TITLES AND ABSTRACTS
Title: Methods of studying free symmetry
Abstract: We introduce a new point of view based on the finiteness of the image of H*(B_G) in H*(X/G) replacing assumptions of finiteness for the space acted on. This leads to diverse homological avenues of approach for studying free actions.
Jon F. Carlson
Title: Idempotent modules and local supports
Abstract: Suppose that G is a finite group scheme and that k is a field of characteristic p > 0. The cohomology ring of G with coefficients in k plays an important role in the structure of the category of kG-modules. We give an explicit description of certain idempotent modules that are associated to closed points in the spectrum Proj H^*(G, k) and that define localizing subcategories of the stable category of all kG-modules. The endomorphism rings of these modules can be computed and provide something of a local support with some interesting properties.
Title: Topology of Feature Spaces and Machine Learning
Abstract: We will describe, with examples, the application of topological constructions to machine learning. The imposition of geometries on data sets and their sets of features allows for powerful visualizations which adds to the transparence and explainability of models, as well as more efficient feature engineering. We will show by example how this works.
Title: Quillen stratification in equivariant homotopy theory
Abstract: This is joint work with Tobias Barthel, Drew Heard, Niko Naumann and Luca Pol. Quillen's stratification theorem describes how the maximal ideal spectrum of the group cohomology ring of a finite group G over an algebraically closed field of characteristic p>0 can be glued together from the group cohomology of its elementary abelian p-subgroups.
In this project we prove a version of Quillen's stratification theorem in equivariant homotopy theory for a finite group G where we work with arbitrary commutative equivariant ring spectra. At the level of the category of equivariant modules over any G-equivariant commutative ring spectrum R, this category is stratified in terms of the geometric fixed points equipped with their Weyl-group actions for all subgroups, providing a classification of localizing tensor ideal subcategories. We then apply our methods to several examples including the case of Borel-equivariant Morava E-theory E_n, for any finite height n and any finite group G.
Title: Floer Homotopy Theory, old and new
Abstract: In 1995 the speaker, Jones, and Segal introduced the notion of “Floer homotopy theory”. The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology. More to the point, the question was asked, “When is the Floer homology isomorphic to the (singular) homology of a naturally occurring spectrum defined from the properties of the moduli spaces inherent in the Floer theory?” Years passed before this notion found some genuine applications to symplectic geometry and low dimensional topology. However in recent years several striking applications have been found, and the theory has been developed on a much deeper level. In this expository talk I will sketch both the interesting algebraic topology and symplectic topology involved in the theory and talk about recent applications by Lipshitz-Sarkar, Manolescu, and Abouzaid-Blumberg.
José Manuel Gómez
Title: Commuting elements in Lie groups (a summary of results)
Abstract: In this talk I will give an overall summary of the main results obtained in my collaboration with Alejandro Adem in the past few years.
Title: Commuting matrices and Hochschild homology
Abstract: In this talk I will discuss the topological structure of the space Hom(Z^n,G) of commuting n-tuples, where G is a unitary group or general linear group over the complex numbers. I will explain how the equivariant homology of Hom(Z^n,G) is related to Hochschild homology, and how this can be used in homology calculations. In particular, I will explain how to obtain the rational homology, stability, and bounds on p-torsion.
Title: Endotrivial modules -- the spheres of modular representation theory
Abstract: An endotrivial module is an invertible element, under tensor product, in the stable module category of kG, for G a finite group and k a characteristic p field. Whereas in the ordinary module category only one-dimensional modules are invertible, the theory becomes much richer after discarding projective modules, and links to other questions in algebra and topology. In my talk I'll survey this story, starting with classical work of Dade, Puig, Alperin, Carlson, and Thevenaz, when G is a p-group, and then moving on to recent results for arbitrary finite groups, where methods from homotopy theory comes into play. Of course group cohomology will be everywhere, and we'll have plenty of opportunities to say cheers and happy birthday to Alejandro along the way.
Title: Minimal Euler characteristics for even-dimensional manifolds
Abstract: We consider the Euler characteristics of closed orientable 2n-manifolds with a given fundamental group G, and highly-connected universal cover. We strengthen the 4-dimensional Hausmann-Weinberger estimates and extend to higher dimensions. As an application we obtain new restrictions for non-abelian finite groups arising as fundamental groups of rational homology 4-spheres. This is joint work with Alejandro Adem.
Title: Large and small group homology
Abstract: Group homology is defined in either algebraic or topological terms, depending on the application and taste. It is therefore somewhat surprising that it can also encode metric properties that one usually studies on smooth manifolds, such as enlargeability or having positive scalar curvature. The key role is played by the image of the homological fundamental class of the given manifold under the classifying map. We will explain this line of thought and present some applications.
Title: The moduli space of quantum toric varieties
Abstract: In this talk I will introduce the moduli space of quantum toric varieties. Then ask questions about its topology. Joint with Katzarkov, Meersseman and Verjovsky.
Title: The loop space homology of a small category
Abstract: In an article published in 2009, Dave Benson described, for a finite group G, the mod p homology of the loop space of the p-completion of BG in purely algebraic terms. In joint work with Carles Broto and Ran Levi, we tried to better understand Benson’s result by generalizing it. Among other things, we showed that when C is a small category, |C| is its geometric realization, R is a commutative ring, and |C|+R is a “plus construction” of |C| with respect to homology with coefficients in R, then H∗(Ω(|C|+R); R) is the homology of any chain complex of projective RC-modules that satisfies certain conditions. Benson’s theorem is then the special case where C is the category associated to a finite group G and R = Fp, so that p-completion appears as a special case of the plus construction.
Title: Geometric Langlands and Co-adjoint orbit
Abstract: Geometric Langlands can be interpreted as a mirror symmetry between the moduli space of Higgs bundle of reductive group G and the same kind of moduli space of its Langlands dual L^G. When we add insertions (as the parabolic structure) into the picture, a pre-condition for mirror symmetry between the moduli space of parabolic Higgs bundles is a mirror symmetry between the co-adjoint orbit of G and that of its Langlands dual L^G. The latter is a part of classical geometric representation theory in the sense that there is no need to consider any moduli space. In the talk, we will outline a program to study such co-adjoint orbit mirror symmetry. This is a joint work with Bohua Fu and Yaoxiong Wen.
Title: The Cheeger-Gromov invariant
Abstract: The Cheeger-Gromov invariant, introduced in the mid-80's uses L^2 cohomology to give a generalization of the Browder-Livesay invariant to manifolds with arbitrary fundamental group. It has had a number of interesting applications (to showing unconditionally some consequences of the Farrell Jones conjecture, and to topological knot cobordism in dimension 3). I would like to take this opportunity to explain this invariant and some of the ways it's been applied, including some recent work I've done with Geunho Lim on this and related invariants (building on earlier work of Cheeger-Gromov, Cha, and Lim) with applications to questions about complexity of manifolds.
Title: Structure theory for braided Hopf algebras
Abstract: Cocommutative Hopf algebras have a very well-developed structure theory via the theorem of Cartier-Kostant-Milnor-Moore: over algebraically closed fields of characteristic zero, they split as a semidirect product of a group ring with the enveloping algebra of the space of primitives, a Lie algebra. Further, the Poincaré-Birkhoff-Witt theorem gives a very explicit description of the graded algebra associated to the filtration of the enveloping algebra of the primitives. This structure theory is in some sense governed by operads, particularly the relations between the associative, Lie, and commutative operads. Further, these constructions lead to computational tools in homological algebra, particularly relating Harrison and Hochschild homology. More structured formulations relate André-Quillen homology to the partition poset and spectral Lie algebras. Unfortunately, in the setting of braided Hopf algebras, most of this theory falls apart due to the fact that the primitives no longer form a Lie algebra. I’ll present ongoing work in which we investigate what sort of structure the primitives support (using the language of braided operads), and how much it can be leveraged to construct analogues of these results in a genuinely braided setting.