Volume 2016, Issue 720

We study the boundary Lt${L_{t}}$ of the Milnor fiber for the reduced holomorphic germs f:(ℂ3,0)→(ℂ,0)${f:(\mathbb{C}^{3},0)\rightarrow(\mathbb{C},0)}$ having a non-isolated singularity at 0. We prove that Lt${L_{t}}$ is a graph manifold by using a new technique of carrousels depending on one parameter. Our results enable us to compare the topology of Lt${L_{t}}$ and of the link of the normalization of f-1⁢(0)${f^{-1}(0)}$.

Let G be a connected real semisimple Lie group having a finite center and a compact Cartan subgroup T with Lie algebra 𝔱0${\mathfrak{t}_{0}}$. Let ω be a G×T${G\times T}$-invariant symplectic form on X=G×𝔱0${X=G\times\mathfrak{t}_{0}}$. We incorporate Dirac cohomology into the geometric quantization of (X,ω)${(X,\omega)}$ and study the resulting multiplicity-free unitary G×T${G\times T}$-representation on a Hilbert space ℋ⁢(X,ω)${\mathcal{H}(X,\omega)}$. We also perform symplectic reduction of (X,ω)${(X,\omega)}$ and show that our quantization method satisfies the principle “quantization commutes with reduction”. As an application we construct various models of discrete series.

We give a formula for the component at p of the p -adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p -adic approximations to p -integral points on such curves when the Mordell–Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p -integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell–Weil group tensored with ℚ$\mathbb{Q}$.

We provide a general method for finding all natural operations on the Hochschild complex of ℰ${\mathcal{E}}$-algebras, where ℰ${\mathcal{E}}$ is any algebraic structure encoded in a prop with multiplication, as for example the prop of Frobenius, commutative or A∞${A_{\infty}}$-algebras. We show that the chain complex of all such natural operations is approximated by a certain chain complex of formal operations , for which we provide an explicit model that we can calculate in a number of cases. When ℰ${\mathcal{E}}$ encodes the structure of open topological conformal field theories, we identify this last chain complex, up quasi-isomorphism, with the moduli space of Riemann surfaces with boundaries, thus establishing that the operations constructed by Costello and Kontsevich–Soibelman via different methods identify with all formal operations. When ℰ${\mathcal{E}}$ encodes open topological quantum field theories (or symmetric Frobenius algebras) our chain complex identifies with Sullivan diagrams, thus showing that operations constructed by Tradler–Zeinalian, again by different methods, account for all formal operations. As an illustration of the last result we exhibit two infinite families of non-trivial operations and use these to produce non-trivial higher string topology operations, which had so far been elusive.

We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology of rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology.

Compact matrix quantum groups are strongly determined by their intertwiner spaces, due to a result by S. L. Woronowicz. In the case of easy quantum groups (also called partition quantum groups ), the intertwiner spaces are given by the combinatorics of partitions, see the initial work of T. Banica and R. Speicher. The philosophy is that all quantum algebraic properties of these objects should be visible in their combinatorial data. We show that this is the case for their fusion rules (i.e. for their representation theory). As a byproduct, we obtain a unified approach to the fusion rules of the quantum permutation group SN+${S_{N}^{+}}$, the free orthogonal quantum group ON+${O_{N}^{+}}$ as well as the hyperoctahedral quantum group HN+${H_{N}^{+}}$. We then extend our work to unitary easy quantum groups and link it with a “freeness conjecture” of T. Banica and R. Vergnioux.

We solve a long standing problem of the classification of all simple modules with finite-dimensional weight spaces over Lie algebra of vector fields on n -dimensional torus for any n . This generalizes the classical result of O.Mathieu on simple weight modules for the Virasoro algebra (n=1${n=1}$). Every such module is either of a highest weight type or is a quotient of a module of tensor fields on a torus, which was conjectured by Eswara Rao.

We develop linear discretization of complex analysis, originally introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We prove convergence of discrete period matrices and discrete Abelian integrals to their continuous counterparts. We also prove a discrete counterpart of the Riemann–Roch theorem. The proofs use energy estimates inspired by electrical networks.