Abstract: I will talk about the following question of Gromov: what closed manifolds can be efficiently wrapped with Euclidean wrapping paper? That is, for what M is there a 1-Lipschitz map $\mathbb R^n \to M$ with positive asymptotic degree? Gromov called such manifolds elliptic. We show that, for example, the connected sum of k copies of CP^2 is elliptic if and only if k ≤ 3. I will try to explain the intuition behind this example, how it extends to a more general dichotomy governed by the de Rham cohomology of M, and why ellipticity is central to the program of understanding the relationship between topology and metric properties of maps.
If I have time, I'll also explain why for a non-elliptic M, a maximally efficient map $\mathbb R^n \to M$ must have components at many different frequencies (in a Fourier-analytic sense), and even then it's at best logarithmically far from having positive asymptotic degree. This is joint work with Sasha Berdnikov and Larry Guth.
Abstract: Instanton Floer homology is introduced by Floer in 1980s. It is a powerful invariant for 3-manifolds and knots and links inside them. In this talk, I will present a surgery formula for instanton theory, which describes the instanton Floer homology of a 3-manifold coming from Dehn surgeries along knots. This formula can be applied in computing the instanton Floer homology of surgery 3-manifolds and study the SU(2)-representations of fundamental groups of 3-manifolds. In particular, using this technique, we could prove that the fundamental group of 3-surgery along any non-trivial knots in S^3 admits an irreducible SU(2) representation, answering a question by Kronheimer and Mrowka proposed in 2004. This is a joint work with John Baldwin, Steven Sivek, and Fan Ye.
For S a closed surface of genus at least 2, Labourie proved that every Hitchin representation of pi_1(S) into PSL(n,R) gives rise to an equivariant minimal surface in the corresponding symmetric space. He conjectured that uniqueness holds as well (this was known for n=2,3), and explained that if true, then the space of Hitchin representations admits a mapping class group equivariant parametrization as a holomorphic vector bundle over Teichmuller space. After giving the relevant background, we will discuss the geometry of large area minimal surfaces in symmetric spaces, and explain how such surfaces can give counterexamples to Labourie’s conjecture. We will then share some new questions about minimal surfaces.
Abstract: The commensurator of a group consists of isomorphisms between its finite index subgroups. Geometrically, the commensurator encodes isometries between finite covers of a Riemannian manifold. I will discuss a question raised independently by Greenberg and Shalom: Can an infinite discrete subgroup of a simple Lie group have dense commensurator and not be a lattice? I will explain the surprising connections between this question and other long-standing open problems, and discuss recent progress on special cases of the question. This is joint work with (subsets of) Brody, Fisher, and Mj.
Abstract: We prove an exact triangle relating the knot instanton homology to the instanton homology of surgeries along the knot. As the knot instanton homology is computable in many instances, this sheds some light on the instanton homology of closed 3-manifolds. We illustrate this with computations in the case of some surgeries on the trefoil which are different from the analogous groups in other Floer theories such as Heegaard Floer and monopole Floer. Finally, we sketch the proof of the triangle.
Abstract: The fundamental theorem of category theory is the Yoneda lemma, which in its simplest form identifies natural transformations between represented functors with morphisms between the representing objects. The ∞-categorical Yoneda lemma is surprisingly hard to prove --- at least in the traditional set-based foundations of mathematics. In this talk we'll describe the experience of developing ∞-category theory in an alternate foundation system based on homotopy type theory, in which constructions determined up to a contractible space of choices are genuinely "well-defined" and elementwise mappings are automatically homotopically-coherently functorial. In this setting the proof the ∞-categorical Yoneda lemma is arguably easier than the 1-categorical Yoneda lemma. We'll end by posing the question as to whether similar foundations would be useful for other "higher structures." This is based on joint work with Mike Shulman and involves computer formalizations written in collaboration with Nikolai Kudasov and Jonathan Weinberger.
Abstract: In the 1970s, Thurston generalized the classification of self-maps of the torus to surfaces of higher genus, thereby completing the work first initiated by Nielsen. This is known as the Nielsen-Thurston Classification Theorem. A well-known proof of this theorem is due to Bers, who reformulated the problem in terms of extremal quasiconformal maps between complex surfaces. In joint work with Camille Horbez, we revisit Bers's approach but from the point of view of hyperbolic geometry. This gives a new proof of the classification theorem, as well as new representatives for pseudo-Anosov homeomorphisms as extremal Lipschitz maps between hyperbolic surfaces. As another application, we also classify the isometries of the Thurston metric on Teichmuller space.
Abstract: In this talk, I will present an explicit formula we found for the Reidemeistertorsion of a closed oriented hyperbolic 3-manifold twisted by the adjoint action ofthe holonomy representation of the fundamental group of the manifold. This is tothe best of our knowledge the first known explicit formula for such quantity. Thisis a joint work with Ka Ho Wong.
Abstract: The moduli space of genus g tropical curves with n marked points is a fascinating topological space, with a combinatorial flavor and deep algebro-geometric meaning. In the algebraic world, forgetting the n marked points gives a fibration whose fibers are configuration spaces of a surface, and Serre's spectral sequence lets one compute the cohomology "in principle". In joint work with Bibby, Chan and Yun, we construct a surprising tropical analog of this spectral sequence, manifesting as a graph complex and featuring the cohomology of compactified configuration spaces on graphs.
Abstract: We describe an approach to Bialynicki-Birula theory for holomorphic C^* actions on complex analytic spaces and Morse-Bott theory for Hamiltonian functions for the induced circle actions. A key principle is that positivity of a suitably defined "virtual Morse-Bott index" at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum. Inspired by Hitchin’s 1987 study of the moduli space of Higgs monopoles over Riemann surfaces, we apply our method in the context of the moduli space of non-Abelian monopoles or, equivalently, stable holomorphic pairs over a closed, complex, Kaehler surface. We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and show that these indices are positive in a setting motivated by a conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type (including symplectic four-manifolds) obey the Bogomolov-Miyaoka-Yau inequality.
Abstract: In this talk, we will consider a generalization of alternating links and their complements in thickened surfaces. We will define a family of these links with a specific type of right-angled structure on their complements (RGCR links) and show that this property is equivalent to the links having two totally geodesic checkerboard surfaces, and equivalent to a set of restrictions on the links' alternating projection diagrams. If time permits, we will then use these diagram restrictions to consider the commensurability classes of RGCR links.
Abstract: The Culler--Vogtmann's Outer space $CV_n$ is a space of marked metric graphs, and it compactifies to a set of $F_n$-trees. Each $F_n$-tree on the boundary of Outer space is equipped with a length measure, and varying length measures on a topological $F_n$-tree gives a simplex in the boundary. The extremal points of the simplex correspond to ergodic length measures. By the results of Gabai and Lenzhen--Masur, the maximal simplex of transverse measures on a fixed filling geodesic lamination on a complete hyperbolic surface of genus $g$ has dimension $3g-4$. In this talk, we give the maximal simplex of length measures on an arational $F_n$-tree has dimension in the interval $[2n-7, 2n-2]$. This is a joint work with Mladen Bestvina, Jon Chaika, and Elizabeth Field.
Abstract: A phenomenon in topology is said to be stable if it occurs in all sufficiently high dimensions. As discovered by Quillen over five decades ago, such phenomena are closely related to number theory, and can often be described in terms of arithmetic objects known as formal groups. Unfortunately, in general this dictionary is not quite one-to-one, and many periodicities one sees on the arithmetic side become broken and more complex in the world of topology. In this talk, I will describe a solution to an old conjecture of Franke that the arithmetic - topology correspondence can be refined to an equivalence of categories when the ambient prime is sufficiently large.