Topology Seminar—Fall 2023

The seminar meets each Tuesday at 3:05 pm in LOV 0231, unless otherwise noted.

Schedule and talks

• Sep 5
  • Organizational Meeting
• Sep 12 & Sep 19
  • Sam Ballas
  • Frame theory on vector bundles
  • Vector valued information can be transmitted by describing the coordinates of vectors in a fixed basis, however this method of transmission is not very robust since corruption of a single coordinate can drastically alter the description of the vector. To combat this it is often desirable to have a spanning set that is larger than a basis (such an object is called a frame) to provide a description of vectors that is more robust with respect to signal corruption. The study of frames for vector fields and their properties is called Frame Theory, and is an active area of current research. In recent work with Tom Needham and Clayton Shonkwiler, we develop a frame theory for vector bundles. The idea being that given a vector bundle one wishes to find a collection of sections that pointwise span the fibers. Finding such objects is somewhat subtle since by definition, non-trivial vector bundles don’t admit global bases of sections. More interestingly, we will see that it is always possible to find a frame with sufficiently many vectors and that the minimal numbers of sections in a frame is obstructed by algebraic topological invariants of the vector bundle.
• Sep 26 & Oct 3
  • Brandon Doherty
  • Cubical models of higher categories without connections
  • We will discuss the cubical Joyal model structures on the categories of cubical sets both with and without connections, by which cubical sets model the theory of (infinity,1)-categories, and the proofs that these model structures are Quillen equivalent to the Joyal model structure on simplicial sets. If time permits, we will also discuss the comical model structures on marked cubical sets which model (infinity,n)-categories for arbitrary n, and the proof that these are equivalent to the corresponding complicial model structures on marked simplicial sets.
• Oct 10
  • Mario Gómez Flores
  • Curvature sets: the spaces of distance matrices of subsets of \(S^1\)
  • For \(n \geq 2\), the \(n\)-th curvature set \(K_n(X)\) of a metric space \(X\) is the set of all \(n\)-by-\(n\) distance matrices of points sampled from \(X\). We study the topological and geometric structures of the curvature sets of \(S^1\) equipped with the geodesic metric. We draw inspiration from Gromov’s original definition and from similar constructions, like configuration spaces and Ran spaces. I will start with a survey of these related objects and general properties of curvature sets. I will then show that \(K_n(S^1)\) is the quotient of the \(n\)-torus by the diagonal action of O(2) and use this representation to compute the homology groups of \(K_n(S^1)\) (which often have torsion) using the Mayer-Vietoris sequence. I will also study \(K_n(S^1)\) geometrically by showing that it is a geometric simplicial complex embedded in \(R^n\) whose simplices are indexed by “orderings” of \(n\) points in \(S^1\). This paper emerged from a project at the 2017 Summer at ICERM program by Peter Eastwood, Anna M. Ellison, and Facundo Mémoli.
• Oct 17
  • Pierre-Louis Blayac (University of Michigan)
  • Divisible convex sets with properly embedded cones
  • A divisible convex set is a convex, bounded, and open subset of an affine chart of the real projective space, on which acts cocompactly a discrete group of projective transformations. These objects have a very rich theory, which involves ideas from dynamical systems, geometric group theory, \((G,X)\)-structures and Riemannian geometry with nonpositive curvature. Moreover, they are an important source of examples of discrete subgroups of Lie groups which are not lattices (although their construction often uses arithmetic lattices). For instance they have links with Anosov representations. In this talk, we will survey known examples of divisible convex sets, and then describe new examples obtained in collaboration with Gabriele Viaggi, of irreducible, non-symmetric, and non-strictly convex divisible convex sets in arbitrary dimensions (at least 3).
• Oct 24
  • Ronno Das (Stockholm University)
  • Hypersurfaces and inclusion-exclusion
  • A generic homogeneous polynomial defines a smooth hypersurface in projective space. Generalizing results of Tommasi and others, we establish homological stability for the space of such smooth polynomials, and more generally smooth sections of line bundles on projective varieties, with or without marked points on the hypersurface. On the way we develop a lift of the inclusion-exclusion formula as a simplicial space, building on work of Vassiliev. Parts of the talk will be based on joint works with Sean Howe and with Alexis Aumonier.
• Oct 31
  • Florian Stecker
  • Spherical homogeneous spaces and Anosov representations acting on them
  • We’ll look at the double quotient \(B\backslash G/H\), where \(G\) is a semisimple Lie group, \(B\) a minimal parabolic subgroup, and \(H\) any closed subgroup which makes the double quotient finite. Then consider certain discrete subgroups \(Γ\) of \(G\) and try to find an open subset of \(G/H\) on which \(Γ\) acts nicely, that is with a manifold quotient. This is joint work with León Carvajales.
• Nov 7
  • Jonathan DeWitt (University of Maryland)
  • Periodic data rigidity of Anosov diffeomorphisms
  • Anosov diffeomorphisms are a class of dynamical systems that exhibit strong chaotic behavior. Every known Anosov diffeomorphism is topologically conjugate to a linear Anosov diffeomorphism, called an Anosov automorphism, which comes from a simple algebraic construction. An invariant associated to periodic points called the periodic data provides a natural obstruction to this conjugacy being smooth. If an Anosov diffeomorphism has the same periodic data as an Anosov automorphism, then it is natural to ask whether that diffeomorphism is conjugate to the automorphism. In this case we say that the automorphism is periodic data rigid, as it is characterized by its periodic data. In this talk we show that a “generic” Anosov automorphism of the torus is periodic data rigid. In order to do this, we show that a linear cocycle over a hyperbolic system with constant periodic data has a hyperbolic splitting whenever the periodic data suggests it should. This is a joint work with Andrey Gogolev.
• Nov 14
  • Jared Miller
  • Exploring Infinite Type Surfaces
  • A surface is said to be of finite type if its fundamental group is finitely generated; otherwise we say it is of infinite type. Infinite type surfaces are, in a sense, much more mysterious than finite type surfaces. There has recently been a surge of interest in mapping class groups for infinite type surfaces, but many questions remain open. In this talk we will discuss the classification of infinite type surfaces and then explore the end-spaces, fundamental domains, and mapping class groups for several examples of infinite type surfaces.
• Nov 21
• Nov 28
• Dec 5
  • Homin Lee (Northwestern)
  • Smooth actions on manifold by higher rank lattices
  • We will discuss about smooth actions on manifolds by higher rank groups, such as lattices in \(\textrm{SL}(n,\mathbb{R})\) with \(n\ge 3\) or \(\mathbb{Z}^{k}\) with \(k\ge 2\).
    The higher rank property of the acting group suggests that the actions are rigid, which means that the action should have an algebraic origin, such as the Zimmer program and the Katok-Spatzier conjecture. One of the main topics is about how we can give an algebraic structure on the acting space which is a smooth manifold.
    We survey some of recent breakthroughs and then focus mainly on actions of higher rank lattices. In particular, we focus on actions on manifold with “positive entropy” by lattices in \(\textrm{SL}(n,\mathbb{R})\), \(n\ge 3\). When the manifold has dimension n, then we will see that the lattice is commensurable to \(\mathrm{SL}(n,\mathbb{Z})\) from certain “algebraic structure” on \(M\) coming from the dynamics.
    Part of the talk is ongoing work with Aaron Brown.