14:30 - 15:00 Registration
15:00 - 16:00 Kyle Hayden (University of Rutgers-Newark)
Title: Exotic aspherical 4-manifolds
Abstract: The Borel conjecture predicts that closed aspherical manifolds (i.e., those whose higher homotopy groups all vanish) are topologically rigid: they are determined up to homeomorphism by their fundamental group. I will discuss the smooth analog of the Borel conjecture (concerning manifolds up to diffeomorphism), which is true in dimensions ≤ 3 but long known to be false in all dimensions ≥ 5. In particular, I will explain joint work with Davis, Huang, Ruberman, and Sunukjian that resolves the remaining 4-dimensional case by detecting exotic smooth structures on certain closed aspherical 4-manifolds. Time permitting, I will discuss a few other recent and potential future applications of these techniques in dimension four.
16:00 - 16:30 Matteus Froese (University of Waterloo)
Title: A Matroid Analogue of the Forested Graph Complex:
Abstract: The moduli space of graphs is a topological space of interest in the study of the homology of $Out(𝐹_𝑛)$, the outer automorphism group of the free group of rank 𝑛. One useful tool for computing this homology is the forested graph complex, a chain complex generated by graphs equipped with oriented spanning forests, modulo automorphisms. In this talk, we introduce a new approach to studying this homology using matroids, a natural generalization of graphs. We use this perspective to give new representatives of the known homology elements in the original forested graph complex. We then construct matroidal analogues of the forested graph complex and compute their homology. These results may lead to more interest in a "moduli space of matroids" and what this may tell us about physics, group theory, and algebraic topology. This is joint work with Michael Borinsky and Rowan Yeats.
16:30 - 17:00 Coffee/tea break
17:00 - 18:00 Spiro Karigiannis (University of Waterloo)
Title: A tale of two Lie groups
Abstract: The classical Lie group SO(4) is well-known to possess a very rich structure, relating in several ways to complex Euclidean spaces. This structure can be used to construct the classical twistor space Z over an oriented Riemannian 4-manifold M, which is a 6-dimensional almost Hermitian manifold. Special geometric properties of Z are then related to the curvature of M, an example of which is the celebrated Atiyah-Hitchin-Singer Theorem. The Lie group Spin(7) is a particular subgroup of SO(8) determined by a special 4-form. Inriguingly, Spin(7) has several properties relating to complex Euclidean spaces which are direct analogues of SO(4) properties, but sadly (or intrestingly, depending on your point of view) not all of them. I will give a leisurely introduction to both groups in parallel, emphasizing the similarities and differences, and show how we can nevertheless at least partially succeed in constructing a “twistor space” over an 8-dimensional manifold equipped with a torsion-free Spin(7)-structure. (I will define what those are.) This is joint work with Michael Albanese, Lucía Martín-Merchán, and Aleksandar Milivojevic. The talk will be accessible to a broad audience.
09:00 - 10:00 Diana Hubbard (CUNY)
Title: Twisting and rank bounds for homological link invariants
Abstract: The fractional Dehn twist coefficient (FDTC) is a number that measures the amount of twisting a mapping class of the surface effects about a chosen boundary component. In this talk I will define the FDTC, discuss its context in low-dimensional topology and geometry, and talk about recent work with Fraser Binns showing that there are bounds on the ranks of link Floer homology (of fibered links) and annular Khovanov homology (of the closures of braids) arising from the FDTC.
10:00 - 10:30 Isaac Broudy (Cornell University)
Title: Simultaneous Laminations
Abstract: Calegari introduced the laminations $\Lambda^\pm_u$ associated to a universal circle. Recently, Landry, Minsky, and Taylor showed that for any closed hyperbolic three-manifold admitting a taut foliation and an almost transverse pseudo-Anosov flow, the boundary of the orbit space of the flow is a universal circle for the foliation. In this talk I will provide a characterization of the $\Lambda^+_u$ and $\Lambda^-_u$ in terms of the stable and unstable prelaminations on the boundary of the orbit space. Additionally, I will present a rigidity result, which proves that $\Lambda^\pm_u$ are determined up to finite ambiguity by the action $\pi_1(M)\curvearrowright \partial\mathcal{O}$.
10:30 - 11:00 Coffee/tea break
11:00 - 12:00 Paul-Hermann Balduf (University of Waterloo/Perimeter Institute)
Title: Topological field theory and graph complex homology
Abstract: Topological field theory is a version of quantum field theory where the observables depend only on the topology of the underlying manifold, but not on further details such as positions or distances. Traditionally, the questions and methods in topological field theory have been rather different from the ones in more conventional quantum field theories such as electrodynamics. In recent years, much work has been done to better understand the connection. I will describe a novel formalism to set up perturbative calculations in topological field theory in terms of integrals over rational functions. Unexpectedly, it turns out that the same integrals also compute cohomology classes in Kontsevich's odd commutative graph complex. Based upon joint work with Davide Gaiotto 2408.03192 and Simone Hu 2503.09558.
12:00 - 13:30 Lunch
13:30 - 14:30 Jesse Madnick (Seton Hall University)
Title: Calibrated Geometry, Hyperbolicity, and Montel’s Theorem
Abstract: Calibrated geometry is the study of area-minimizing submanifolds, often in Riemannian manifolds with special geometry (such as Kahler, Calabi-Yau, G2-, or Spin(7) geometry). While much is known in the Kahler case, calibrated geometry remains largely mysterious in virtually all other settings. In complex geometry, “hyperbolicity” refers to the interplay of three a priori unrelated ideas: (1) negative curvature; (2) the scarcity of holomorphic lines, and (3) the non-degeneracy of certain invariant pseudo-distances. In complex analysis, Montel’s theorem relates these hyperbolicity notions to the metric properties of the space of holomorphic disks. In this talk, we generalize these hyperbolicity notions, as well as the theorems that connect them (including Montel’s theorem), to arbitrary calibrated manifolds. This is joint work with Kyle Broder (Queensland), Anton Iliashenko (BIMSA), and Spiro Karigiannis (Waterloo).
14:30 - 15:30 Steven Boyer (UQAM)
Title: Circular orders and the topology of 3-manifolds
Abstract: The L-space conjecture connects the left-orderabilty of the fundamental group of a 3-manifold W to the existence of a taut foliation on W on the one hand, and to the Heegaard Floer homology of W on the other. In this talk, I'll discuss some recent results on a conjectured connection, due to Idrissa Ba and Adam Clay, of the circular-orderabilty of the fundamental group of W to foliations and Heegaard Floer theory on its finite cyclic covers. This is joint work with Cameron Gordon and Ying Hu.
15:30 - 16:30 Collaboration Time
16:30 - 17:30 Bena Tshishiku (Brown University)
Title: Open book decompositions and Morse complexity
Abstract: An open book decomposition of a manifold is a structure generalizing a fibration over the circle. For 3-manifolds open book decompositions are closely related to contact structures. In general we do not have good methods to determine which closed aspherical manifolds have open book decompositions. We study this problem for locally symmetric manifolds and explain a connection to Morse complexity, a semi-norm on homology introduced by Gromov, for which we give new computations. This is joint work with Fedor Manin and Shmuel Weinberger.
09:00 - 10:00 Piotr Przytycki (McGill University)
Title: K(pi,1) conjecture for 3-dimensional Artin groups
Abstract: This is joint work with Jingyin Huang. I will outline our proof of the K(pi,1) conjecture for Artin groups of dimension at most 3. The key Artin group is the one whose Coxeter group is the isometry group of the icosahedral tiling of the hyperbolic space with dodecahedral vertex links. A related square complex satisfies a new type of combinatorial nonpositive curvature.
10:00 - 10:30 Theofanis Chatzidiamantis-Christoforidis (Western University)
Title: Fixed Points in Synthetic Homotopy Theory
Abstract: Homotopy type theory (HoTT) is a formal system in which one can do model-independent, or synthetic, homotopy theory. In particular, anything proved in HoTT can be interpreted in any (∞, 1)-topos. We investigate what it means for a space to have the property "every self-map has a fixed point" in HoTT, inspired by work of Szymik in the category of topological spaces. We will introduce the language of HoTT, compare the property we are investigating to those considered by Szymik, and prove that every classifying space of a finite, simple, non-Abelian group satisfies our fixed point property. Along the way, we compute the homotopy groups of the space of maps into a classifying space in HoTT. This talk is based on joint work with Dan Christensen.
10:30 - 11:00 Coffee/tea break
11:00 - 11:30 Brian Udall (Rice University)
Title: Asymptotic rigidity in big mapping class groups
Abstract: The study of variations of Thompson's groups and Houghton's groups has a rich and interesting history. The "big" mapping class groups of certain infinite surfaces and graphs contain such variations, known as the asymptotically rigid subgroups. The first example is due to Funar-Kapoudjian, who constructed a finitely presented group containing the mapping class group of every genus 0 compact surface. In this talk I will give an outline of the ideas around asymptotic rigidity, with a focus on variations on Houghton's groups. In analogy with the classical setting and with surface Houghton groups, I'll discuss joint work with Hill-Kwak-West on graph Houghton groups, showing that for a graph with n ends all accumulated by loops, its Houghton group is type F_{n-1}, but not type F_n. Open questions and more examples will also be given.
11:30 - 12:30 Yvon Verberne (Western University)
Title: Spherical and affine Artin groups: Finite quotients, the isomorphism problem, and profinite rigidity.
Abstract: In this talk, we will begin by introducing the spherical and affine Artin groups. We will discuss how to determine the smallest nonabelian finite quotients, and how we can leverage this result to prove that the isomorphism problem is solvable amongst the spherical and affine Artin groups. Additionally, we discuss how to apply our result on the smallest nonabelian quotients of spherical Artin groups to prove that spherical Artin groups are profinitely rigid. This work is joint with Sam Hughes, Thomas Ng, Kaitlin Ragosta, and Nancy Scherich.
13:30 - Lunch and Collaboration Time