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: Aditya Dwarkesh

Abstract: In this talk, we will introduce the notion of a principal connection and discuss various associated features, such as its curvature. This will be done with a view towards building a link between the theory of connections on a principal bundle, and that of characteristic classes on the base space.

Speaker: Martin Frankland

Abstract: A common theme in homotopy theory is to record not just whether two objects are equivalent, but also *how* they are equivalent. For instance, we can consider maps between spaces, homotopies between maps, homotopies between homotopies, and so on. An infinity-category is a way of encoding such data into mapping spaces between objects.

Quillen introduced model categories as a framework for doing homotopy theory. While we can do a lot with model categories, it is sometimes convenient to work directly at the level of infinity-categories. In this talk, I will motivate infinity-categories using examples.

Speaker: Martin Frankland

Abstract: We will review how to do homotopy theory with model categories. As motivation for infinity-categories, we will discuss the benefits and drawbacks of working with model categories.

Speaker: Martin Frankland

Abstract: I will provide an introduction to simplicial sets, which will serve as foundation for the rest of the lecture series. We will discuss examples and a few constructions, such as the function complex and geometric realization.

Speaker: Gabriel Valenzuela

Abstract: TBD

The Zoom link is posted above.

Title: Fibration category structures for the discrete homotopy n-types of graphs

Abstract: In classical homotopy theory, graphs are treated as 1-dimensional CW complexes. But since the classical notions of continuous maps and their homotopies do not respect the discrete nature of graphs, this fails to capture the full combinatorial richness of graph theory. Discrete homotopy theory, introduced around 20 years ago by H. Barcelo and collaborators, building on the work of R. Atkin from the mid-seventies, is a homotopy theory specifically designed to study discrete objects like graphs. This theory has found a wide range of applications, including in matroid theory, hyperplane arrangements, and topological data analysis. 

Currently, a central open problem in the field is to determine whether the cubical nerve functor, which associates a cubical Kan complex to a graph is a DK-equivalence of relative categories. If true, this would allow the import of important results like the Blakers-Massey theorem from classical homotopy theory to the discrete realm. In this talk, based on joint work with C. Kapulkin (arXiv:2408.05289), we will describe a new line of attack towards this problem by establishing the theory of discrete homotopy n-types.

Speaker: Matthew Alexander

Abstract: TBD

Speaker: TBD

Abstract: TBD

The Zoom link is posted above.

Speaker: Francis Bischoff

Abstract: TBD

Speaker: Martin Frankland

Abstract: TBD

Speaker: Aditya Dwarkesh

Abstract: TBD