GAP Seminar @ USTC (2018 Fall)

Welcome to GAP seminar

This seminar aims at filling up the gap between graduate level math and research math, and enlarging the scope of graduate students as well as more advanced researchers. In each talk, the speaker will focus on a currently active research topic. He/she will spend one hour or so to introduce the background of the topic, including basic conceptions/examples, important known results and major problems etc. The last part of the talk will usually be more technical and is related to the speaker's own work. The subjects of future talks are mainly chosen from geometry(G), algebra, analysis(A), and mathematical physics(P).

Upcoming Talks

Date: 01/04, 16:00-17:30
Speaker: TBA
Title: TBA
Abstract: TBA

Date: 01/11, 16:00-17:30
Speaker: TBA
Title: TBA
Abstract: TBA

Past Talks

Archive of GAP Seminar: Fall 2013 | Spring2014 | Fall 2014 | Spring 2015 | Fall 2015 | Spring 2016 | Fall 2016 | Spring 2017 | Fall 2017 | Spring 2018

Date: 12/21, 16:00-17:30
Speaker: Haoyu Hu (南京大学)
Title: Ramification of étale sheaves on positive characteristic varieties
Abstract: The ramification theory is an branch of algebraic number theory very close to the class field theory. Classically, we know the ramification phenomena well on schemes of "dimension 1". In this talk, I will introduce Abbes and Saito's ramification theory for local fields of imperfect residue fields and the theory of singular support and characteristic cycles of étale sheaves on higher dimensional smooth varieties due to Beilinson and Saito. After that, I focus on a recent joint work with J.-B. Teyssier that applies these theories to a boundedness property of nearby cycles and étale cohomology groups.

Date: 12/14, 16:00-17:30
Speaker: Sz-Sheng Wang (清华大学)
Title: The movable fan of certain Calabi--Yau threefolds of Picard number two
Abstract: For a Calabi--Yau threefold (CY3), it is known that the movable cone is covered by the nef cones of all its birational minimal models. This arrangement is called the movable fan. We study the movable fan of certain CY3s in projective bundles which admits a determinantal contraction to a CY3s with only ordinary double points. Our construction unifies certain examples in literature. This is joint work with Ching-Jui Lai.

Date: 12/12, 16:00-17:30
Speaker: Xi Chen (剑桥大学)
Title: Nonlinear detection of Hermitian connections in Minkowski space
Abstract: I will describe how to recover a Hermitian connection from the source-to-solution map of a cubic non-linear wave equation in Minkowski space; the equation is naturally motivated by the Yang-Mills-Higgs equations. Through a linearization trick, we convert the cubic non-linearity to the nonlinear interaction of three linear waves. The collision of the three waves creates new propagating singularities and makes a broken geodesic. By the microlocal analysis of intersecting Lagrangian distributions, we reduce the recovery to a geometric problem: recovering a connection from its broken non-abelian X-ray transform along light rays. This is joint work with M. Lassas (Helsinki), L. Oksanen (UCL), G. Paternain (Cambridge).

Date: 12/07, 15:00-16:30 (Different Time)
Speaker: Qi Ding (复旦大学)
Title: Ricci有下界流形中的面积极小超曲面
Abstract: 首先我们介绍一下背景知识,包括介绍欧式空间中面积极小超曲面的性质,和Ricci有下界的流形结构。然后我们简要回顾一下这一方向的历史,最后介绍一下我们近来的研究工作。

Date: 11/30, 14:00-15:30 (Different Time)
Speaker: Zhi Jiang (复旦大学)
Title: Birational geometry of varieties with large irregularities
Abstract: Varieties with large irregularities are special and always have nice properties. I will introduce some methods due to Green-Lazarsfeld, Chen-Hacon, Pareschi-Popa which are quite useful to study such varieties.

Date: 11/23, 16:00-17:30
Speaker: Bo Liu (华东师范大学)
Title: Localization of eta invariants
Abstract: The famous Atiyah-Singer index theorem announced in 1963 computed the index of elliptic operators, which is defined analytically, in a topological way, by using the characteristic classes. In 1968, Atiyah and Segal established a localization formula for the equivariant index which computes the equivariant index via the contribution of the fixed point sets of the group action. It is natural to ask if the localization property holds for the more complex spectral invariants, which are not computable in a local way and not a topological invariant.In this talk, we will establish a version of localization formula for equivariant eta-invariants,which were introduced in the 1970's as the boundary contribution of index theorem for compact manifolds with boundary and are formally equal to the number of positive eigenvalues of the Dirac operator minus the number of its negative eigenvalues, by using differential K-theory, a new research field in this century. This is a joint work with Xiaonan Ma.

Date: 11/02, 16:00-17:30
Speaker: Haojie Chen (浙江师范大学)
Title: Kodaira dimensions of almost complex manifolds
Abstract: The Kodaira dimension is a fundamental invariant in complex geometry and algebraic geometry. It gives a rough classification scheme of complex manifolds up to birational equivalence. We will talk about our recent generalization of plurigenera and Kodaira dimensions to almost complex manifolds. We will then discuss some structural results including the birational invariance on almost complex 4-manifolds. As an application, we show that the Kodaira dimenson of the standard almost complex structure on the six sphere is 0 and all the plurigenera are 1, which are different from those of a hypothetical complex structure. This talk is based on joint work with Weiyi Zhang.

Date: 10/26, 16:00-17:30
Speaker: Di Yang (中国科学技术大学)
Title: Frobenius manifold and integrable hierarchies
Abstract: In this introductory talk, I review the construction of integrable hierarchy associated with a Frobenius manifold. Particular consideration will be addressed on semisimple Frobenius manifolds. If time permits, I will also describe some recent developments of the study.

Date: 10/19, 16:00-17:30
Speaker: Peng Wang (福建师范大学)
Title: Symmetric minimal surfaces in S^3 as conformally-constrained Willmore minimizers in S^n
Abstract: The Willmore conjecture states that the Clifford torus minimizes uniquely the Willmore energy \int (H^2+1) dM among all tori in S^3, which is solved recently by Marques and Neves in 2012. For higher genus surfaces, it was conjectured by Kusner that the Lawson minimal surface, \xi_{m,1}: M-->S^3, minimizes uniquely among all genus m surfaces in S^n. The conjecture reduces to the Willmore conjecture for tori if m=1, since \xi_{1,1} is the Clifford torus. In this talk, we will prove this conjecture under the assumption that the (conformal) surfaces in S^n have the same conformal structure as \xi_{m,1}, i.e, they are conformal maps from the same Riemann surface.

Date: 10/12, 16:00-17:30
Speaker: Guozhen Wang (复旦大学)
Title: Stable motivic homotopy theory over real and complex fields
Abstract: We will give an introduction to the general construction of the stable motivic homotopy category and the formulation of Grothendieck's six operations. In particular, there are a bunch of functors between the motives over the real number fields and those over the complex fields. This gives new methods to understand the classical homotopy categories and equivariant homotopy category for the group $Z/2Z$. I will show how these techniques can lead to new methods for the computations of the homotopy groups of the equivariant and non-equivariant sphere spectrum.

Date: 09/28, 16:00-17:30
Speaker: Jingang Xiong (北京师范大学)
Title: On extinction behaviors of the fast diffusion equations
Abstract: It had been known in 1980s that solutions of the fast diffusions $\partial_t u=\Delta u^m$ in $R^n\times R$ will be extinct in the finite time, when the initial data belong to some suitable good spaces and $0< m < \frac{n-2}{n}$. In this talk, I will report my recent results with Tianling Jin on certain extinct behaviors which capture singular solutions of the corresponding elliptic equations. In particular, the Fowler solutions of the Yamabe equation can be captured.

Date: 09/28, 14:00-15:30 (Different Time!)
Speaker: Kang Zuo (University of Mainz and USTC)
Title: Arithmetic Simpson correspondence
Abstract: We propose an arithmetic Simpson correspondence for Higgs bundles over arithmetic schemes. It predicts that the monodromy of the Yang-Mills-Higgs connection on a rank-2 graded stable Higgs bundle on the projective line of degree -1 and with logarithmic singularities at four punctured points lies in an algebraic number ring if and only if the zero of the Higgs field is the image of a torsion point on the elliptic curve as double cover of the projective line ramified at those four points. We did construct 26 pieces complete solutions for monodromies lying in the integer ring and Higgs fields having zeros of order 1, 2, 3, 4 and 6. This is a joint project with J. Lu, X. Lu, R.R. Sun and J. B. Yang.

Date: 09/21, 16:00-17:30
Speaker: Quan Xu (清华大学)
Title: Syntomic regulators in p-adic integration
Abstract: In this talk, we will fast recall the classical complex regulators and then shift our focus to p-adic regulators, especially the syntomicregulators. Furthermore, we also clarify the relation between syntomic regulators and p-adic integration. In particular, we are going to see how they works for smooth curves with good reduction. In the end, we will investigate the syntomic regulator for semi-stable curves in p-adic integration, including a conjecture and partial results.

Date: 09/14, 16:00-17:30
Speaker: Ziyu Zhang (Leibniz University Hannover)
Title: Formality conjecture and moduli spaces of sheaves on K3 surfaces
Abstract: The formality conjecture for K3 surfaces, formulated by D.Kaledin and M.Lehn, states that on a complex projective K3 surface, the differential graded algebra RHom(F,F) is formal for any coherent sheaf F polystable with respect to an ample line bundle. In this talk, I will explain how to combine techniques from twistor spaces, dg categories and Fourier-Mukai transforms to prove this conjecture, and how to generalize it to derived objects. Based on joint work with Nero Budur.

Date: 09/07, 16:00-17:30
Speaker: Hitoshi Moriyoshi (Nagoya university)
Title: Combinatorial Gauss-Bonnet theorem and the Alexander-Spanier cohomology
Abstract: For a smooth surface the celebrated Gauss-Bonnet theorem tells that the integration of the Gauss curvature is equal to the Euler number of surface times 2\pi. Also on a combinatorial surface, namely a polyhedral surface, there is a similar theorem to the above, that is, the sum of Angle defect at each vertex amounts to the Euler number of surface times 2\pi. This theorem goes back at least as far as Descartes. Thus the Angle defect seems to be a counterpart of the Gauss curvature on a polyhedral surface. In this talk we justify it by introducing a notion of the Alexander-Spanier cohomology, which also make possible a generalization of the theorem in higher dimensional case. The main subjects in my talk are polyhedrons, which are definitely accessible for everyone.