Geometry and Physics Seminar | |
Student No.： | 50 |
Time： | Tue 10:30-12:00, 2017-3-7～2017-6-15 |
Instructor： | Si Li, Zhengyu Zong |
Place： | Conference room 3, Floor 2, Jin Chun Yuan West Building |
Starting Date： | 2017-3-7 |
Ending Date： | 2017-6-15 |
Date: 2017-6-27
Speaker: Jingyu Zhao (Institute of Advanced Study)
Time: Tue 9:00-10:30
Place: Conference room 4, Floor 2, Jin Chun Yuan West Building
Title: Degeneration of the noncommutative Hodge-to-de Rham spectral sequences
Abstract: For a compact Kahler manifold, the Hodge-to-de Rham spectral sequence degenerates at E^1 page. This can be proved using Hodge theory which is highly transcendental. In 1982, Deligne and Illusie gave a purely algebraic proof of the degeneration of the Hodge-to-de Rham spectral sequences using reduction to characteristic p methods. For a DG category, there is a noncommutative Hodge-to-de Rham spectral sequence which relates the Hochschild homology and periodic cyclic homology of A. It is conjectured by Kontsevich and Soibelman and proved by Kaledin in 2016 that the noncommutative Hodge-to-de Rham spectral sequences degenerates for smooth and proper DG categories. In this lecture, we give an outline of the proof of degeneration following Kaledin’s work. The key step is to prove a non-commutative analogue of the Cartier isomorphism in the work of Deligne and Illusie.
Speaker: Yining Zhang (Indiana University)
Time: 10:30-12:00
Place: Conference room 4, Floor 2, Jin Chun Yuan West Building
Title: Dual Hodge decompositions and derived Poisson brackets
Abstract: After briefly recall the definition of cyclic homology and classical Hodge decomposition, I will state a theorem by B. Feigin and B. Tsygan. Using this theorem, I shall construct a Hodge type decomposition (dual Hodge decomposition) of (reduced) cyclic homology of universal enveloping algebras of Lie algebras. In fact this decomposition can be viewed as being Koszul dual to the Hodge decomposition of cyclic homology of cocommutative coalgebras. Next, I will introduce the notion of noncommutative Poisson bracket in the sense of Crawley-Boevey and its homological generalization derived Poisson bracket. As a consequence, if an algebra admits a derived Poisson structure, then its cyclic homology would be a graded Lie algebra. Lastly, I want to talk about the interactions between this Lie bracket on cyclic homology and dual Hodge decomposition. If time permits, I will mentions some applications to string topology. This is based on a joint work with Yuri Berest and Ajay C. Ramadoss.
Speaker: Zhengfang Wang (BICMR)
Time: Thu 9:00-10:30
Place: Conference room 4, Floor 2, Jin Chun Yuan West Building
Title: A-infinity and L-infinity algebra structures arising from (dg) Frobenius algebras
Abstract: Given a (differential graded) Frobenius algebra, one can construct an unbounded complex, called Tate-Hochschild complex, which is defined as a mapping cone of a certain morphism from the Hochschild chain complex (with some shift) to the Hochschild cochain complex. We construct a cyclic (or called Calabi-Yau) A-infinity algebra structure, which extends the classical Hochschild cup and cap products, on the Tate-Hochschild complex. This A-infinity algebra has an explicit formula with m_n=0 for n>3. We will also construct an $L$-infinity algebra structure by extending the classical Gerstenhaber bracket. Moreover, we prove that the cohomology ring of the Tate-Hochschild complex is a Batalin-Vilkovisky (BV) algebra with the BV operator extending the Connes' boundary operator. At the end, we will talk about how to relate the Tate-Hochschild complex to string topology. This talk is based on the recent joint work with M. Rivera.
Speaker: Bohan Fang (BICMR)
Time: Thu 10:30-12:00
Place: Conference room 1, Floor 1, Jin Chun Yuan West Building
Title: Oscillatory integrals on the T-dual cycles
Abstract: The mirror of a complete toric variety is a Landau-Ginzburg model. The oscillatory integral of the superpotential function on a Lagrangian cycle mirror to a coherent sheaf is a B-model genus 0 invariant. I will describe a way to compute it, and the answer is a genus 0 Gromov-Witten descendant potential with certain Gamma class of the coherent sheaf inserted. This result is related to Iritani's result identifying the integral structures on both sides. When the toric variety is P^1, this result can be extended to compute all genus descendants from Eynard-Orantin's topological recursion theory on the mirror of P^1.
Date: 2017-6-19
Speaker: Daping Weng (Yale University)
Time: Mon, 9:00-10:30
Place: Conference room 4, Floor 2, Jin Chun Yuan West Building
Title: Cluster Donaldson-Thomas Transformation of Grassmannian.
Abstract: On the one hand, there is a 3d Calabi Yau category with stability conditions associated to a quiver without loops or 2-cycles with generic potential, and one can study its Donaldson-Thomas invariants. On the other hand, such a quiver also defines a cluster Poisson variety, which often has geometric realizations. In certain cases the Donaldson-Thomas invariants of the former can be captured by a birational automorphism of the latter. In this talk I will describe the cluster Poisson structure on the moduli space of configurations of points on a projective space, and state my result on the geometric realization of the corresponding cluster Donaldson-Thomas transformation.
Speaker: Ping Xu (Penn State University)
Time: Mon, 10:30-12:00
Place: Conference room 4, Floor 2, Jin Chun Yuan West Building
Title: Formal exponential maps and L-infinity structure
Abstract: Exponential maps arise naturally in the contexts of Lie theory and smooth manifolds. The infinite jets of these classical exponential maps are related to the Poincar\'e--Birkhoff--Witt isomorphism and the complete symbols of differential operators. We will investigate the question how to extend these maps to graded manifolds by introducing formal exponential maps in a purely algebraic way. As an application, we will show there is an L-infinity structure in connection with the Atiyah class of a dg manifold.
Date: 2017-6-13
Speaker: Yang Shen (Zhejiang University)
Time: Tue 9:00-10:30
Title: Moduli spaces as ball quotients
This is a joint work with Professor Kefeng Liu.
Speaker: Jingyu Zhao (Institute of Advanced Study)
Time: Tue 10:30-12:00
Title: Getzler’s Gauss-Manin connection on equivariant Floer cohomology
Date: 2017-6-6
Speaker: Xuwen Zhu (Stanford University)
Time: Tue 10:30-12:00
Place: Conference room 4, Floor 2, Jin Chun Yuan West Building
Title: The deformation theory of constant curvature metrics with conical singularities
Abstract: We would like to understand the deformation theory of constant curvature metrics with prescribed conical singularities on a compact Riemann surface. In the positive curvature case, when some or all of the cone angles are bigger than $2\pi$, the analysis is much more complicated than the small angle case. We discover that one key ingredient of the obstructed deformation is related to splitting of cone points. We construct a resolution of the configuration space, and prove a new regularity result that the family of constant curvature conical metrics has a nice compactification as the cone points coalesce, and moreover, the fibrewise family of constant curvature metrics is polyhomogeneous on this compactification. And we hope to apply this new construction to describe the moduli space of spherical conic metrics with no angle constraints. This is joint work with Rafe Mazzeo.
Date: Tue 2017-4-25
Speaker: Jie Tu (CMS, Zhejiang University )
Time: 10:30-12:00
Place: Conference room 3, Floor 2, Jin Chun Yuan West Building
Title: The Deformation of Pairs (X,E) Lifting from Base Family
Abstract: In this talk, I will introduce my joint work with K.F. Liu on the analytic deformation theory of pairs (X,E), where X is a compact complex manifold and E is holomorphic vector bundle over X.
The splitting of holomorphic cotangent bundle via any integrable connection de- composes the Beltrami differential of pairs into the horizontal part and the vertical part. The horizontal part is the Beltrami differential of base family {Xt}. When the vertical part is vanishing under the decomposition by a Nakano semi-positive Chern connection ∇, i.e. {(Xt , Et )} is lifting from base family {Xt } via ∇, we get a infinitesimal extension of ∂ ̄-closed bundle valued (n, q)-form by the recursive method.
Speaker: Chunle Huang 黄春乐（Zhejiang University）
Time: 10:30-12:00
Place: Conference room 3, Floor 2, Jin Chun Yuan West Building
Title: Logarithmic vanishing theorems on compact K\"{a}hler manifolds
Abstract: We will first establish an $L^2$-type Dolbeault isomorphism for logarithmic differential forms by H\"{o}rmander's $L^2$-estimates. By using this isomorphism and the construction of smooth Hermitian metrics, we obtain a number of new Akizuki-Kodaira-Nakano type vanishing theorems for sheaves of logarithmic differential forms on compact K\"ahler manifolds with simple normal crossing divisors, which generalize several classical vanishing theorems, including Norimatsu's vanishing theorem, Gibrau's vanishing theorem, Le Potier's vanishing theorem and a version of the Kawamata-Viehweg vanishing theorem.
Date: 2017-3-28
Speaker: Baosen Wu 吴宝森（YMSC）
Time: 10:30-12:00
Place：Conference room 3, Floor 2, Jin Chun Yuan West Building
Title: Construction of stable sheaves via Serre correspondence
Abstract: Serre correspondence is a classical tool to construct rank 2 vector bundles or coherent sheaves with prescribed Chern classes. However, testing the stability of such bundles is in general very difficult. We shall work on Calabi-Yau threefolds and use deformation to show the stability of such bundles or sheaves.
Speaker: 曹亚龙
Time: 10:00-11:30
Place: Conference room3, Floor 2, Jin Chun Yuan West Building
Title: Gopakumar-Vafa type invariants for Calabi-Yau 4-folds
Abstract:
As an analogy of Gopakumar-Vafa conjecture for CY 3-folds, Klemm-Pandharipande proposed GV type invariants on CY 4-folds using GW theory and conjectured their integrality. In this talk, we propose a sheaf theoretical interpretation to these invariants using Donaldson-Thomas theory on CY 4-folds. This is a joint work with Davesh Maulik and Yukinobu Toda.
Date: 2017-3-7
Speaker: Sz-Sheng Wang 王赐圣（YMSC）
Title: Towards decompositions of small transitions of Calabi-Yau threefolds
Abstract: Let \pi : \hat{X} \to X be a small projective resolution of a Calabi--Yau 3-fold X, which has terminal singularities. If X can be smoothed to a Calabi--Yau manifold \tilde{X}, then the process of going from \hat{X} to \tilde{X} is called a small transition.