Let $n\in {\mathbb N}$. Houghton's group $H_n$ is the group of permutations of $\{1,\dotsc, n\}\times {\mathbb N}$, that eventually act as a translation in each copy of ${\mathbb N}$. We prove the solvability of the conjugacy problem and conjugator search problem for $H_n$, $n\geq 2$.

We establish two-weight norm inequalities for singular integral operators defined on spaces of homogeneous type. We do so first when the weights satisfy a double bump condition and then when the weights satisfy separated logarithmic bump conditions. Our results generalize recent work on the Euclidean case, but our proofs are simpler even in this setting. The other interesting feature of our approach is that we are able to prove the separated bump results (which always imply the corresponding double bump results) as a consequence of the double bump theorem.

For a Nevanlinna-Pick problem with more than one solution, Rolf Nevanlinna proved that all extremal solutions are inner functions. If the interpolation points are contained in finitely many cones terminating at the unit circle, it is shown that all extremal solutions are Blaschke products.

For $h=3$ and $h=4$ we prove the existence of infinite $B_h$ sequences $\mathcal{B}$ with counting function

$$\mathcal{B}(x)= x^{\sqrt{(h-1)^2+1}-(h-1) + o(1)}.$$

This result extends a construction of I. Ruzsa for $B_2$ sequences.

The problem of reduction of singularities for germs of codimension one foliations in dimension three has been solved by Cano in Reduction of the singularities of codimension one singular foliations in dimension three. The author divides the proof in two steps. The first one consists in getting pre-simple points and the second one is the passage from pre-simple to simple points. In arbitrary dimension of the ambient space the problem is open. In this paper we solve the second step of the problem.

A method is given for resolving a matrix of preference scores into a well-specified mixture of options. This is done in agreement with several desirable properties, including the continuity of the mixing proportions with respect to the preference scores and a condition of compatibility with the Condorcet-Smith majority principle. These properties are achieved by combining the classical rating method of Zermelo with a projection procedure introduced in previous papers of the same authors.

We extend Deligne's notion of determinant functor to Waldhausen categories and (strongly) triangulated categories. We construct explicit universal determinant functors in each case, whose target is an algebraic model for the $1$-type of the corresponding $K$-theory spectrum. As applications, we answer open questions by Maltsiniotis and Neeman on the $K$-theory of (strongly) triangulated categories and a question of Grothendieck to Knudsen on determinant functors. We also prove additivity theorems for low-dimensional $K$-theory of (strongly) triangulated categories and obtain generators and (some) relations for various $K_{1}$-groups. This is achieved via a unified theory of determinant functors which can be applied in further contexts, such as derivators.

We study holomorphic maps of $\mathbf{C}^2$ tangent to the identity at a fixed point which have degenerate characteristic directions. With the help of some new invariants, we give sufficient conditions for the existence of attracting domains in these degenerate characteristic directions.

We develop a theory of 'non-uniformly local' tent spaces on metric measure spaces. As our main result, we give a remarkably simple proof of the atomic decomposition.