Spring 2024


Date Speaker Title Abstract
Jan 23 Sam Ballas Bibundles I Groupoids are objects that simultaneously generalize groups, sets, and relations. Together with functors and natural transformations groupoids form a 2-category. While very nice, this 2-category suffers from defects if one wants to introduce topology to the picture. Roughly speaking, this category has too few morphisms. Bibundles are an attempt to rectify this situation by enlarging the class of morphisms. In this talk I will try to introduce the basic concepts described above and show how bibundles help to solve certain gluing problems that arise from topological groupoids. I am not at all an expert in this area, so I will try to keep everything pretty basic and accessible to those who only know very basic category theory.
Jan 30 Sam Ballas Bibundles II This will be the second part of my talk on bibundles. This time we will see that bibundles are a generalization of functors between groupoids that behave well with respect to natural gluing constructions.
Feb 6 Aziz Guelen (Ohio State) Orthogonal Möbius Inversion and Grassmannian Persistence Diagrams We introduce the notion of orthogonal Möbius inversion, which can be applied to functions that take inner product spaces as values. When applied to the birth-death spaces, or to the space of harmonic persistent cycles (i.e. the kernel of the persistent Laplacian), we prove that one produces canonical representatives for each bar in the barcode of a filtration. Furthermore, we establish that these representatives are stable with respect to the edit distance.
Feb 13 Mario Gomez The Four Point Condition as the Tropicalization of Ptolemy's Inequality Ptolemy's inequality, named after the Greek astronomer and mathematician Claudius Ptolemy, is a theorem that relates the six distances of a quadrilateral in Euclidean space. The case of equality, known as Ptolemy's theorem, allowed him to write a precise trigonometric table in the 2nd century AD. In my talk, I will go over the history and importance of the theorem and show a proof of the generalizations to non-Euclidean geometries given by J. Valentine in the 70s. The proof involves other classical determinants in metric geometry which can be used, for example, to argue why the Earth is spherical. I will also show that the degenerate case of the inequality as curvature goes to negative infinity is the 4-point condition, i.e. the condition for 0-hyperbolicity. This is a joint result with F. Mémoli.
Feb 20 Phil Bowers Negatively curved spaces and groups and the Cannon conjecture I will introduce \(\delta\)-negatively curved spaces and groups (word hyperbolic groups) and state the Cannon conjecture and discuss Cannon’s outline for proving the conjecture. Time permitting, I will outline the efforts of Cannon, Floyd, and Perry, and separately Cannon and Swenson to prove the conjecture.
Feb 27 Phil Bowers Negatively curved spaces and groups and the Cannon conjecture In this talk I will define the boundary of a negatively curved space carefully, describe the combinatorial half spaces and combinatorial disks defined on the boundary of a negatively curved group, and outline Cannon’s plan for proving the Cannon conjecture using the Combinatorial Riemann Mapping Theorem.
Mar 5 Phil Bowers Negatively curved spaces and groups and the Cannon conjecture This will be the third part in the series outlining Cannon's approach to resoslving the Cannon conjecture.
Mar 19 Florian Stecker Eigenvalues of subgroups of Lie groups Let Gamma be a subgroup of GL(n,R). Every matrix in Gamma has n eigenvalues, defining a point in R^n. I want to talk about a theorem by Benoist which describes the shape of the set of these points, assuming almost nothing about the group Gamma. Concretely, it shows that the eigenvalues form a convex cone. Finally, I want to talk about some questions I'm trying to answer about this cone, in the specific context of Anosov representations of triangle groups.
Mar 26 Jared Miller Conjugating Representations in PGL(k, C) into PGL(k, R) Properties of the space of representations of a surface group into a given simple Lie group is a very active area of research and is particularly relevant to higher Teichmüller theory. In this talk we study representations of finitely generated groups into PGL(k, C) and determine necessary and sufficient conditions for such a representation to be conjugate into PGL(k, R). In this way, we identify representations in the larger representation variety which are conjugate in PGL(k, C) to a representation in hom(pi_1 (S), PGL(k, R))/PGL(k, R).
Apr 2
Apr 9 Oishee Banerjee On configuration spaces and sieves Configuration space of a space X is the space of all finite subsets of X. Conf X comes up in the study of various other kinds of spaces as well. For example, in the 70’s and 80’s homotopy theorists studied and formulated relations between the Lie algebra cohomology of vector fields on manifold X, configuration space of X, and certain function space on X. In the last couple of decades, some algebraic geometers and number theorists joined in as well. But if you ask anyone “When is a space configuration-like?” the best answer you would get is “You know it when you see it”. Because what makes some space configuration-like has never been made quite precise. What’s more, (a modification of) the sieve method (from analytic number theory) turns out to be very efficient at unraveling the configuration-like properties of a space. We show that there’s a very simple thread that connects all these seemingly different ideas, and it lies in (homotopical) algebra.
Apr 16
Apr 23 Brandon Doherty Symmetry properties of the cubical Joyal model structure Via the cubical Joyal model structure, cubical sets having faces, degeneracies and connections can be viewed as models for (infinity,1)-categories; in this model, homotopies are most naturally defined using the geometric product, rather than the cartesian product. This is an alternative monoidal product having convenient properties, but with the drawback that it is not symmetric. In this talk, based on work in progress joint with Tim Campion, we discuss a comparison between the less structured cubical sets on which the cubical Joyal model structure is defined and cubical sets with symmetries, which allows us to prove that the geometric product is symmetric up to natural weak equivalence in the cubical Joyal model structure. If time permits, we will also discuss applications of this comparison to the construction of a Quillen-equivalent model structure on symmetric cubical sets, and the potential application of similar techniques to proving that the cubical Joyal model structure is monoidal with respect to the cartesian product.

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