Schedule 2024

Jack Romo - University of Leeds

Title: Homotopy Bicategories of (infinity,2)-categories

Abstract: Across the multitude of definitions for a higher category, a dividing line can be found between two major camps of model. On one side lives the ‘algebraic’ models, like Bénabou’s bicategories, tricategories following Gurski and the models of Batanin and Leinster, Trimble and Penon. On the other end, one finds the ‘non-algebraic’ models, including more homotopy-theoretic ones like quasicategories, Segal n-categories, complete n-fold Segal spaces and more. The bridges between these models remain somewhat mysterious. Progress has been made in certain instances, as seen in the work of Tamsamani, Leinster, Lack and Paoli, Cottrell, Campbell, Nikolaus and others. Nonetheless, the correspondence remains incomplete; indeed, for instance, there is no fully verified means in the literature to take an `algebraic’ homotopy n-category of any known model of $(\infty, n)$-category for general $n$. One might see this as an extension of the fundamental n-groupoid of a homotopy type, a statement I will make precise. In this talk, I will explore current work in the problem of taking homotopy bicategories of non-algebraic $(\infty, 2)$-categories, including a construction of my own. If time permits, I will discuss some of the applications of this problem to topological quantum field theories.

Baylee Schutte - University of Aberdeen

Title: Projective Span of Wall Manifold

Abstract: The projective span (pspan) of a smooth manifold is defined to be the maximal number of linearly independent tangent line fields. To initiate a study of projective span, we have calculated the pspan of Wall manifolds, which are certain mapping tori of Dold manifolds. (Classicaly, Wall manifolds were used by C. T. C. Wall in his determination of the oriented cobordism ring). In this talk, we explore how the theory of Clifford algebras and their modules can be used to construct quasi-invariant vector fields on a cover of a Wall manifold, for such vector fields descend to line fields on the quotient. This is joint with Mark Grant and based on arXiv:2311.14107

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Jackson Morris - University of Washington

Title: Symplectic Orientations

Abstract: In stable homotopy theory, there is a notion of a complex-oriented cohomology theory. One can view these theories as those having Thom classes for all complex vector bundles, and one can further see that these spectra receive a map from the complex cobordism spectrum MU. Quillen showed that MU held the universal formal group law, and moreover any complex oriented theory has an associated formal group law related to MU via the map above. The work of Devinatz-Hopkins-Smith showed that understanding this complex orientation data determines much of the global structure of the category of spectra. In motivic homotopy theory, the goal of understanding this global structure is more difficult. In particular, we can run the above game and talk about complex orientations, but it does not give us a complete answer. The question becomes: what type of orientation data can we use to help us here? In this talk, I will talk about the classical theory of complex orientations and their connections with formal group laws, before moving to the motivic case and talking about work on symplectic orientations and formal ternary laws