For a division ring $D$, denote by $\mathcal M_D$ the $D$-ring
obtained as the completion of the direct limit $\varinjlim_n
M_{2^n}(D)$
with respect to the metric induced by its unique rank function.
We prove that, for any ultramatricial $D$-ring $\mathcal B$ and
any
non-discrete extremal pseudo-rank function $N$ on $\mathcal B$,
there is an isomorphism of $D$-rings $\overline{\mathcal B} \cong \mathcal
M_D$, where $\overline{\mathcal B}$ stands
for the completion of $\mathcal B$ with respect to the pseudo-metric
induced by $N$.
This generalizes a result of von Neumann. We also show a corresponding
uniqueness result for $*$-algebras over fields $F$ with positive
definite involution, where the
algebra $\mathcal M_F$ is endowed with its natural involution
coming from the $*$-transpose involution on each of the factors
$M_{2^n}(F)$.
Given a link $L$, the Blanchfield pairing $\operatorname{Bl}(L)$ is a pairing
which is defined on the torsion submodule of the Alexander module
of $L$. In some particular cases, namely if $L$ is a boundary
link or if the Alexander module of $L$ is torsion, $\operatorname{Bl}(L)$
can be computed explicitly; however no formula is known in general.
In this article, we compute the Blanchfield pairing of any link,
generalizing the aforementioned results. As a corollary, we obtain
a new proof that the Blanchfield pairing is hermitian. Finally,
we also obtain short proofs of several properties of $\operatorname{Bl}(L)$.
We define Schwartz functions, tempered functions and tempered
distributions on (possibly singular) real algebraic varieties.
We prove that all classical properties of these spaces, defined
previously on affine spaces and on Nash manifolds, also hold
in the case of affine real algebraic varieties, and give partial
results for the non-affine case.
Order three elements in the exceptional groups of type $G_2$
are classified up to conjugation over arbitrary fields. Their
centralizers are computed, and the associated classification
of idempotents in symmetric composition algebras is obtained.
Idempotents have played a key role in the study and classification
of these algebras.
Over an algebraically closed field, there are two conjugacy classes
of order three elements in $G_2$ in characteristic not $3$ and
four of them in characteristic $3$. The centralizers in characteristic
$3$ fail to be smooth for one of these classes.
We prove many simultaneous congruences mod 2 for elliptic and
Hilbert modular forms
among forms with different Atkin--Lehner eigenvalues. The proofs
involve the notion of quaternionic $S$-ideal classes and the
distribution of Atkin--Lehner signs among
newforms.
We prove that digital sequences modulo $m$ along squares are
normal,
which covers some prominent sequences like the sum of digits
in base $q$ modulo $m$, the Rudin-Shapiro sequence and some generalizations.
This gives, for any base, a class of explicit normal numbers
that can be efficiently generated.
We define a homology theory of virtual links built out of the
direct sum of the standard Khovanov complex with itself, motivating
the name doubled Khovanov homology. We demonstrate that
it can be used to show that some virtual links are non-classical,
and that it yields a condition on a virtual knot being the connect
sum of two unknots. Further, we show that doubled Khovanov homology
possesses a perturbation analogous to that defined by Lee in
the classical case and define a doubled Rasmussen invariant.
This invariant is used to obtain various cobordism obstructions;
in particular it is an obstruction to sliceness. Finally, we
show that the doubled Rasmussen invariant contains the odd writhe
of a virtual knot, and use this to show that knots with non-zero
odd writhe are not slice.
Let $\mathcal{C}$ be a curve of genus at least $2$ embedded in $E_1
\times \cdots \times E_N$ where the $E_i$ are elliptic curves
for $i=1,\dots, N$. In this article we give an explicit sharp
bound for the Néron-Tate height of the points of $\mathcal{C}$ contained
in the union of all algebraic subgroups of dimension
$\lt \max(r_\mathcal{C}-t_\mathcal{C},t_\mathcal{C})$
where $t_\mathcal{C}$, respectively $r_\mathcal{C}$, is the minimal dimension
of a translate, respectively of a torsion variety, containing
$\mathcal{C}$.
As a corollary, we give an explicit bound for the height of
the rational points of special curves, proving new cases of
the explicit Mordell Conjecture and in particular making explicit
(and slightly more general in the CM case) the Manin-Dem'janenko
method in products of elliptic curves.