Thursday, Aug. 4, 2022, 10:30 a.m. through Thursday, Aug. 4, 2022, 11:30 a.m.
Jose Perea (Northeastern University)
A vector bundle can be thought of as a family of vector spaces parametrized by a fixed topological space. Vector bundles have rich structure, and arise naturally when trying to solve synchronization problems in data science. I will show in this talk how the classical machinery (e.g., classifying maps, characteristic classes, etc) can be adapted to the world of algorithms and noisy data, as well as the insights one can gain. In particular, I will describe a class of topology-preserving dimensionality reduction problems, whose solution reduces to embedding the total space of a particular data bundle. Applications to computational chemistry and dynamical systems will also be presented.
Wednesday, Aug. 3, 2022, 2 p.m. through Wednesday, Aug. 3, 2022, 2:45 p.m.
Ulrike Tillmann (University of Oxford)
Persistent homology is a main tool in topological data analysis. So it is natural to ask how strong this quantifier is and how much information is lost. There are many ways to ask this question. Here we will concentrate on the case of level set filtrations on simplicial sets. Already the example of a triangle yields a rich structure with the Möbius band showing up as one of the fibres. Our analysis forces us to look at the persistence map with fresh eyes.
The talk will be based on joint work with Jacob Leygonie.
Wednesday, Aug. 3, 2022, 1 p.m. through Wednesday, Aug. 3, 2022, 1:45 p.m.
Niny Arcila-Maya (Duke University)
Topological Azumaya algebras are topological shadows of more complicated algebraic Azumaya algebras defined over, for example, schemes. Tensor product is a well-defined operation on topological Azumaya algebras. Hence given a topological Azumaya algebra A of degree mn, where m and n are positive integers, it is a natural question to ask whether A can be decomposed according to this factorization of mn. In this talk, I explain the definition of a topological Azumaya algebra over a topological space X, and present a result about what conditions should m, n, and X satisfy so that A can be decomposed.
Wednesday, Aug. 3, 2022, 10:30 a.m. through Wednesday, Aug. 3, 2022, 11:30 a.m.
Emily Riehl (Johns Hopkins University)
Mathematics students learn a powerful technique for proving theorems about an arbitrary natural number: the principle of mathematical induction. This talk introduces a closely related proof technique called path induction, which can be thought of as an expression of Leibniz's indiscernibility of identicals: if x and y are identified, then they must have the same properties, and conversely. What makes this interesting is that the notion of identification referenced here is given by Per Martin-Löf's intensional identity types, which encode a more flexible notion of sameness than the traditional equality predicate in that an identification can carry data, for instance of an explicit isomorphism or equivalence. The nickname path induction for the elimination rule for identity types derives from a new homotopical interpretation of type theory, in which the terms of a type define the points of a space and identifications correspond to paths. In this homotopical context, indiscernibility of identicals is a consequence of the path lifting property of fibrations. Path induction is then justified by the fact that based path spaces are contractible.
Wednesday, Aug. 3, 2022, 9 a.m. through Wednesday, Aug. 3, 2022, 10 a.m.
Jenny Wilson (University of Michigan)
In this talk I will give an overview of two related projects. The first project concerns the high-degree rational cohomology of the special linear group of a number ring R. Church--Farb--Putman conjectured that, when R is the integers, these cohomology groups vanish in a range close to the virtual cohomological dimension. The groups SL_n(R) satisfy a twisted analogue of Poincare duality called virtual Bieri--Eckmann duality, and their rational cohomology groups are governed by SL_n(R)-representations called the Steinberg modules. I will discuss a recent proof of the codimension two case of the Church--Farb--Putman conjecture using the topology of certain simplicial complexes related to the Steinberg modules. The second project concerns Rognes’ connectivity conjecture on a family of simplicial complexes (the common basis complexes) with implications for algebraic K-theory. I will describe work-in-progress proving Rognes' conjecture for fields, and its connections to SL_n(R) and the Steinberg modules. This talk includes past and ongoing work joint with Benjamin Brück, Alexander Kupers, Jeremy Miller, Peter Patzt, Andrew Putman, and Robin Sroka.
Tuesday, Aug. 2, 2022, 3:30 p.m. through Tuesday, Aug. 2, 2022, 4:15 p.m.
Erik Carlsson (University of California, Davis)
I'll discuss a method for approximating the super-level set persistent homology of a Gaussian kernel density estimator for a point cloud data set, which is related to the witness complex. Instead of selecting elements of the data set, the witnesses are generated using quadratic programming, and the shifted Voronoi diagram (aka the power diagram) of a specific choice of landmark points. Interestingly, issues related to scalability in higher dimensions lead to considering the Lagrangian dual problem of the QP. This is joint work with J. Carlsson.
Tuesday, Aug. 2, 2022, 3 p.m. through Tuesday, Aug. 2, 2022, 3:45 p.m.
John Berman (University of Massachusetts)
I will review the theory of ramification in number theory and then show that being totally ramified or unramified is equivalent to a natural condition in higher algebra. This leads to a much simplified calculation of THH of a ring of integers in a number field, relying on ramified descent (a kind of weaker etale descent).
Tuesday, Aug. 2, 2022, 2:30 p.m. through Tuesday, Aug. 2, 2022, 3:15 p.m.
Ling Zhou (The Ohio State University)
Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by softwares, such as Ripser. The cup product operation which is available at cohomology level gives rise to a graded ring structure that extends the usual vector space structure and is therefore able to extract and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the cup-length, which is useful for discriminating spaces. In this talk, we lift the cup-length into the persistent cup-length function for the purpose of capturing ring-theoretic information about the evolution of the cohomology (ring) structure across a filtration. We show that the persistent cup-length function can be computed from a family of representative cocycles and devise a polynomial time algorithm for its computation. We furthermore show that this invariant is stable under suitable interleaving-type distances.
Tuesday, Aug. 2, 2022, 1:30 p.m. through Tuesday, Aug. 2, 2022, 2:15 p.m.
Chad Giusti (University of Delaware)
Tuesday, Aug. 2, 2022, 10:30 a.m. through Tuesday, Aug. 2, 2022, 11:30 a.m.
Anibal Medina-Mardones (Max Planck Institute for Mathematics)
In order to incorporate ideas from algebraic topology in concrete contexts such as topological data analysis and topological lattice field theories, one needs effective constructions of concepts defined only abstractly or axiomatically. In this talk, I will discuss such constructions for certain invariants derived from the cup product on the cohomology of spaces or, more specifically, from an E∞-structure on their cochains. Together with allowing for the concrete computation of finer cohomological invariants in persistent homology -Steenrod barcodes- these effective constructions also reveal combinatorial information connected to convex geometry and higher category theory.