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Place: N818
Title: KHI <= HFK
Abstract: This is a work in progress with John Baldwin, Steven Sivek, and Fan Ye. Our goal is to establish a spectral sequence from the hat version of knot Floer homology (HFK) to instanton knot homology (KHI). Both HFK and KHI are influential Floer homology theories for knots in the 3-sphere, each yielding significant results in the study of 3-dimensional topology. Despite their successes, their constructions are fundamentally different: HFK is derived from Heegaard diagrams associated to the knot, while KHI arises from a set of partial differential equations. Hence, it is natural and challenging to study the relation between these two versions of Floer homologies. In this talk, I will outline an approach to bridge these two homology theories. I will begin by reviewing relevant known results and then discuss ideas for constructing the desired spectral sequence, building on these foundational insights.
Place: N818
Title: Symplectic packings in higher dimensions
Abstract: The problem of symplectically packing k symplectic balls into a larger one has been solved in dimension four, i.e. there is now a combinatorial criteria of when this is possible. However, not much is known about symplectic packing problems in higher dimensions. We take a step in this direction in dimension six, by considering a “stabilized” packing problem, i.e. we consider symplectically packing a disjoint union of four dimensional balls times a closed Riemann surface into a bigger ball times the same Riemann surface. We show this is possible if and only if the corresponding four dimensional ball packing is possible. The proof is a mixture of geometric constructions, pseudo-holomorphic curves, and h-principles. This is based on work with Kyler Siegel.
Place: N204
Title: The topology of positive scalar curvature
Abstract: Given a smooth manifold M, geometers and topologists ask the question “which geometries, i.e. which kind of Riemannian metrics” can one put on M? In recent years, this problem has been studied intensively for metrics with positive scalar curvature. The main questions are:
Place: N818
Title: 曲面同胚的边界不可压缩协边
Abstract: 我们探讨曲面同胚的边界不可压缩协边， 它的群结构和例子。
Place: N818
Title: Dolbeault cohomology on almost complex 4 manifolds and symplectic manifolds
Abstract: We will introduce the notions of two Dolbeault type cohomologies on almost complex manifolds. With such cohomology, we will give the ddc-lemma on compact almost complex 4manifolds and related applications and problems on compact almost complex 4 manifolds and symplectic4 manifolds.
Place: N818
Title: Topological classification of manifolds with positive isotropic curvature
Abstract: In this talk I will discuss the topological classification of compact manifolds with positive isotropic curvature. This curvature condition was introduced by Micallef and Moore in 1988, and played an important role in the proof of the differentiable sphere theorem by Brendle and Schoen. First I’ll briefly survey some of the previous works by various authors on Riemannian manifolds with positive isotropic curvature. Then I’ll introduce my recent work on the topological classification of compact manifolds of dimension n ≥ 12 with positive isotropic curvature. The main tool is Ricci flow with surgery, which was used by Perelman to attack the Poincare conjecture and Thurston’s geometrization conjecture. Techniques from topology are also used extensively.
Place: N818
Title: On (n-2)-connected 2n-dimensional Poincare complexes
Abstract: Poincare complexes, roughly speaking, are CW complexes which satisfy Poincare duality for arbitrary local coefficients. They share many similar properties with manifolds. In this talk, I will focus on (n-2)-connected 2n-dimensional Poincare complexes with torsion-free homology. I will discuss their classification, and compare our results with known results concerning (n-2)-connected 2n-manifolds.
Place: N818
Title: Topological complexity of enumerative problems and classifying spaces of PU_n
Abstract: We study the topological complexity, in the sense of Smale, of three enumerative problems in algebraic geometry: finding the 27 lines on cubic surfaces, the 28 bitangents and the 24 inflection points on quartic curves. In particular, we prove lower bounds for the topological complexity of any algorithm solving the three problems and for the Schwarz genera of their associated covers. The key is to understand certain cohomology classes of the classifying spaces of the projective unitary groups PU_n. This work is joint with Xing Gu.
Place: N818
Title: On the existence of critical Z/2 eigensections on S2
Abstract: On the 2-sphere , the Z/2 eigensection is a generalization of the Laplacian eigenfunction. Specifically, critical Z/2 eigensections serve as flat models for Z/2 harmonic 1-forms, which are analogous to quadratic differentials on 3-manifolds. Recently, Taubes and Wu have investigated the existence of critical Z/2 eigensections and constructed several examples. In this talk, we will further discuss existence of infinitely many critical eigenvalues. This is joint work with S. He.
Place: N818
Title: Eisenstein series and cusp counting in hyperbolic manifolds
Abstract: We study the Eisenstein series in a complete infinite volume hyperbolic manifold. We show that each full rank cusp corresponds to a cohomology class via the Eisenstein series construction. Moreover, by computing the intertwining operator, we show that different cusps give rise to linearly independent classes. As a consequence, the number of full rank cusps is bounded by the dimension of the cohomology group. This is joint work with Beibei Liu.
Place: N818
Title: Montesinos’ trick and exotic contact submanifolds
Abstract: The first half of the talk will describe a generalization of the famous ``Montesinos Trick’’ from 3d topology. It follows from studying an explicit family of a branched coverings of affine varieties using classical techniques in the style of Lefschetz, Milnor, and Rolfsen. Then I’ll use the trick to describe exotic embeddings of the standard contact 2n-1 sphere into the standard contact 2n+1 sphere for n at least 2.
Place: N820
Title: An invitation to the topology of diffeomorphism groups of high dimensional manifolds 4
Abstract: We will discuss our recent progress on mapping class groups and block diffeomorphism groups of certain high dimensional manifolds.
Place: N820
Title: An invitation to the topology of diffeomorphism groups of high dimensional manifolds 3
Abstract: In this talk, we will first focus on discussing some relations between diffeomorphism groups and embedding spaces. We will also survey some finiteness properties of diffeomorphism groups and some properties of Miller-Morita-Mumford classes of manifold bundles.
Place: N820
Title: An invitation to the topology of diffeomorphism groups of high dimensional manifolds 2
Abstract: We will review some classical methods of computing homotopy groups of diffeomorphism groups. We will end this talk by discussing our recent progress on the fundamental groups of diffeomorphism groups of certain high dimensional manifolds.
Place: N820
Title: An invitation to the topology of diffeomorphism groups of high dimensional manifolds 1
Abstract: Let M be a compact smooth manifold. The diffeomorphism group Diff(M) of M is an important object in algebraic and geometric topology. In this talk, we will introduce two different models of Diff(M) and some general properties of Diff(M).