also show that there is a
unique Steenrod module structure on any unstable cyclic module that has
dimension one in each complex degree (half the topological degree) with a fixed
alpha-number (sum of 'digits') and is zero in other degrees.

11.
A statistical approach to persistent homology - Bubenik, Peter; Kim, Peter T.
Assume that a finite set of points is randomly sampled from a subspace of a
metric space. Recent advances in computational topology have provided several
approaches to recovering the geometric and topological properties of the
underlying space. In this paper we take a statistical approach to this problem.
We assume that the data is randomly sampled from an unknown probability
distribution. We define two filtered complexes with which we can calculate the
persistent homology of a probability distribution. Using statistical estimators
for samples from certain families of distributions, we show that we can recover
the persistent homology of the underlying distribution.

12.
A cohomological interpretation of Brion's formula - Hüttermann, Thomas
A subset $P$ of $mathbb{R}^n$ gives rise to a formal Laurent series with
monomials corresponding to lattice points in $P$ . Under suitable
hypotheses, this series represents a rational function $R(P)$;
this happens, for example, when $P$ is bounded in which case
$R(P)$ is a Laurent polynomial. Michel Brion [2] has discovered
a surprising formula relating the Laurent polynomial $R(P)$ of a
lattice polytope $P$ to the sum of rational functions corresponding
to the supporting cones subtended at the vertices of $P$ . The
result is re-phrased and generalised in the language of cohomology
of line bundles on complete toric varieties. Brion's formula
is the special case of an ample...

13.
Heller triangulated categories - Künzer, Matthias
Let $\mathcal{E}$ be a Frobenius category. Let $\underset {=} {\mathcal{E}}$ denote its stable category.
The shift functor on $\underline {E}$ induces, by pointwise application,
an inner shift functor on the category of acyclic complexes
with entries in $\underset {=} {\mathcal{E}}$. Shifting a complex by 3 positions yields an
outer shift functor on this category. Passing to quotient modulo
split acyclic complexes, Heller remarked that inner and
outer shift become isomorphic, via an isomorphism satisfying
yet a further compatibility. Moreover, Heller remarked that
a choice of such an isomorphism determines a Verdier triangulation
on $\underset {=} {\mathcal{E}}$, except for the octahedral axiom. We generalise the
notion of acyclic complexes such that...

14.
A chain coalgebra model for the James map - Hess, Kathryn; Parent, Paul-Eugène; Scott, Jonathan
Let $EK$ be the simplicial suspension of a pointed simplicial
set $K$. We construct a chain model of the James map,
$\alpha_K : CK \to \Omega CEK$. We compute the cobar diagonal on
$\Omega CEK$, not assuming that $EK $is 1-reduced, and show that
$\alpha_K$ is comultiplicative. As a result, the natural isomorphism
of chain algebras $TCK \cong \Omega CK$ preserves diagonals.
In an appendix, we show that the Milgram map,
$\Omega (A \otimes B) \to \Omega A \otimes \Omega B$, where $A$ and $B$ are coaugmented
coalgebras, forms part of a strong deformation retract of
chain complexes. Therefore, it is a chain equivalence even
when $A$ and $B$ are...

15.
$DG$-models of projective modules and Nakajima quiver
varieties - Eshmatov, Farkhod
Associated to each finite subgroup $\Gamma$ of ${\tt SL}_2(\mathbb{C})$ there is a family
of noncommutative algebras $O^\tau(\Gamma)$, which is a deformation of
the coordinate ring of the Kleinian singularity $\mathbb{C}^2/\Gamma$. We study
finitely generated projective modules over these algebras. Our
main result is a bijective correspondence between the set of isomorphism
classes of rank one projective modules over $O^\tau$ and
a certain class of quiver varieties associated to $\Gamma$s. We show that
this bijection is naturally equivariant under the action of a “large”
Dixmier-type automorphism group $G$. Our construction leads to
a completely explicit description of ideals of the algebras $O^\tau$ .

16.
Thick subcategories of the derived category of a hereditary
algebra - Brüning, Kristian
We classify thick subcategories of the bounded derived category of a hereditary
abelian category A in terms of subcategories of A. The proof can
be applied to characterize the localizing subcategories of the full derived
category of A. As an application we prove an algebraic analog of the
telescope conjecture for the derived category of a representation finite
hereditary artin algebra.

17.
Erratum to `Category of $A_\infty$-categories' - Lyubashenko, Volodymyr
The erroneous statement (HHA 5 (2003), no. 1, 1–48) that
the collection of unital $A_\infty$-categories, all $A_\infty$-functors, and all
$A_\infty$-transformations (resp. equivalence classes of natural $A_\infty$-
transformations) form a $\mathcal{K}-2$-category $\mathcal{K}^u A_\infty$(resp. ordinary
2-category $^u A_\infty$) is corrected as follows. All 2-category axioms
are satisfied, except that $1_e \cdot f$ does not necessarily equal $1_{ef}$
for all composable 1-morphisms $e, f$. The axiom $e \cdot 1_f = 1_{ef}$
does hold. The mistake does not affect results on invertible
2-morphisms and quasi-invertible 1-morphisms in $^u A_\infty$.

18.
A few localisation theorems - Kahn, Bruno; Sujatha, R.
Given a functor $T : \mathcal{C} \to \mathcal{D}$ carrying a class of morphisms
$S \subset \mathcal C$ into a class $S^\prime \subset \mathcal{D}$, we give sufficient conditions for $T$
to induce an equivalence on the localised categories. These conditions
are in the spirit of Quillen’s Theorem A. We give some
applications in algebraic and birational geometry.

19.
From loop groups to 2-groups - Baez, John C.; Stevenson, Danny; Crans, Alissa S.; Schreiber, Urs
We describe an interesting relation between Lie 2-algebras,
the Kac-Moody central extensions of loop groups, and the
group String($n$). A Lie 2-algebra is a categorified version of
a Lie algebra where the Jacobi identity holds up to a natural
isomorphism called the 'Jacobiator.' Similarly, a Lie 2-group is
a categorified version of a Lie group. If $G$
is a simply-connected
compact simple Lie group, there is a 1-parameter family of Lie
2-algebras $\mathfrak{g}_k$ each having $\mathfrak{g}$ as its Lie algebra of objects, but
with a Jacobiator built from the canonical 3-form on $G$. There
appears to be no Lie 2-group having $\mathfrak{g}_\mathcal{k}$ as its Lie 2-algebra,
except when $k = 0$....

20.
On higher nil groups of group rings - Juan-Pineda, Daniel
Let $G$ be a finite group and $mathbb{Z}[G]$ its integral group ring.
We prove that the nil groups $N^j K_2 \mathbb{Z}[G])$ do not vanish for
all $j \geq 1$ and for a large class of finite groups. We obtain from
this that the iterated nil groups $N^j K_i (\mathbb{Z}[G])$ are also nonzero
for all $i \geq 2, j \geq i - 1$