Place: N602
Title: Contact geometry of symplectic log Calabi-Yau surfaces
Abstract: A symplectic log Calabi-Yau surface is a pair of a closed symplectic 4-manifold and a symplectic divisor representing the anti-canonical class. The symplectic divisor is either a torus or a circular spherical divisor by the adjunction formula. In this talk, We describe the classification of symplectic LCY surfaces and discuss their contact aspects (joint works with Cheuk Mak, Jie Min and Shengzhen Ning).
Place: N602
Title: Floer cohomology of compositions of Lagrangian Dehn twists
Abstract: There is a conjecture due to Paul Seidel, that asserts the composition of a sequence of Lagrangian Dehn twists can be computed as a mapping cone between the Floer chain complex of the identity, as well as a cube complex formed by the Hochschild complex of a directed subcategory with spherical objects. We give two proofs of this conjecture, one is purely algebraic, and the other relies on clean surgery and should be of independent interest. This result was previously announced by Sikimeti M’au and Tim Perutz, but either approach we present in this talk is different from their solution. This is partly an upcoming work of Shuo Zhang, and partly a joint work in progress with Cheuk-Yu Mak and Shuo Zhang.
Place: N602
Title: Relative Calabi-Yau structures for microlocal sheaves/Fukaya categories
Abstract: For a Liouville manifold with a Liouville hypersurface at infinity, one can associate a pair of (partially) wrapped Fukaya categories to the pair of Liouville manifolds. The result of Ganatra-Pardon-Shende shows that the Fukaya categories are equivalent to certain categories coming from microlocal sheaf theory. We consider cotangent bundles with Weinstein hypersurfaces, study duality and exact sequences that arise from the pair of categories and show that this pair admits a strong relative Calabi-Yau structure, such that the wrap-once functor gives the inverse dualizing bimodule. This is a non-commutative analogue of the Poincare-Lefschetz duality on manifolds with boundary. This is joint work in preparation with Chris Kuo.
Place: N602
Title: On the equivariant pushforward of non-equivariant homology
Abstract: Guillou and May gave a model of G-equivariant spectra as spectral homotopy analogues of Mackey functors. If one applies this construction to HZ-modules, one gets modules over the commutative G-spectral ring given by the equivariant pushforward of HZ. This, however, is not a G-equivariant generalized Eilenberg-Mac Lane spectrum in the ordinary sense. In this talk, we will explore this phenomenon, propose its explanation, and also discuss how the derived category of modules over the pushforward can be modeled in terms of more familiar pieces, and calculated with. My talk will reflect on discussions with several people, and on some concrete joint work in progress with Bar Roytman.
Place: N602
Title: On aspherical symplectic fillings of the prequantization bundles
Abstract: A prequantization bundle is a negative circle bundle over a symplectic manifold, equipped with a contact form induced by an S1-invariant connection. In this talk, I will present a result concerning the diffeomorphism types of the symplectic fillings of a prequantization bundle over a surface, under certain topological and symplectic assumptions. Furthermore, I will discuss an extension of the result to higher dimensions. Under the similar assumptions, the result states that the homology of an aspherical filling is isomorphic to the homology of the disk bundle.
Place: N602
Title: An upper bound of LS category of relative Sullivan algebras
Abstract: Lusternik-Schnirelmann category (LS category) is an invariant of topology spaces. It measures how many contractible open sets can cover this space. The LS category of a fibration can be bounded by such categories of its base and fiber. In rational homotopy theory, some fibrations can be represented by relative Sullivan algebras, and there is an algebraic version of LS category defined by such algebras. Felix, Halperin and Thomas asked whether the LS category of a relative Sullivan algebra is also bounded by the categories of its base algebra and fiber algebra. We will give a positive answer of this.
Place: N933
Title: Equivariant geometric bordism, representations and graphs
Abstract: Based upon a classical result of Conner and Floyd on involutions on manifolds, using the idea of the GKM theory, we show that an effective G-action on a smooth closed manifold fixing a finite set can always produce a G-labelled graph, on which all tangential G-representations at fixed points are equipped, where G is an elementary abelian 2-group. We further obtain a complete characterization of G-representations at fixed points of such an G-action in terms of its G-labelled graph, so that the ring generated by the equivariant geometric unoriented bordism classes of all such G-manifolds can be described in terms of G-labelled graphs. This enriches the GKM theory. In addition, we also obtain a complete characterization of G-representations at fixed points of such an G-action in terms of own way of G-representations. This is based on joint works with Bo Chen, Hao Li and Qiangbo Tan.
Place: N933
Title: Some 3-manifold invariants derived from the intersecting kernels of Heegaard splittings
Abstract: In the talk, I will desccribe some some quotient groups of the intersecting kernels of Heegaard splittings and show they are independent of the choice of the Heegaard splittings, therefore they are invariants of 3-manifolds M, in which the Reidmester-Singer Theorem on the stable equivalence of Heegaard splittings plays an essential role.
Place: N602
Title: Galois Symmetry and Manifolds
Abstract: It is known that two complex algebraic varieties can be algebraically isomorphic but not homeomorphic. Such examples can be obtained by changing the coefficients of the defining equations by some automorphism of the ground field. In this talk, I will illustrate some new results in understanding how the entire Galois group of Q-bar, the algebraic closure of the rationals Q, changes the underlying manifold structures of smooth complex varieties defined by equations with coefficients in Q-bar.
Place: N602
Title: Vanishing lines and periodicities in higher real K-theories
Abstract: Higher real K-theories generalize topological K-theory, reflecting the beautiful periodic phenomena of stable homotopy groups of spheres and detecting numerous important elements within. In this talk, I will demonstrate that the homotopy fixed point spectral sequences, which compute their homotopy groups, exhibit strong horizontal vanishing lines. I will present the concrete filtration of these vanishing lines. Additionally, I will discuss the periodicities of these higher real K-theories, which can be viewed as a generalization of the Bott periodicity of KO at higher heights. This talk is based on joint work with Mike Hill, Guchuan Li, Xiaolin Danny Shi, Guozhen Wang, and Zhouli Xu.
Place: N602
Place: N602
Place: N602
Title: Some calculations about the homology of big mapping class groups
Abstract: I will first given an introduction to surfaces of infinite type. Then I survey some recent developments on the homology of big mapping class groups. In particular, I will discuss the question when a homology class of big mapping class groups could have compact support. This is a joint work with Martin Palmer.
Place: N402
Title: Vanishing cycles for symplectic foliations
Abstract: The main objects of the talk will be symplectic foliations, and more precisely a subclass of these called “strong”. Strong symplectic foliations are meant to be one of the possible rigid generalizations of taut foliations to high dimensions, and indeed have quite a rigid nature, with techniques such as pseudo-holomorphic curves à la Gromov and asymptotically holomorphic sequences of sections à la Donaldson working well in this setting. I will present a joint work in progress with Klaus Niederkrüger and Lauran Toussaint that aims at giving a new obstruction for a symplectic foliation to be strong. This comes in the form of a symplectic high-dimensional version of vanishing cycles for smooth codimension 1 foliations on 3-manifolds, and the proof relies on pseudo-holomorphic curve techniques, in a way which is parallel to the case of the Plastikstufe introduced by Niederkrüger ‘06 in the contact case.
Place: N602
Title: Partial-dual genus polynomial and its categorification
Abstract: The partial-dual genus polynomial of a ribbon graph is the generating function that enumerates all partial duals of the ribbon graph. In this talk, I will give a quick introduction to this polynomial and discuss the categorification of it. This is a joint work with Ziyi Lei.
Place: N802
Title: Cobordisms with controlled fundamental groups
Abstract: Given n-dimensional manifolds X and Y, a cobordism between them is an (n+1)-dimensional manifolds W whose boundary is the disjoint union of X and Y. With no extra constraints on W, existence problem has been resolved by the work of Pontryagin and Thom. However, the problem is more difficult if one puts extra constrains on the topology of W (e.g. its homology or homotopy groups). In this talk, I will discuss our recent work about pi-1 injective cobordisms, which is a generalization of boundary-incompressible 3-manifolds. We will also give some applications about finite group actions on 4-manifolds with isolated fixed points. This is a joint work with 林剑锋 (清华大学）
Place: N802
Title: Confined subgroups in groups with contracting elements
Abstract: In this talk, we study confined subgroups in groups with contracting elements. This class of subgroups generalizes normal subgroups and has recently received many interests in semi-simple Lie groups. We shall focus on growth of confined subgroups and their relation with conservative action on boundary. Our results apply to fundamental groups of Riemannian manifolds with negative curvature and mapping class groups. This represents a joint work with Inheyok Choi, Ilya Gekhtman, and Tianyi Zheng.
Place: N802
Title: Symplectic structures from almost symplectic structures
Abstract: In this talk, we will consider a stabilized version of the fundamental existence problem of symplectic structures. Given a formal symplectic manifold, i.e. a closed manifold M with a non-degenerate 2-form and a non-degenerate second cohomology class, we investigate when its natural stabilization to M x T^2 can be realized by a symplectic form. We show that this can be done whenever the formal symplectic manifold admits a symplectic divisor. It follows that the product with T^2 of an almost symplectic blow up admits a symplectic form. Another corollary is that if a formal symplectic 4-manifold, which either satisfies that its positive/negative second betti numbers are both at least 2, or that is simply connected, then Mx T^2 is symplectic. This is joint work with Fabio Gironella, Fran Presas, Lauran Touissant.
Place: N802
Title: suspension splitting of 5-manifolds
Abstract: In this talk we will briefly review recent research in the homotopy types of suspended manifolds, which have rich applications to the characterization of classical invariants in geometry, topology and physics. In particular, we shall talk about the suspension splitting of a closed orientable non-simply-connected 5-manifold and its applications. This is a joint work with Zhongjian Zhu.