Henry Adams: Hausdorff vs Gromov-Hausdorff distances

The goal of this talk is to show how tools from topology can be used to bound quantities arising in metric geometry. I'll begin by introducing the Hausdorff and Gromov-Hausdorff distances, which are ways to measure the "distance" between two metric spaces. Though Hausdorff distances are easy to compute, Gromov-Hausdorff distances are not. In the setting when X is a sufficiently dense subset of a closed Riemannian manifold M, we show how to lower bound the Gromov-Hausdorff distance between X and M by 1/2 the Hausdorff distance between them. The constant 1/2 can be improved depending on the dimension and curvature of the manifold, and obtains the optimal value 1 in the case of the circle. Our proofs begin by converting discontinuous functions between metric spaces into simplicial maps between nerve complexes. We then produce topological obstructions to the existence of such maps using the nerve lemma and the fundamental class of the manifold. Joint with Florian Frick, Sushovan Majhi, Nicholas McBride, available at https://arxiv.org/abs/2309.16648.

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Christin Bibby: Supersolvable posets and fiber-type arrangements

We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups. These bundles are akin to those of Fadell--Neuwirth for configuration spaces, and their existence is detected by a combinatorial property of an associated finite partially ordered set. We obtain a combinatorially determined class of K(pi,1) spaces, and under a stronger combinatorial condition prove a factorization of the Poincar\'e polynomial when the Lie group is noncompact. In the case of toric arrangements, this provides an analogue of Falk--Randell's formula relating the Poincar\'e polynomial to the lower central series of the fundamental group. This is joint work with Emanuele Delucchi.

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Elden Elmanto: Motivic cohomology of affine cones 

In the 80s, Beilinson conjectured a general theory of motivic cohomology as an algebro-geometric counterpart to singular cohomology in topology. For equicharacteristic schemes, such a theory was offered last year in joint work with Matthew Morrow.

I will give a working demo of this theory by resolving an old question of Srinivas which amounts concretely to: any corank zero vector bundle on the cone of a smooth projective variety over an algebraically closed field of positive characteristic splits off a rank one trivial summand.

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Hana Kong: Cofiber of Tau Method for C-Motivic Modular Forms and Multiplicative Structure 

In this talk, I will discuss joint work with Isaksen, Li, Ruan, and Zhu, on the analysis of the C-motivic Adams-Novikov spectral sequence applied to the C-motivic modular forms spectrum mmf, as well as the classical topological modular forms spectrum tmf. Our analysis resolves a previously unaddressed aspect regarding the multiplicative properties of the homotopy groups of tmf. 

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Piotr Pstragowski: Spectral weight filtrations

In the 1970s, Deligne used the resolution of singularities to define a canonical filtration on rational cohomology of a complex variety, called the weight filtration. Unlike the rational cohomology groups themselves, the filtration is an honest invariant of the algebraic structure, and they depend on more than the homotopy type of the variety. I will talk about joint work with Peter Haine in which we show that such a weight filtration exists on cohomology with respect to any complex-orientable cohomology theory. 

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Martina Rovelli: Limits and colimits of (∞,n)-categories

The universal properties of many objects in mathematics are encoded as those of limits and colimits of certain diagrams valued in an ordinary category. With the increasing popularity of fields where ordinary category theory falls short and which rely on higher category theory instead, in order to capture the correct universal property of objects of interest it then becomes importat to establish a coherent and practical theory of limits and colimits for diagrams valued in an n-category or an (∞, n)-category. We’ll recall what has been done for the case n = 1, and describe work in progress with Moser and Rasekh about the formulation a theory of (∞,n)-limits for the case of general n.

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Nick Salter: The equicritical stratification and stratified braid groups

Thinking about the configuration space of n-tuples in the complex plane as the space of monic squarefree polynomials, there is a natural equicritical stratification according to the multiplicities of the critical points. There is a lot to be interested in about these spaces: what are their fundamental groups (“stratified braid groups”)? Are they K(pi,1)’s? How much of the fundamental group is detected by the map back into the classical braid group? They are also amenable to study from a variety of viewpoints (most notably, they are related both to Hurwitz spaces and to spaces of meromorphic translation surface structures on the sphere). I will discuss some of my results thus far in this direction. Portions of this are joint with Peter Huxford.

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Maru Sarazola: A model structure for Grothendieck fibrations

The Grothendieck construction plays a central role in category theory: it gives an equivalence between the category of pseudo-functors from a fixed category C to Cat, and the category of Grothendieck opfibrations over C. This allows us to move between the settings of indexed categories and fibered categories, gaining access to tools and insights from both contexts. Unfortunately, the word "pseudo" cannot be omitted here, and it forces us to pass to the world of 2-categories to keep track of these higher coherence data. The goal of this talk is to explain how we can use homotopy theory to remain in a strict 1-categorical world, as long as we are willing to get a Quillen equivalence instead of a categorical equivalence. As a crucial step, this involves the construction of a new model structure on the slice category Cat/C whose fibrant objects are the Grothendieck fibrations. Based on joint work with Lyne Moser.

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Lori Ziegelmeier: U-match decomposition: working with large boundary matrices in persistent homology

Large simplicial complexes, some containing hundreds of billions of simplices, have become important tools for modeling and statistics in the field of data science. This is especially true of persistent homology, which quantifies the topology of noisy data sets across multiple scales of measurement. The size of these simplicial complexes prohibits storing the associated boundary matrices in computer memory. To circumvent this issue, we introduce a matrix factorization scheme called U-match decomposition which can rapidly and independently generate relevant individual rows and columns on the fly, similar to those of the boundary matrix. This decomposition scheme has been implemented in the Open Applied Topology (OAT) software package. We introduce three problems in which U-Match has shown promise: (1) calculating cycle representatives and optimizing these representatives, (2) uncovering the structure in an application to knowledge networks, and (3) developing an algorithm to compute persistent relative homology.