## 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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