Time: Tuesdays 3:00 - 4:20 PM, unless otherwise noted.
Location: on Zoom.
The Zoom link will be emailed to members of the Department of Mathematics and Statistics. Please contact Martin Frankland if you would like to receive the announcements but are not a member of the department.
In this learning seminar, we will study operads and their applications in topology and algebra.
The lecture series will be interspersed with guest research talks, which are posted on our Youtube channel.
Date | Speaker | Title and abstract | References |
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Wednesday September 15 11:30 AM |
Aziz Kharouf (University of Haifa / Bilkent University) |
Higher order Toda brackets Toda brackets are a type of higher homotopy operation. Like Massey products, they are not always defined, and their value is indeterminate. Nevertheless, they play an important role in algebraic topology and related fields. Toda originally constructed them as a tool for computing homotopy groups of spheres. Adams later showed that they can be used to calculate differentials in spectral sequences. After reviewing the construction and properties of the classical Toda bracket, we shall describe a higher order version, there are two ways to do that. We will provide a diagrammatic description for the system we need to define the higher order Toda brackets, then we will use that to give an alternative definition using the homotopy cofiber. |
Recording |
September 21 | Martin Frankland | Introduction to operads Operads are a tool to encode algebraic structure. They were introduced in the 1960s to describe the structure found in loop spaces. They have since found applications in topology, algebra, and mathematical physics. This talk will be an introduction to operads and algebras over them. We will look at examples, including the associative, commutative, and Lie operads. |
Notes from the talk |
September 28 | Larry So | Stasheff associahedra, A loop space is a good example of an |
Notes from the talk |
October 5 | Larry So | The little n-cubes operad and iterated loop spaces Like an |
Notes from the talk |
October 12 | Carlos Gabriel Valenzuela | The linear isometries operad and Li-Spaces as infinite loop spaces Many infinite loop spaces can be expressed in terms of the linear isometries operad. In this seminar I will talk about its construction as an |
Notes from the talk |
October 19 | Martin Frankland | S-modules and structured ring spectra Using the linear isometries operad, I will sketch the construction of L-spectra and S-modules, a model for stable homotopy theory with a well-behaved smash product. One salient feature is that S-algebras and commutative S-algebras model |
A. Elmendorf, I. Kriz, M. Mandell, and J.P. May. Rings, modules, and algebras in stable homotopy theory (1997). Notes from the talk A pleasantry |
October 26 | Arnaud Ngopnang Ngompe | Homotopy invariant algebraic structure Homotopy invariance is an important property not satisfied by all operads. In our talk, we give counterexamples and discuss some examples involving |
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November 2 | Maximilien Péroux (University of Pennsylvania) |
Equivariant variations of topological Hochschild homology Topological Hochschild homology (THH) is an important variant for ring spectra. It is built as a geometric realization of a cyclic bar construction. It is endowed with an action of circle. This is because it is a geometric realization of a cyclic object. The simplex category factors through Connes’ category From work in progress with Gabe Angelini-Knoll and Mona Merling, I present a generalization of |
Recording |
November 9 | Reading Week. No seminar. | ||
November 16 | Don Stanley | Model categories related to operads We look at model categories on the category of operads, on the category algebras over a fixed operad and on the category of modules over a fixed algebra over a fixed operad. We discuss how these can be useful. |
Notes from the talk |
November 23 | Matthew Alexander | Modular operads from moduli spaces Modular operads are particular, heavily-structured operads which show up in many areas of physics. The story of these operads winds its way through Feynman diagrams, quantum field theory, and string theory. In this talk we shall motivate the construction of modular operads by considering their application to the theory of moduli spaces of Riemann surfaces. Along the way, we shall encounter ways in which physics has inspired (or created) solutions to problems involving Riemann surfaces. |
Notes from the talk |
November 30 | Nimanthi Yaseema | Koszul duality for algebras If an algebra |
For more information, contact Martin Frankland.