Orientation theory in arithmetic geometry

This work is devoted to study orientation theory in arithmetic geometric within the motivic homotopy theory of Morel and Voevodsky. The main tool is a formulation of the absolute purity property for an \emph{arithmetic cohomology theory}, either represented by a cartesian section of the stable homotopy category or satisfying suitable axioms. We give many examples, formulate conjectures and prove a useful property of analytical invariance. Within this axiomatic, we thoroughly develop the theory of characteristic and fundamental classes, Gysin and residue morphisms. This is used to prove Riemann-Roch formulas, in Grothendieck style for arbitrary natural transformations of cohomologies, and a new one for residue morphisms. They are applied to rational motivic cohomology and \'etale rational $\ell$-adic cohomology, as expected by Grothendieck in \cite[XIV, 6.1]{SGA6}.Comment: 81 pages. Final version, to appear in the Actes of a 2016 conference in the Tata Institute. Thanks a lot goes to the referee for his enormous work (more than 100 comments) which was of great help. Among these corrections, he indicated to me a sign mistake in formula (3.2.14.a) which was very hard to detec

Tilting theory via stable homotopy theory

We show that certain tilting results for quivers are formal consequences of stability, and as such are part of a formal calculus available in any abstract stable homotopy theory. Thus these results are for example valid over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory and also in the equivariant, motivic or parametrized variant thereof. In further work, we will continue developing this calculus and obtain additional abstract tilting results. Here, we also deduce an additional characterization of stability, based on Goodwillie's strongly (co)cartesian n-cubes. As applications we construct abstract Auslander-Reiten translations and abstract Serre functors for the trivalent source and verify the relative fractionally Calabi-Yau property. This is used to offer a new perspective on May's axioms for monoidal, triangulated categories.Comment: minor improvements in the presentation (the definition of a strong stable equivalence made more precise, references updated and added

Ring objects in the equivariant derived Satake category arising from Coulomb branches (with an appendix by Gus Lonergan)

This is the second companion paper of arXiv:1601.03586. We consider the morphism from the variety of triples introduced in arXiv:1601.03586 to the affine Grassmannian. The direct image of the dualizing complex is a ring object in the equivariant derived category on the affine Grassmannian (equivariant derived Satake category). We show that various constructions in arXiv:1601.03586 work for an arbitrary commutative ring object. The second purpose of this paper is to study Coulomb branches associated with star shaped quivers, which are expected to be conjectural Higgs branches of $3d$ Sicilian theories in type $A$ by arXiv:1007.0992.Comment: 38 pages; v2. 66 pages, proofs in some results in Sec.5 are corrected. A new appendix by Gus Lonergan on a new proof of the commutativity of the convolution product; v3. the appendix is mentioned in the title; v4. A remark (5.22) on quantization of results in Sect.5 is added; v5. the definition of $\mathcal{S}_N^\circ$ is adde

A Finite Algorithm for the Realizabilty of a Delaunay Triangulation

The \emph{Delaunay graph} of a point set $P \subseteq \mathbb{R}^2$ is the plane graph with the vertex-set $P$ and the edge-set that contains $\{p,p'\}$ if there exists a disc whose intersection with $P$ is exactly $\{p,p'\}$. Accordingly, a triangulated graph $G$ is \emph{Delaunay realizable} if there exists a triangulation of the Delaunay graph of some $P \subseteq \mathbb{R}^2$, called a \emph{Delaunay triangulation} of $P$, that is isomorphic to $G$. The objective of \textsc{Delaunay Realization} is to compute a point set $P \subseteq \mathbb{R}^2$ that realizes a given graph $G$ (if such a $P$ exists). Known algorithms do not solve \textsc{Delaunay Realization} as they are non-constructive. Obtaining a constructive algorithm for \textsc{Delaunay Realization} was mentioned as an open problem by Hiroshima et al.~\cite{hiroshima2000}. We design an $n^{\mathcal{O}(n)}$-time constructive algorithm for \textsc{Delaunay Realization}. In fact, our algorithm outputs sets of points with {\em integer} coordinates

固有値分解とテンソル分解を用いた大規模グラフデータ分析に関する研究

筑波大学 (University of Tsukuba)201

Notes on Feynman Integrals and Renormalization

I review various aspects of Feynman integrals, regularization and renormalization. Following Bloch, I focus on a linear algebraic approach to the Feynman rules, and I try to bring together several renormalization methods found in the literature from a unifying point of view, using resolutions of singularities. In the second part of the paper, I briefly sketch the work of Belkale, Brosnan resp. Bloch, Esnault and Kreimer on the motivic nature of Feynman integrals.Comment: 39

Hyperplane sections and derived categories

We give a generalization of the theorem of Bondal and Orlov about the derived categories of coherent sheaves on intersections of quadrics revealing its relation to projective duality. As an application we describe the derived categories of coherent sheaves on Fano 3-folds of index 1 and degrees 12, 16 and 18.Comment: 76 page