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16.04.2014 // Introduction  Victoria Hoskins
We provide some motivation coming from algebraic topology and state the classification of topological vector bundles up to homotopy. Then we give an overview of some of the ideas involved in the construction of a A^{1}homotopy theory for schemes and outline the key ideas and techniques needed to construct the unstable A^{1}homotopy category.

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23.04.2014 // Classification of principal bundles in algebraic topology  Vincent Trageser
In this talk, we give the homotopical classification of principal bundles in algebraic topology. For a topological group G, we show there is a classifying space BG such that homotopy classes of maps form a space X to BG are in bijection with isomorphism classes of principal Gbundles on X. In particular, we focus on the general linear group where we describe the classifying space as an infinite grassmannian.

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30.04.2014 // An introduction to homotopical algebra  Elena Martinengo
In this talk, we give the basic notions from homotopical algebra needed for the construction of the A^{1}homotopy category. To invert a certain class of morphisms in a category it is often useful to introduce a model structure. We give a quick introduction to model categories, homotopy categories and state some important results from homotopical algebra. We focus on the key examples of topological spaces and simplicial sets.

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07.05.2014 // Sheaves in the Nisnevich topology  Eva Martínez
We define and compare the Zariski, étale and Nisnevich topologies. We explain how the Nisnevich topology incorporates useful features of the Zariski and étale topologies. We prove that, for the Nisnevich topology, it suffices to check the sheaf condition on 'distinguished squares'. Then we define fibres of sheaves in the Nisnevich topology.

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14.05.2014 // The simplicial homotopy category of sheaves 
Anna Wißdorf
We consider the category of simplicial sheaves in the Nisnevich topology and give this a model category structure whose associated homotopy category is called the simplicial homotopy category of sheaves. We also briefly discuss Quillen functors and give some examples. Finally, we describe the B.G. property for a simplicial presheaf and note that fibrant simplicial sheaves in the Nisnevich topology have the B.G. property.

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21.05.2014 // The A^{1}homotopy category of smooth kschemes  Nikolai Beck
We start by recalling the naive notions of A^{1}homotopy and then define A^{1}weak equivalences in the category of simplicial sheaves. Morel and Voevodsky prove that there is a model structure on this category whose weak equivalences are A^{1}weak equivalences and so we construct the A^{1}homotopy category as the associated homotopy category. We also describe the A^{1}localisation functor and discuss morphism groups in the A^{1}homotopy category.

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28.05.2014 // K_{0} and K_{1} for rings  Ángel Muñoz Castañeda
We recall the definition of the Grothendieck group K_{0} and describe the Grothendieck groups of schemes and rings. We define K_{1} of a ring via a short exact sequence of groups and prove Whitehead's lemma. Finally, we define K_{0} and K_{1}regularity of a ring and prove that a regular ring is K_{0} and K_{1}regular.

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04.06.2014 // Excision for Picard groups and KaroubiVillamayor Ktheory  Ángel Muñoz Castañeda
We show that distinguished squares of schemes give rise to an exact sequence relating their Picard groups and prove that K_{0}regularity implies Picregularity. We define the KaroubiVillamayor Ktheory groups of a ring and relate these to K_{0} and K_{1} of the ring. Then, for certain cartesian squares of rings, we obtain long exact sequences intertwining these groups.

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11.06.2014 // Anodyne extensions I  Joana Cirici
In this talk, we introduce an alternative construction of the A^{1}homotopy category given by Morel that involves inverting certain 'anodyne extensions' in the category of presheaves on smooth affine schemes. This construction of the A^1homotopy category will be used to prove the homotopic classification of vector bundles in the following talks.

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18.06.2014 // Anodyne extensions II  Victoria Hoskins
We complete the definition of elementary anodyne extensions by defining the class of fundamental geometric anodyne extensions and describe a key example: a section of a vector bundle over a kspace. We describe the cofibrant objects and prove that the naive A^1homotopy equivalences are weak equivalences. Finally, we give a technical result for elementary anodyne extensions.

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25.06.2014 // Homotopic properties of the canonical torsor on the Grassmannians  Arijit Dey
We consider vector bundles and GL_{n}torsors and focus on the example of the Grassmannian Gr_{n} of nplanes. By taking the colimit over n, we obtain the infinite Grassmannian Gr. We prove that the canonical torsor over the infinite Grassmannian Gr is a fibration and that Gr is fibrant and similarly that the infinite projective space is fibrant.

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02.07.2014 // Homotopic classification of vector bundles on smooth kschemes
Finally, we reach the goal of our seminar and prove the homotopic classification of vector bundles! We start by proving the result for smooth affine schemes via a computation on naive homotopy classes of maps. Then we deduce the nonaffine case from this by using the JouanolouThomason theorem and strong A^{1}homotopy invariance of K_{0}.

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In the final two weeks of the semester, we have two special talks.

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09.07.2014 // Geometric invariant theory over the reals  Matthew Spong
For the action of a real reductive group G on a real affine space W, the map from W to the spectrum of the ring of invariant polynomials is not always surjective. Therefore, we describe Luna's construction of a topological quotient of W by the action of G. Following work of RichardsonSlodowy and Schwarz, we equip this topological quotient with a real semialgebraic structure. If time permits, we look at one or two examples of such quotients and determine their semialgebraic structures.

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09.07.2014 // Moduli spaces of decorated principal bundles over a curve (Phd Defense) Nikolai Beck
