It is shown that the coniveau filtration on the cohomology
of smooth projective varieties is preserved up to shift
by pushforwards, pullbacks and products.

We show that there exists a closed infinite dimensional subspace
of $H^\infty(B^n)$ such that every function of norm one is
universal for some sequence of automorphisms of $B^n$.

A relation between the anticyclic structure of the dendriform operad
and the Coxeter transformations in the Grothendieck groups of the
derived categories of modules over the Tamari posets is obtained.

We prove that every real
algebraic integer $\alpha$ is expressible by a
difference of two Mahler measures of integer polynomials.
Moreover, these polynomials can be chosen in such a way that they
both have the same degree as that of $\alpha$, say
$d$, one of these two polynomials is irreducible and
another has an irreducible factor of degree $d$, so
that $\alpha=M(P)-bM(Q)$ with irreducible polynomials
$P, Q\in \mathbb Z[X]$ of degree $d$ and a
positive integer $b$. Finally, if $d \leqslant 3$, then one can take $b=1$.

Given an odd surjective Galois representation $\varrho\from \G_\Q\to\PGL_2(\F_3)$ and a
positive integer~$N$, there exists a twisted modular curve $X(N,3)_\varrho$
defined over $\Q$ whose rational points classify the quadratic $\Q$-curves of degree $N$
realizing~$\varrho$. This paper gives a method to provide an explicit plane quartic model for
this curve in the genus-three case $N=5$.

Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times
\SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional
$CW$-complex of the homotopy type of an $n$-sphere. We study the
automorphism group $\Aut (G)$ in order to compute the number of
distinct homotopy types of spherical space forms with respect to free
and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$
is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self
homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with
free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as
well.

We prove that the elliptic surface
$y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1)$ has geometric Mordell--Weil
rank $15$. This completes a list of Kuwata, who gave explicit examples
of elliptic $K3$-surfaces with geometric Mordell--Weil ranks
$0,1,\dots, 14, 16, 17, 18$.

Let $A$ be a stable, separable, real rank zero $C^{*}$-algebra, and
suppose that $A$ has an AF-skeleton with only finitely many extreme
traces.
Then the corona algebra ${\mathcal M}(A)/A$ is
purely infinite in the sense of Kirchberg and R\o rdam, which implies that
$A$ has the corona factorization property.

The original Sato--Tate Conjecture concerns the angle distribution
of the eigenvalues arising from non-CM elliptic curves. In this paper,
we formulate a modular analogue of the Sato--Tate Conjecture and prove
that the angles arising from non-CM holomorphic Hecke
eigenforms with non-trivial central characters are not distributed
with respect to the Sate--Tate measure
for non-CM elliptic curves. Furthermore, under a reasonable conjecture,
we prove that the expected distribution is uniform.

Dans le papier Beyond Endoscopy une id\'ee pour entamer la
fonctorialit\'e en utilisant la formule des traces a \'et\'e
introduite. Maints probl\`emes, l'existence d'une limite convenable
de la formule des traces, est eqquiss\'ee dans cette note
informelle mais seulement pour $GL(2)$ et les corps des fonctions
rationelles sur un corps fini et en ne pas resolvant
bon nombre de questions.

Using ideas of S. Wassermann on non-exact $C^*$-algebras and
property T groups, we show that one of his examples of non-invertible
$C^*$-extensions is not semi-invertible. To prove this, we
show that a certain element vanishes in the asymptotic tensor
product. We also show that a modification of the example gives
a $C^*$-extension which is not even invertible up to homotopy.

In his last letter to Hardy, Ramanujan
defined 17 functions $F(q)$, where $|q|<1$. He called them mock theta
functions, because as $q$ radially approaches any point $e^{2\pi ir}$
($r$ rational), there is a theta function $F_r(q)$ with $F(q)-F_r(q)=O(1)$.
In this paper we establish the relationship between two families of mock
theta functions.

We prove Beurling's theorem for rank $1$ Riemannian symmetric
spaces and relate its consequences with the characterization of
the heat kernel of the symmetric space.

Let $p$ be a prime greater than or equal to 17 and
congruent to
2 modulo 3. We use results of Beukers and Helou on
Cauchy--Liouville--Mirimanoff
polynomials to show that
the intersection of the Fermat curve of degree $p$ with the
line $X+Y=Z$ in the projective plane
contains no algebraic points of degree
$d$ with $3 \leq d \leq 11$.
We prove a result on
the roots of these polynomials and show that, experimentally,
they seem to satisfy
the conditions of a mild extension of
an irreducibility theorem of P\'{o}lya and Szeg\"{o}.
These conditions are conjecturally
also necessary for irreducibility.