Topology and Geometry Seminar

Speaker: Francis Bischoff

Abstract: Can you embed the real projective plane in three-dimensional space? Motivated by this and similar questions, I will introduce the correspondence between real line bundles and hypersurfaces, the first Stiefel-Whitney class, and Poincare duality. In particular, I will explain why every hypersurface of a smooth manifold is the regular level set of a 'twisted function' in a canonical way and will show you why this implies that it is a regular level set of an ordinary function if and only if it intersects every loop in an even number of points. 

Speaker: Martin Frankland

Abstract: Characteristic classes are certain cohomology classes that we assign to vector bundles. In this talk, we will first familiarize ourselves with cohomology. We will go over singular cohomology of spaces, the cup product, the cap product, along with examples. Then we will focus on manifolds and discuss de Rham cohomology, the fundamental class, Poincaré duality, and the intersection product.

Speaker: Martin Frankland

Abstract: In this second part, we will look at more examples of cohomology rings. We will then focus on manifolds and discuss de Rham cohomology, orientations, Poincaré duality, and the intersection product.

Speaker: Martin Frankland

Abstract: In this third part, we will discuss orientations of manifolds, Poincaré duality, and the intersection product.

Title: Broken Toric Varieties and Balloon Animal Maps

Abstract: We will see the definition and some examples of broken toric varieties and balloon animal maps between them. After an overview of some of the many different areas in which they appear, we look at how their geometry can be studied via complexes of sheaves on an associated complex of polytopes.  This yields results such as a version of the Decomposition Theorem and identifying the weight and Leray filtrations on the cohomology groups of a broken toric variety.

Speaker: Aditya Dwarkesh

Abstract: In the first talk of this series, we saw how the first Stiefel-Whitney class is a complete invariant of real line bundles. Building on this, we will now study how to build a complete invariant for complex line bundles: the Chern class. Among other things, this will require us to study a new kind of cohomology theory known as the Čech cohomology. Finally, if time permits, we will also touch upon the question of complete invariants for holomorphic line bundles.

Title: Universal differentials in the bar spectral sequence

Abstract: The synthetic analogue of the bar comonad controls the universal differentials in the bar spectral sequence of algebras over spectral operads. This can be viewed as a deformation of Koszul duality of such algebras. I will explain ongoing work with Burklund and Senger on identifying the universal differentials in the bar spectral sequence for spectral Lie algebras over F_p. This will also shed light on the mod p homology and Lubin–Tate theory of labeled configuration spaces via a theorem of Knudsen.

The Zoom link is posted above.

Speaker: Francis Bischoff

Abstract: In this talk, I will introduce the notion of principal bundles, reductions of structure group, and their relationship to vector bundles with additional geometric structures. I will then discuss the classification of vector bundles (and principal bundles more generally) in terms of homotopy classes of maps into a classifying space. 

Speaker: Martin Frankland

Abstract: In the previous talks, we constructed the first Stiefel-Whitney class of a real vector bundle and argued that it is the obstruction to orientability. In this talk, we will introduce higher Stiefel-Whitney classes, which are invariants of a vector bundle living in the cohomology of the base space. Postponing an explicit construction, we will assume that the Stiefel-Whitney classes satisfy four axioms. From there, we will deduce some computations and applications, for instance to embeddings of manifolds.

Speaker: Carlos Gabriel Valenzuela Ruiz

Abstract: In this talk we’ll continue where we left off last week. We’ll explore a direct and non-trivial application of the Stiefel-Whitney classes we constructed in the previous talk. Then I’ll present and prove the splitting principle and perform some computations with it, in particular, we’ll prove the uniqueness of the S-W classes.

Speaker: Matt Alexander

Abstract: In this talk, we will look at how the classification of principal bundles can be reduced to looking at maps into certain classifying spaces. We will give a simplicial construction of these spaces, and then introduce Grassmannians and show how the classifying spaces of O(n) and U(n) principal bundles arise as colimits of Grassmannians. Finally, we will see how the cohomology ring of certain classifying spaces ties into the Chern and Stiefel-Whitney classes that we have already seen.

Speaker: Francis Bischoff

Abstract: TBD

Speaker: TBD

Abstract: TBD