| August 25 |
Organizational Meeting |
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| September 1 |
David Mehrle |
An introduction to equivariant homotopy theory |
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Equivariant homotopy theory has played a major role in many of the recent triumphs of algebraic topology, such as the resolution of the Kervaire invariant problem and the disproof of the telescope conjecture. The power -- and the difficulty -- of equivariant homotopy theory lies in its ability to capture more information about spaces than homotopy group or cohomology rings alone. This, however, requires that we replace the familiar homotopy groups by new algebraic objects called Mackey functors. I'll give a brief introduction to equivariant homotopy theory and the algebra of Mackey functors, and share some of my work on equivariant algebra.
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| September 8 |
David Mehrle |
Free incomplete Tambara functors are almost never flat |
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The fact that free algebras like $\mathbb{Z}[x]$ are flat as $\mathbb{Z}$-modules is incredibly useful in homological algebra. Building on last week's talk, we investigate this property in the context of Mackey functors, where the free algebras are called free incomplete Tambara functors. We find that this property holds only very rarely in the equivariant context, and that it is controlled entirely by combinatorial considerations. This is joint work with Mike Hill and J.D. Quigley.
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| September 15 |
Ben Knudsen |
An invitation to graph braid groups |
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For the past 20 years, configuration spaces of graphs have been studied intensively in algebraic topology, geometric group theory, topological robotics, and algebraic geometry. Nevertheless, there are many questions unanswered and much work to be done. This talk will survey the fundamental results and open questions in this area, assuming no prior familiarity. Depending on time, we may touch on some past, present, and future joint work with An, Drummond-Cole, and Ramos.
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| September 22 |
Lucas Williams (Purdue) |
Invariants for Families of Periodic Points |
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In this talk we investigate invariants that count periodic points of a map. Given a self map $f$ of a compact manifold we could detect $n$-periodic points of $f$ by computing the Reidemeister trace of $f^n$ or by computing the equivariant Fuller trace. In 2020 Malkiewich and Ponto showed that the collection of Reidemeister traces of $f^k$ for varying $k\mid n$ and the equivariant Fuller trace are equivalent as periodic point invariants, and they conjecture that for families of endomorphisms the Fuller trace will be a strictly richer invariant for $n$-periodic points.
In this talk we will explain our new result which confirms Malkiewich and Ponto's conjecture. We do so by proving a new Pontryagin-Thom isomorphism between equivariant parameterized cobordism and the spectrum of sections of a particular parametrized spectrum and using this result to carry out geometric computations.
Time permitting, we will discuss how this homeomorphism of a manifold gives rise to an element of the kernel of the ghost map on $\pi_1(-)$ of topological restriction homology.
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| September 29 |
Jesse Keyes (Kentucky) |
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| October 6 |
No Seminar |
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| October 13 |
Jacob Cleveland |
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| October 20 |
No Seminar |
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| October 27 |
Mark Shoemaker |
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| November 3 |
Rachel Pries |
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| November 10 |
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| November 17 |
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| November 24 |
No Seminar: Thanksgiving |
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| December 1 |
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| December 8 |
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