\'Etale triviality of finite equivariant vector bundles

Let H be a complex Lie group acting holomorphically on a complex analytic space X such that the restriction to X_{\mathrm{red}} of every H-invariant regular function on X is constant. We prove that an H-equivariant holomorphic vector bundle E over X is $H$-finite, meaning f_1(E)= f_2(E) as H-equivariant bundles for two distinct polynomials f_1 and f_2 whose coefficients are nonnegative integers, if and only if the pullback of E along some H-equivariant finite \'etale covering of X is trivial as an H-equivariant bundle.

Learning from the Patient - The Cooperative Endeavor of Analytic Psychotherapy

Communication involves intimate human interaction that creates shared realities and forges connections. Nowhere is this more apparent and more necessary than in the therapeutic relationship, where the communication is, essentially, the curative agent. By following the patient’s lead, the many nuances of the various levels of communication are harnessed to promote insight and understanding. In this way, that analytic space becomes grounds for new restorative experiences and healing.

A generalization of Milnor’s formula

The Milnor number $$\mu _f$$ μ f of a holomorphic function $$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$ f : ( C n , 0 ) → ( C , 0 ) with an isolated singularity has several different characterizations as, for example: 1) the number of critical points in a morsification of f, 2) the middle Betti number of its Milnor fiber $$M_f$$ M f , 3) the degree of the differential $${\text {d}}f$$ d f at the origin, and 4) the length of an analytic algebra due to Milnor’s formula $$\mu _f = \dim _{\mathbb {C}}{\mathcal {O}}_n/{\text {Jac}}(f)$$ μ f = dim C O n / Jac ( f ) . Let $$(X,0) \subset ({\mathbb {C}}^n,0)$$ ( X , 0 ) ⊂ ( C n , 0 ) be an arbitrarily singular reduced analytic space, endowed with its canonical Whitney stratification and let $$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$ f : ( C n , 0 ) → ( C , 0 ) be a holomorphic function whose restriction f|(X, 0) has an isolated singularity in the stratified sense. For each stratum $${\mathscr {S}}_\alpha $$ S α let $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) be the number of critical points on $${\mathscr {S}}_\alpha $$ S α in a morsification of f|(X, 0). We show that the numbers $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) generalize the classical Milnor number in all of the four characterizations above. To this end, we describe a homology decomposition of the Milnor fiber $$M_{f|(X,0)}$$ M f | ( X , 0 ) in terms of the $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) and introduce a new homological index which computes these numbers directly as a holomorphic Euler characteristic. We furthermore give an algorithm for this computation when the closure of the stratum is a hypersurface.

Nonproper intersection products and generalized cycles

We develop intersection theory in terms of the $${{\mathscr {B}}}$$ B -group of a reduced analytic space. This group was introduced in a previous work as an analogue of the Chow group; it is generated by currents that are direct images of Chern forms and it contains all usual cycles. However, contrary to Chow classes, the $${{\mathscr {B}}}$$ B -classes have well-defined multiplicities at each point. We focus on a $${{\mathscr {B}}}$$ B -analogue of the intersection theory based on the Stückrad–Vogel procedure and the join construction in projective space. Our approach provides global $${{\mathscr {B}}}$$ B -classes which satisfy a Bézout theorem and have the expected local intersection numbers. We also introduce $${{\mathscr {B}}}$$ B -analogues of more classical constructions of intersections using the Gysin map of the diagonal. These constructions are connected via a $${{\mathscr {B}}}$$ B -variant of van Gastel’s formulas. Furthermore, we prove that our intersections coincide with the classical ones on cohomology level.

Proteomic Organellar Mapping Reveals Modulation of Peroxisomes by HIV-1

The interactions between HIV-1 and the host cell have been extensively studied, but cell-wide organellar changes caused by the virus have not been defined. Here, we aimed to detail perturbations to the host cell spatial proteome following HIV protein expression. Subcellular fractionation followed by mass spectrometric analysis of Jurkat T cells, in which HIV-expression was uniformly induced, identified thousands of cytoplasmic and membrane-bound proteins and enabled their placement within a multi-dimensional analytic space based on the 7 fractions. Spatial proteomic data were analyzed via a support vector machine bagging classifier to assess the movement of proteins between subcellular compartments. The proteins were also examined for absolute movement within 7-dimensional Euclidean space. We found that while some proteins moved between organelles, others remained classified within the same organelle but moved as a group relative to other organelles. The expression of HIV-1 affected the distribution of peroxisomal proteins, causing them to move closer in analytic space to proteins of the endoplasmic reticulum and lysosomes, but only when the viral protein Nef was expressed. These data identify Nef-dependent changes in peroxisomes as a novel perturbation of the host cell by HIV-1.

Lagrangian cobordism and tropical curves

We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relation to rational equivalence of 0-cycles on the mirror rigid analytic space.

Lipschitz Normal Embedding Among Superisolated Singularities

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. A complex analytic germ is said Lipschitz normally embedded (LNE) if its outer and inner metrics are bilipschitz equivalent. LNE seems to be fairly rare among surface singularities; the only known LNE surface germs outside the trivial case (straight cones) are the minimal singularities. In this paper, we show that a superisolated hypersurface singularity is LNE if and only if its projectivized tangent cone has only ordinary singularities. This provides an infinite family of LNE singularities, which is radically different from the class of minimal singularities.