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Current Schedule

This Week

Wednesday September 2 at 4:00 PM in RH 104
Lucas Williams (Purdue), “Invariants for Families of Periodic Points”

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 $k|n$ 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.

Upcoming Weeks

Wednesday September 9 at 4:00 PM in RH 104
Mihai Marian (IU), “A Khovanov theory for strongly invertible knots”
Wednesday September 30 at 4:00 PM in RH 104
Eleftherios Chatzitheodoridis (VCU), “Trees associated with unitary partition complexes”

In a recent paper, Heuts and Moerdijk develop a poset of trees whose classifying space is homotopy equivalent to the n-th partition complex, that is, the classifying space of the poset of partitions of a set with n elements. The latter poset features in work of Arone, Dwyer, and Lesh, and it finds a unitary analog introduced by Arone and Lesh. In joint work with Julie Bergner, Pedro Brunialti Lima de Andrade, and Josh Turner, we develop a topological poset of trees whose classifying space is homotopy equivalent to the n-th unitary partition complex, that is, the classifying space of the topological poset of partitions of n-dimensional complex space in pairwise orthogonal subspaces. Our work builds on the study of unitary partition complexes by Bergner, Joachimi, Lesh, Stojanoska, and Wickelgren, and it also entails some new results in the theory of topological categories that may be of independent interest.