## Graduiertenkolleg 1821 Cohomological Methods in Geometry Summer School 2013 Regulators and Differential Algebraic K-Theory

### Who should be interested in the program

• people in algebraic K-theory because the regulator to Hodge cohomology allows to deduce non-trivial information on K-groups
• people in Arakelov geometry because the interplay between between K-groups and Hodge cohomology comes up in arithmetic intersection theory
• people in algebraic and differential topology because differential cohomology theories provide finer invariants of smooth manifolds, e.g. higher torsion invariants

### What you should know

As part of the program, we are going to review the basic definitions of things like algebraic K-groups, Hodge cohomology and stable homotopy theory which are needed to formulate the main results. Depending on their background, we recommend that people are thoroughly familiar with one these aspects.

Algebraic topology

• Very important: homotopy groups, classifying spaces, singular cohomology, fibre sequences
• Good to know: spectra, homotopy pullbacks, limits and colimits, H-spaces, Serre spectral sequence
Differential topology
• Very important: differential forms, Chern classes of vector bundles,
• Good to know: metrics and unitary connections on vector bundles, curvature, Chern character forms
K-theory
• Very important: K_0 of the category of vector bundles, topological K-theory, a rough idea of K_0 and K_1 for rings,
• Good to know: definition and some properties of higher algebraic K-theory of rings and varieties
Hodge theory for complex manifolds
• Very important: Hodge's Theorem, Dolbeault complex
• Good to know: the category of mixed Hodge structures, existence of a mixed Hodge structure on cohomology of complex algebraic varieties
Number theory and arithmetic geometry
• Good to know: rings of integers, Dirichlet unit theorem
Algebraic geometry
• Very important: spectrum of a ring, sheaves
• Good to know: sites, sheaf cohomology, some homological algebra

What the summer school is about
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