14:00 - 14:30 Registration (Physics and Astronomy 100)
14:30 - 15:30 Emmy Murphy (University of Toronto)
Title: Local symplectic mapping classes and symplectic pseudo-isotopy
Abstract: Given a contact manifold Y, a symplectic pseudo-isotopy is a symplectomorphism of R x Y, which is the identity at $-\infty$ and cylindrical at $+\infty$. These groups fit into a number of Serre fibrations with other relevant spaces, such as contactomorphism groups, embedding spaces of Lagrangians, or Symp_cpt(R^{2n}). We discuss some recent progress in this direction, related to Weinstein handlebody theory and Legendrian flexibility.
15:30 - 16:00 Vivian He (University of Toronto)
Title: Square-tiled surfaces and curve counting.
Abstract: Fix a non-positively curved surface X of genus g. We prove that the number of simple closed curves on X of length at most L is at least on the order of CL^{6g-6}, where C is a constant independent of the metric. Our proof uses combinatorial tools from square-tiled surfaces and generalizes Rivin's curve counting result on hyperbolic surfaces.
16:00 - 16:30 Coffee/tea break
16:30 - 17:30 Dan Christensen (Western University)
Title: Using universes to describe Eilenberg-Mac Lane spaces, Euler classes and cup products
Abstract: Motivated by homotopy type theory, but using traditional topological terminology, I will illustrate how taking seriously the idea of a "space of spaces" leads to new insights into classical topics. I'll begin by discussing "central" spaces, and how they can be used to give new models of Eilenberg-Mac Lane spaces whose elements are themselves oriented Eilenberg-Mac Lane spaces. Using these, we can give a new description of the Euler class of an oriented sphere bundle, and a concrete description of the cup product operation in integral cohomology. The take-away message will be about the general techniques, rather than these specific results, with the hope that the use of universes in topology will spread.
9:00 - 10:00 Duncan McCoy (L’Université du Québec à Montréal)
Title: Cusps of arithmetic hyperbolic manifolds.
Abstract: In contrast to dimensions two and three, where hyperbolic manifolds are ubiquitous, hyperbolic manifolds in higher dimensions are much harder to construct and their topology less well understood. The Margulis thick-thin decomposition implies that a non-compact hyperbolic manifold can be decomposed into a compact piece along with a collection of cusps, which are subsets diffeomorphic to the product of a flat manifold with an open interval. One natural question is thus to ask which combinations of flat manifolds can be realized as cusp cross-sections in some hyperbolic manifold. I will discuss some aspects of this question with an emphasis on the case of arithmetic hyperbolic manifolds. In particular, I will explain how one can characterize which flat manifolds arise as a cusp cross-section in a given commensurability class of arithmetic hyperbolic manifolds and, time permitting, some mildly interesting examples. This is joint work with Connor Sell.
10:00 - 10:30 Yikai Teng (Rutgers University-Newark)
Title: End Khovanov homology and Lagrangian exotic planes
Abstract: In this talk, we define the end Khovanov homology, which is an invariant for properly embedded surfaces in $\mathbb{R}^4$ up to ambient diffeomorphisms. Moreover, we apply this invariant to detect the first known example of an exotic Lagrangian plane in $\mathbb{R}^4$.
10:30 - 11:00 Changjie Chen (CRM, Université de Montréal)
Title: Morse theory on moduli of curves
Abstract: In 1997, Sarnak conjectured that the determinant of the Laplacian is a Morse function on the space of unit area Riemannian metrics on a given real surface, and hence induces a Morse function on its moduli space. On the other hand, the systole function, defined as the length of a shortest closed geodesic with respect to the base metric, is topologically Morse on the moduli space M_{g,n}, proven by Akrout, though it does not yield a Morse theory. In this talk, I will introduce a family of Morse functions, denoted sys_T, defined as weighted exponential averages of all geodesic-length functions, on the Deligne-Mumford compactification (M_{g,n} bar). These functions are compatible with the Deligne-Mumford stratification and the Weil-Petersson metric, and their critical points can be characterized by a combinatorial property. I will talk about an index gap theorem for sys_T and its homological consequences, in the form of a stability theorem for the homology of moduli spaces of stable curves. If time permits, I will explain how sys_T connects to Sarnak’s conjecture.
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11:00 - 11:30 Coffee/tea break
11:30 - 12:30 Alexander Kupers (University of Toronto)
Title: Disc-presheaves and smooth structures
Abstract: The configuration spaces of points with framings in a manifold, together with natural point-forgetting and -splitting maps, assemble to an object known as a Disc-presheaf. To what extent is this Disc pre-sheaf sensitive to the smooth structure? I will explain joint work with Ben Knudsen, Manuel Krannich, and Fadi Mezher on this problem.
12:30 - 14:00 Lunch break + Conference Photo
14:00 - 15:00 Kristen Hendricks (Rutgers University)
Title: Heegaard Floer homology and equivariant cobordism
Abstract: The integer homology cobordism group is a major object of interest in low-dimensional topology; in the past decade, rapid progress has been made on understanding its structure using new versions of invariants from gauge and Floer theory. In this talk we consider (two versions of) the equivariant analog of the homology cobordism group, and its forgetful map to the ordinary homology cobordism group. In particular we show how a new equivariant surgery formula for Ozsváth and Szabó's three-manifold invariant Heegaard Floer homology can be used to construct a Z^infty summand within this kernel, extending a result of Dai-Hedden-Mallick. This is joint work with A. Mallick, M. Stoffregen, and I. Zemke.
15:00 - 16:00 Coffee/tea break + collaboration time
16:00 - 17:00 Michael Wong (University of Ottawa)
Title: A first step towards the sutured Floer homology of a profinite cyclic cover
Abstract: The Alexander polynomial can be understood as encoding the first homology of the infinite cyclic cover of the exterior of a knot. Is there a reasonable analogous understanding of its categorification, knot Floer homology? In particular, what is the sutured Floer homology of the infinite cyclic cover of the exterior of a knot? Based on a computation using bordered Floer bimodules, it seems that the answer, if it made sense, would take the form of an infinite tensor product of bimodules. But such objects do not behave well at all. Inspired by a parallel story in number theory, we are led to consider instead profinite tensor products (corresponding to profinite cyclic covers), which, once defined, will have much better properties. In this talk, I will explain the story above, and outline the construction of a profinite tensor product as well as some potential future directions. This is joint work in progress with David Treumann.
9:00 - 10:00 Jean Pierre Mutanguha (McGill University)
Title: Canonical decompositions of free-by-cyclic groups
Abstract: Free-by-cyclic groups can be defined as mapping tori of free group automorphisms. I will discuss a dynamical decomposition of automorphisms that produces a canonical decomposition of the corresponding free-by-cyclic groups. This will involve the partial order on attracting laminations for an automorphism. The decomposition of free-by-cyclic groups can be considered an extension of the JSJ decomposition of 3-manifolds. This talk is on joint work with Spencer Dowdall, Yassine Geurch, Radhika Gupta, and Caglar Uyanik.
10:00 - 10:30 Geunyoung Kim (McMaster University)
Title: Local modifications of surfaces and their 4–dimensional Montesinos trick
Abstract: We study how local modifications of surfaces in 4-manifolds affect their double branched covers. The moves we consider include annular rational tangle replacement, ribbon move, and tubing. We prove that these operations correspond to torus surgery, Pochette surgery, and 1 or 2-surgery on the double branched covers. This correspondence can be regarded as a 4-dimensional analogue of Montesinos’ trick, which relates crossing changes on knots to Dehn surgeries in 3-manifolds. This is joint work with Puttipong Pongtanapaisan.
10:30 - 11:00 Shintaro Fushida-Hardy (University of Waterloo)
Title: A Construction of Lagrangian Surfaces
Abstract: We describe an approach for finding Lagrangian surfaces in symplectic 4-manifolds, and discuss an open problem that could be resolved by an appropriate construction. This is joint work with Laura Wakelin.
11:00 - 11:30 Coffee/tea break
11:30 - 12:30 Maggie Miller (University of Texas at Austin)
Title: Fibered knots in 4D
Abstract: Fibrations of knots in the 3-sphere have surprising connections to 4-dimensional topology. I’ll talk about some open problems in dimension four, their connections to classical knot theory, and how to do some interesting constructions in 4D.
12:30 - 14:00 Lunch break
14:00 - 16:00 Collaboration time