Motivic Homotopy Theory and Refined Enumerative Geometry.

Cover -- Title page -- Contents -- Preface -- List of talks -- Lecture series -- Research talks -- List of participants -- SL-oriented cohomology theories -- 1. Introduction -- 2. Vector bundles with additional structure -- 3. Orientations and twisted cohomology -- 4. ^{ } Thom isomorphisms for -oriented cohomology -- 5. Orienting when ^{0,0}(-) is a sheaf -- 6. Interlude on the Hopf element and the connecting homomorphism -- 7. Characteristic classes and Hopf element -- References -- The homotopy Leray spectral sequence -- 1. Introduction -- Notations and conventions -- 2. Homotopy t-structure and duality -- 3. Fiber homology and Gersten complexes -- 4. The homotopy Leray spectral sequences -- 5. Applications -- References -- Examples of wild ramification in an enriched Riemann-Hurwitz formula -- 1. Introduction -- 2. Notation -- 3. Examples -- 4. Main Theorem -- References -- Lectures on Chow-Witt groups -- Introduction -- 1. Preliminaries -- 2. The Rost-Schmid complex and Chow-Witt groups -- 3. Products and general pull-backs -- 4. Some computations -- References -- Chow-Witt rings of split quadrics -- 1. Introduction -- 2. Conventions and notation -- 3. Singular cohomology of real quadrics -- 4. The blow-up setup of Balmer-Calmès -- 5. \I-cohomology: additive structure -- 6. \I-cohomology: multiplicative structure -- 7. Geometric bidegrees and real realization -- 8. Chow-Witt rings -- 9. Milnor-Witt cohomology -- References -- Lectures on quadratic enumerative geometry -- Introduction -- Lecture 1. Euler characteristics -- References: Lecture 1 -- Lecture 2. Riemann-Hurwitz formulas -- References: Lecture 2 -- Lecture 3. Pontryagin classes, splitting principles and Becker-Gottlieb transfers -- References: Lecture 3 -- Lecture 4. Reduction to the normalizer -- References: Lecture 4 -- Remarks on motivic Moore spectra -- 1. Introduction.

2. Preliminaries on ₀, ₁, and ₂ -- 3. Elementary properties of motivic Moore spectra -- 4. Toda brackets -- 5. Multiplications -- 6. Slices of motivic Moore spectra -- References -- Oriented Schubert calculus in Chow-Witt rings of Grassmannians -- 1. Introduction -- 2. Recollection on Chow-Witt rings of Grassmannians -- 3. Multiplicative structure via even Young diagrams -- 4. Computations with torsion classes -- 5. Recollection on Witt groups and cohomology -- 6. Oriented Schubert classes -- 7. Geometric interpretation of oriented intersection multiplicities -- 8. Main results and consequences -- 9. Sample enumerative applications -- References -- Back Cover.

This volume contains the proceedings of the Workshop on Motivic Homotopy Theory and Refined Enumerative Geometry, held from May 14-18, 2018, at the Universität Duisburg-Essen, Essen, Germany. It constitutes an accessible yet swift introduction to a new and active area within algebraic geometry, which connects well with classical intersection theory. Combining both lecture notes aimed at the graduate student level and research articles pointing towards the manifold promising applications of this refined approach, it broadly covers refined enumerative algebraic geometry.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.