Let $K/k$ be a finite unramified Galois extension of number fields with Galois group $G$. This determines two homomorphisms from the ideal class group $\mathrm{Cl}_k$ of $k$: the capitulation map $\mathrm{Cl}_k \to \mathrm{Cl}_K$ and the Artin map $\mathrm{Cl}_k \twoheadrightarrow G^{\mathrm{ab}}$ onto the abelianization $G^{\mathrm{ab}}$ of $G$. We call (ker (capitulation), $\mathrm{Cl}_k$, Artin) the capitulation triple of $K / k$.

Artin's transition to group theory shows that any triple $(X, Y, \zeta)$ which arises in this way satisfies the group-theoretic property of being a transfer triple for $G$, defined as follows: there exist a group extension $A \rightarrowtail H \twoheadrightarrow G$ with $A$ finite abelian and an isomorphism $\eta : Y \stackrel{\sim}{\to} H^{\mathrm{ab}}$ such that $\eta (X)$ is the kernel of the transfer homomorphism $H^{\mathrm{ab}}\to A$, and $\zeta$ is the composite of $\eta$ with $H^{\mathrm{ab}}\to G^{\mathrm{ab}}$.

When $G$ is abelian, we show that a triple $(X, Y, \zeta)$ is a transfer triple for $G$ if and only if $|G| X = 0$ and $|G|$ divides $|X|$. Whether all transfer triples for $G$ can be realized arithmetically remains an unsolved problem.

For a braided tensor category ${\cal C}$ and a subcategory ${\cal K}$ there is a notion of a centralizer $C_{\cal C}({\cal K})$, which is a full tensor subcategory of ${\cal C}$. A pre-modular tensor category is known to be modular in the sense of Turaev if and only if the center ${\cal Z}_2({\cal C})\equiv C_{\cal C}({\cal C})$ (not to be confused with the center ${\cal Z}_1$ of a tensor category, related to the quantum double) is trivial, that is, consists only of multiples of the tensor unit, and $\dim{\cal C}\ne 0$. Here $\dim{\cal C} = \sum_i d(X_i)^2$, the $X_i$ being the simple objects.

We prove several structural properties of modular categories. Our main technical tool is the following double centralizer theorem. Let ${\cal C}$ be a modular category and ${\cal K}$ a full tensor subcategory closed with respect to direct sums, subobjects and duals. Then $C_{\cal C}(C_{\cal C}({\cal K})) = {\cal K}$ and $\dim{\cal K}\cdot\dim C_{\cal C}({\cal K}) = \dim{\cal C}$.

We give several applications.

(1) If ${\cal C}$ is modular and ${\cal K}$ is a full modular subcategory, then ${\cal L}=C_{\cal C}({\cal K})$ is also modular and ${\cal C}$ is equivalent as a ribbon category to the direct product: ${\cal C}\simeq{\cal K}\boxtimes{\cal L}$. Thus every modular category factorizes (non-uniquely, in general) into prime modular categories. We study the prime factorizations of the categories $D(G)$-Mod, where $G$ is a finite abelian group.

(2) If ${\cal C}$ is a modular $*$-category and ${\cal K}$ is a full tensor subcategory then $\dim{\cal C}\ge\dim{\cal K}\cdot\dim {\cal Z}_2({\cal K})$. We give examples where the bound is attained and conjecture that every pre-modular ${\cal K}$ can be embedded fully into a modular category ${\cal C}$ with $\dim{\cal C}=\dim{\cal K}\cdot\dim {\cal Z}_2({\cal K})$.

(3) For every finite group $G$ there is a braided tensor $*$-category ${\cal C}$ such that ${\cal Z}_2({\cal C})\simeq\mbox{Rep}\,G$ and the modular closure/modularization $\overline{\cal C}$ is non-trivial.

We first prove that the Whitehead group of a torsion-free virtually solvable linear group vanishes. Next we make a reduction of the fibered isomorphism conjecture from virtually solvable groups to a class of virtually solvable $\mathbb {Q}$-linear groups. Finally we prove an L-theory analogue for elementary amenable groups.

Let $F$ be a $p$-adic field and $K/F$ a finite, cyclic, tamely ramified field extension. We show, without any restriction, that base change from $F$ to K, in the sense of Arthur and Clozel, is compatible with the operation of $K/F$-lifting for simple characters defined by the authors in an earlier paper. This extends and completes the previous work, and represents a substantial step towards the goal of describing tamely ramified base change in terms of the structure theory of representations of Bushnell and Kutzko. We also give a consequence for the Langlands correspondence. The main step in the proof is to show that lifting of simple characters is compatible with automorphic induction, in the sense of Henniart and Herb, through an unramified extension. This is achieved via a new and independently interesting result concerning the conductor of a pair of supercuspidal representations. It gives a complete description of the minima of (a suitably normalized version of) the conductor when one of the representations is fixed. All results concerning the conductor are also valid for non-Archimedean local fields of positive characteristic.

It is shown how many of the partial theta function identities in Ramanujan's lost notebook can be generalized to infinite families of such identities. Key in our construction is the Bailey lemma and a new generalization of the Jacobi triple product identity. By computing residues around the poles of our identities we find a surprising connection between partial theta function identities and Garrett–Ismail–Stanton-type extensions of multisum Rogers–Ramanujan identities.

Let $F:(0, \infty) \times [0, \infty) \rightarrow \mathbb{R}$ be a function of Carathéodory type. We establish the inequality $$ \int_{\mathbb{R}^{N}} F( | x |, u(x) ) dx \leq \int_{\mathbb{R}^{N} } F( | x |, u^{\ast}(x)) dx. $$ where $u^{\ast}$ denotes the Schwarz symmetrization of $u$, under hypotheses on $F$ that seem quite natural when this inequality is used to obtain existence results in the context of elliptic partial differential equations. We also treat the case where $\mathbb{R}^N$ is replaced by a set of finite measure. The identity $$ \int_{\mathbb{R}^{N}} G(u(x)) dx = \int_{\mathbb{R}^{N}} G(u^{\ast}(x)) dx $$ is also discussed under the assumption that $G : [0,\infty) \rightarrow \mathbb{R}$ is a Borel function.

The paper extends to complex Hamiltonian systems previous work of the authors on the Sims extension of the Titchmarsh–Weyl theory for Sturm–Liouville equations with complex potentials, and analyses the spectral properties of associated non-self-adjoint operators.

The concept of generalized functions taking values in a differentiable manifold is extended to a functorial theory. We establish several characterization results which allow a global intrinsic formulation both of the theory of manifold-valued generalized functions and of generalized vector bundle homomorphisms. As a consequence, a characterization of equivalence that does not resort to derivatives (analogous to scalar-valued cases of Colombeau's construction) is achieved. These results are employed to derive a point value description of all types of generalized functions valued in manifolds and to show that composition can be carried out unrestrictedly. Finally, a new concept of association adapted to the present setting is introduced.

The aim of this paper is to provide an analysis of non-linear approximation in the $L_p$-norm $p = d / (d - 1)$ of functions of bounded variation on $\mathbb{R}^d$ with $d > 1$ by polynomials in the Haar system. The exponent $p$ is the natural exponent as it is the correct exponent in the Sobolev inequality. The approximation schemes that we discuss in this paper are mostly related to Haar thresholding and $m$-term approximation. These problems for $d = 2$ are studied in detail in a paper by Cohen, DeVore, Petrushev and Xu. The main aim of this paper is to extend their results to the case $d \geq 2$.

We obtain the optimal order of the $m$-term Haar approximation and prove the stability of Haar thresholding in the BV-norm. As one of the main tools, we establish the boundedness of certain averaging projections in BV.

Our results are of three types. First, we describe a general procedure of adjoining polynomial variables to $A_\infty$-ring spectra whose coefficient rings satisfy certain restrictions. A host of examples of such spectra is provided by killing a regular ideal in the coefficient ring of $MU$, the complex cobordism spectrum. Second, we show that the algebraic procedure of adjoining roots of unity carries over in the topological context for such spectra. Third, we use the developed technology to compute the homotopy types of spaces of strictly multiplicative maps between suitable $K(n)$-localizations of such spectra. This generalizes the famous Hopkins–Miller theorem and gives strengthened versions of various splitting theorems.

Let $K$ be a knot with an unknotting tunnel $\gamma$ and suppose that $K$ is not a $2$-bridge knot. There is an invariant $\rho = p/q \in \mathbb{Q}/2 \mathbb{Z}$, with $p$ odd, defined for the pair $(K, \gamma)$.

The invariant $\rho$ has interesting geometric properties. It is often straightforward to calculate; for example, for $K$ a torus knot and $\gamma$ an annulus-spanning arc, $\rho(K, \gamma) = 1$. Although $\rho$ is defined abstractly, it is naturally revealed when $K \cup \gamma$ is put in thin position. If $\rho \neq 1$ then there is a minimal-genus Seifert surface $F$ for $K$ such that the tunnel $\gamma$ can be slid and isotoped to lie on $F$. One consequence is that if $\rho(K, \gamma) \neq 1$ then $\mathrm{genus}(K) > 1$. This confirms a conjecture of Goda and Teragaito for pairs $(K, \gamma)$ with $\rho(K, \gamma) \neq 1$.