Accurate and comprehensive extraction of information from high-dimensional single cell datasets necessitates faithful visualizations to assess biological populations. A state-of-the-art algorithm for non-linear dimension reduction, t-SNE, requires multiple heuristics and fails to produce clear representations of datasets when millions of cells are projected. We develop opt-SNE, an automated toolkit for t-SNE parameter selection that utilizes Kullback-Leibler divergence evaluation in real time to tailor the early exaggeration and overall number of gradient descent iterations in a dataset-specific manner. The precise calibration of early exaggeration together with opt-SNE adjustment of gradient descent learning rate dramatically improves computation time and enables high-quality visualization of large cytometry and transcriptomics datasets, overcoming limitations of analysis tools with hard-coded parameters that often produce poorly resolved or misleading maps of fluorescent and mass cytometry data. In summary, opt-SNE enables superior data resolution in t-SNE space and thereby more accurate data interpretation.

For two DG-categories A and B we define the notion of a spherical Morita quasi-functor A → B. We construct its associated autoequivalences: the twist T ∈ Aut D(B) and the co-twist F ∈ Aut D(A). We give sufficiency criteria for a quasi-functor to be spherical and for the twists associated to a collection of spherical quasi-functors to braid. Using the framework of DG-enhanced triangulated categories, we translate all of the above to Fourier-Mukai transforms between the derived categories of algebraic varieties. This is a broad generalisation of the results on spherical objects in [ST01] and on spherical functors in [Ann07]. In fact, this paper replaces [Ann07], which has a fatal gap in the proof of its main theorem. Though conceptually correct, the proof was impossible to fix within the framework of triangulated categories.

We show that the adjunction counits of a Fourier-Mukai transform Φ : D(X 1 ) → D(X 2 ) arise from maps of the kernels of the corresponding Fourier-Mukai transforms. In a very general setting of proper separable schemes of finite type over a field we write down these maps of kernels explicitly -facilitating the computation of the twist (the cone of an adjunction counit) of Φ. We also give another description of these maps, better suited to computing cones if the kernel of Φ is a pushforward from a closed subscheme Z ⊂ X 1 × X 2 . Moreover, we show that we can replace the condition of properness of the ambient spaces X 1 and X 2 by that of Z being proper over them and still have this description apply as is. This can be used, for instance, to compute spherical twists on non-proper varieties directly and in full generality.On the other hand, the identity functor Id is the Fourier-Mukai transform D(X 1 ) → D(X 1 ) with kernelConsider now the left adjunction counitIn general, morphisms between Fourier-Mukai kernels map neither injectively nor surjectively to natural transformations between the Fourier-Mukai transforms. Thus there is no a priori reason for (1.4) to come 1 arXiv:1004.3052v3 [math.AG]

The paper provides new examples of an explicit submanifold in Bridgeland stabilities space of a local Calabi-Yau.More precisely, let X be the standard resolution of a transversal slice to an adjoint nilpotent orbit of a simple Lie algebra over C. An action of the affine braid group on the derived category D b (Coh(X)) and a collection of t-structures on this category permuted by the action have been constructed in [BR] and [BM] respectively. In this note we show that the t-structures come from points in a certain connected submanifold in the space of Bridgeland stability conditions. The submanifold is a covering of a submanifold in the dual space to the Grothendieck group, and the affine braid group acts by deck transformations.We also propose a new variant of definition of stabilities on a triangulated category, which we call a "real variation of stability conditions" and discuss its relation to Bridgeland's definition. The main theorem provides an illustration of such a relation. We state a conjecture by the second author and A. Okounkov on examples of this structure arising from symplectic resolutions of singularities and its relation to equivariant quantum cohomology. We verify this conjecture in our examples.To Borya Feigin with gratitude and best wishes on his anniversary 1

Abstract. We introduce a relative version of the spherical objects of Seidel and Thomas [ST01]. Define an object E in the derived category D(Z × X) to be spherical over Z if the corresponding functor from D(Z) to D(X) gives rise to autoequivalences of D(Z) and D(X) in a certain natural way. Most known examples come from subschemes of X fibred over Z. This categorifies to the notion of an object of D(Z × X) orthogonal over Z. We prove that such an object is spherical over Z if and only if it possesses certain cohomological properties similar to those in the original definition of a spherical object. We then interpret this geometrically in the case when our objects are actual flat fibrations in X over Z.

Given a differentially graded (DG)-category ${{\mathcal{A}}}$, we introduce the bar category of modules ${\overline{\textbf{{Mod}}}-{\mathcal{A}}}$. It is a DG enhancement of the derived category $D({{\mathcal{A}}})$ of ${{\mathcal{A}}}$, which is isomorphic to the category of DG ${{\mathcal{A}}}$-modules with ${A_{\infty }}$-morphisms between them. However, it is defined intrinsically in the language of DG categories and requires no complex machinery or sign conventions of ${A_{\infty }}$-categories. We define for these bar categories Tensor and Hom bifunctors, dualisation functors, and a convolution of twisted complexes. The intended application is to working with DG-bimodules as enhancements of exact functors between triangulated categories. As a demonstration, we develop a homotopy adjunction theory for tensor functors between derived categories of DG categories. It allows us to show in an enhanced setting that given a functor $F$ with left and right adjoints $L$ and $R$, the functorial complex $FR \xrightarrow{F{\operatorname{act}}{R}} FRFR \xrightarrow{FR{\operatorname{tr}} - {\operatorname{tr}}{FR}} FR \xrightarrow{{\operatorname{tr}}} {\operatorname{Id}}$ lifts to a canonical twisted complex whose convolution is the square of the spherical twist of $F$. We then write down four induced functorial Postnikov systems computing this convolution.

We study the exotic t-structure on Dn, the derived category of coherent sheaves on two-block Springer fibre (i.e. for a nilpotent matrix of type (m + n, n) in type A). The exotic t-structure has been defined by Bezrukavnikov and Mirkovic for Springer theoretic varieties in order to study representations of Lie algebras in positive characteristic. Using work of Cautis and Kamnitzer, we construct functors indexed by affine tangles, between categories of coherent sheaves on different two-block Springer fibres (i.e. for different values of n). After checking some exactness properties of these functors, we describe the irreducible objects in the heart of the exotic t-structure on Dn and enumerate them by crossingless (m, m + 2n) matchings. We compute the Ext's between the irreducible objects, and show that the resulting algebras are an annular variant of Khovanov's arc algebras. In subsequent work we will make a link with annular Khovanov homology, and use these results to give a characteristic p analogue of some categorification results using two-block parabolic category O (by Bernstein-Frenkel-Khovanov, Brundan, Stroppel, et al).

10 11 12 Accurate and comprehensive extraction of information from high-dimensional single cell 13 datasets necessitates faithful visualizations to assess biological populations. A state-of-the-art 14 algorithm for non-linear dimension reduction, t-SNE, requires multiple heuristics and fails to 15 produce clear representations of datasets when millions of cells are projected. We developed 16 opt-SNE, an automated toolkit for t-SNE parameter selection that utilizes Kullback-Liebler 17 divergence evaluation in real time to tailor the early exaggeration and overall number of 18 gradient descent iterations in a dataset-specific manner. The precise calibration of early 19 exaggeration together with opt-SNE adjustment of gradient descent learning rate dramatically 20 improves computation time and enables high-quality visualization of large cytometry and 21 transcriptomics datasets, overcoming limitations of analysis tools with hard-coded parameters 22