Topology and Geometry Seminar
Fall 2024
Topic: Characteristic classes
Time: Thursday 1-2:15 pm
Location: Classroom Building CL 416
The topic for our learning seminar this term is characteristic classes. We will also have occasional invited research talks over Zoom.
Zoom Link: https://uregina-ca.zoom.us/j/97896109097?pwd=RkI2UkZsMlYyZTBzejhEY1R4RCt4Zz09
Schedule of talks
September 19: Real line bundles, submanifolds, and Stiefel-Whitney classes
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.
September 26: A crash course on cohomology
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.
October 3: A crash course on cohomology, part 2
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.
October 10: A crash course on cohomology, part 3
Speaker: Martin Frankland
Abstract: In this third part, we will discuss orientations of manifolds, Poincaré duality, and the intersection product.
October 24: Evan Sundbo (University of Toronto)
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.
October 31: Complex line bundles, divisors, and the degree map
Speaker: Aditya Dwarkesh
Abstract: TBD
November 7: Adela YiYu Zhang (University of Copenhagen)
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.
November 14: Principal bundles and vector bundles
Speaker: Manak Singh
Abstract: TBD
November 21: Characteristic classes: Whitney classes, axiomatic approach
Speaker: Martin Frankland
Abstract: TBD
November 28: Characteristic classes: Computations, splitting principle, localization
Speaker: Carlos Gabriel Valenzuela Ruiz
Abstract: TBD
December 5: Classifying spaces
Speaker: Matt Alexander
Abstract: TBD
December 12: Connections and curvature
Speaker: Francis Bischoff
Abstract: TBD
To be scheduled: Chern-Weil theory
Speaker: TBD
Abstract: TBD
We gratefully acknowledge that this seminar is supported by the Pacific Institute for the Mathematical Sciences.