Past Workshops

Abstract: Knots associated to overtwisted manifolds are less explored. There are two types of knots in an overtwisted manifold - loose and non-loose. Non-loose knots are knots with tight complements where as loose knots have overtwisted complements. While we understand loose knots, non-loose knots remain a mystery. The classification and structure problems of these knots vary greatly compared to the knots in tight manifolds. Especially we are interested in how satellite operations on a knot in overtwisted manifold changes the geometric property of the knot. In this talk, I will discuss under what conditions cabling operation on a non-loose knot preserves non-looseness. This is a joint work with Etnyre, Min and Mukherjee.

Abstract: One of the interesting problems in 3-dimensional contact geometry is to classify Legendrian knots and study their various properties since they have several applications such as an interaction with Heegaard Floer homology, a rich source of producing new contact manifolds, etc. There has been several attempts to classify Legendrian knots in contact 3-spheres. For example, Eliashberg and Fraser classified Legendrian unknots in every contact 3-spheres, Entyre and Honda classified Legendrian torus knots in the standard 3-sphere, and Geiges and Onaran classified Legendrian left-handed trefoil in overtwisted contact 3-spheres. In this talk, we will classify Legendrian torus knots in overtwisted 3-spheres. This is a joint work with John Entyre and Anubhav Mukherjee.

Abstract: We investigate the conjectural relation between the asymptotic translation lengths of the mapping class group action on the curve complexes and normal subgroups of the mapping class groups. We present a moral support to the conjecture coming from fibered hyperbolic 3-manifolds, and also discuss the relation to the invariant homology. This talk is based on joint works with subsets of the set {Eiko Kin, Dongryul M. Kim, Hyunshik Shin, Philippe Tranchida, Chenxi Wu}.

Abstract: When one glues a finite number of tetrahedra together along their faces at random, the probability that the resulting complex is a manifold tends to zero as the number of tetrahedra grows. However, the only non-manifold points are the vertices of this complex. So, if we truncate the tetrahedra at their vertices, we obtain a random manifold with boundary. This talk will be about the geometry and topology of that manifold. This is joint work with Jean Raimbault.

Abstract: A major goal in complex dynamics is to understand dynamical moduli spaces; that is, conjugacy classes of holomorphic dynamical systems. One of the great successes in this regard is the study of the moduli space of quadratic polynomials; it is isomorphic to the complex plane. This moduli space contains the famous Mandelbrot set, which has been extensively studied over the past 40 years. Understanding other dynamical moduli spaces to the same extent tends to be more challenging as they are often higher-dimensional. In this talk, we consider the moduli space of quadratic rational maps, which is isomorphic to C^2. We will focus on special algebraic curves, called "Milnor curves" in this space. In general, it is unknown if Milnor curves are irreducible over C. Because these curves are smooth, this is equivalent to asking whether they are connected. We will exhibit the first infinite collection of Milnor curves that are connected. This is joint work with X. Buff and A. Epstein.

Abstract: We apply Menke's JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces. For large families of contact structures on Seifert fibered spaces over S^2, we reduce the problem of classifying symplectic fillings to the same problem for universally tight or canonical contact structures. We show that fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a symplectic 2-handle to a filling of a lens space. This is joint work with Austin Christian.

Abstract: The Legendrian invariant in knot Floer homology, defined by Lisca, Ozsváth, Stipsicz and Szabó, is torsion for knots in overtwisted structures, and it is non-zero only if the knot is strongly non-loose as a transverse knot. Using a correspondence between the knot invariants and invariants of contact surgeries, I will show that strongly non-loose transverse realizations of negative torus knots are classified by their invariants and that their U-torsion order equals one.

Abstract: The surgery number of a 3-manifold M is the minimal number of components in a surgery description of M. Computing surgery numbers is in general a difficult task and is only done in a few cases. In this talk, I want to report on the same question for contact manifolds. In particular, we will study a method to compute contact surgery numbers for contact structures on some Brieskorn spheres. This talk is based on joint work with John Etnyre and Sinem Onaran.

Abstract: Recently, two kinds of real-valued homology cobordism invariants of oriented homology 3-spheres were introduced by Daemi and Nozaki-Sato-Taniguchi. Their methods are based on quantitative constructions of instanton Floer homology. We develop local equivalence theory in the setting of “quantitative instanton Floer theory”. From this viewpoint, we give several new real-valued homology cobordism invariants and prove a connected sum formula for Daemi’s invariants. As applications, we give several new facts on the homology cobordism group. This is joint work with Aliakbar Daemi and Kouki Sato.

Abstract: A classical question of Casson asks which 3-manifolds bound 4-manifolds with the rational homology of a ball (also known as "rational balls"). Motivated by results on singular curves in the complex projective plane, we study Casson's question for 3-manifolds obtained as positive, integral surgeries along positive torus knots and their cables. For torus knot we obtain a complete answer, using a combination of Donaldson's theorem and Heegaard Floer homology. This is joint work with Paolo Aceto, Kyle Larson, and Ana G. Lecuona (arXiv:2008.06760).

Abstract: We introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution over any homology ball, rather than a particular contractible manifold. As an application, we define a modification of the homology cobordism group which takes into account an involution on each homology sphere, and prove that this admits an infinite-rank subgroup of strongly non-extendable corks. Using our invariants, we establish several new families of corks and prove that various known examples are strongly non-extendable. This is joint work with Matt Hedden and Abhishek Mallick.