2022/10/19: 王维佳 (清华大学), Modular Regulator and Special Double L-values（In 2006, Manin introduced and studied multiple L-functions of cusp forms but currently not many multiple L-values are well-understood. Beilinson (1986) defined the Eisenstein symbol in the motivic cohomology of the universal elliptic curve and the work of Deninger–Scholl (1989) shows the Petersson inner product of its regulator gives us L-values. In this talk I will present how to relate this modular regulator with double L-values and iterated integrals of Eisenstein series.）
2022/11/30: 束红非（BIMSA），TBA/WKB correspondence and Resurgent Quantum Mechanics （In this talk, we derive a system of thermodynamics Bethe ansatz (TBA) equations governing the exact WKB periods in one-dimensional Quantum Mechanics with arbitrary polynomial potentials. These TBA equations provide a solution to the Riemann-Hilbert problem of WKB periods in resurgent Quantum Mechanics. Combined with the exact quantization condition, our TBA equations provide a powerful method to solve the spectral problem. If time allows, we will also mention recent achievements on the higher order ODE.）
2023/03/08: 王智拓（哈尔滨工业大学数学研究院）， Construction Renormalization of the two dimensional Grosse-Wulkenhaar model（The Grosse-Wulkenaar model is a non-integrable matrix model with quartic interaction. It is the first renormalizable model in non-commutative quantum field theory and is closely related to the Kontsevich model for the study of the two dimensional quantum gravity. In this talk I will present some results about the construction of the two-dimensional Grosse-Wulkenhaar model with renormalization group analysis.）
2023/03/28：周贝加（北京大学），Fredholm Theory for Real Pseudoholomorphic Curves in Symplectization （In the symplectization of a contact manifold, people can consider the Symplectic Field Theory (SFT) in all odd dimensions and Embedded Contact Homology (ECH) in dimension 3. Because there is Real Gromov-Witten Theory and Real Seiberg-Witten Theory, it is a natural idea that Real SFT and Real ECH should exist. The first step to construct those theories is to get Fredholm index for moduli space of real pseudoholomorphic curves in symplectization. By this formula of Fredholm index, we can get parallel index inequalities for real pseudoholomorphic curves as in ECH, which is crucial to define Real ECH. I will give and explain the formula of Fredholm index and index inequalities. Other parts of the project of define Real SFT and Real ECH will be mentioned as well.）
2023/09/28：王维佳（清华大学），A regularization approach to automorphic forms （In this talk I will explain work in progress with H. Zhang, in which we develop a new regularization technique using Écalle’s transseries. This has led to numerous applications to study automorphic forms. I will talk about some of our results and the key steps in these works.）
2023/10/18：范辉军（北京大学），Progress on the study of Gauged linear Sigma models （Gauged linear sigma model (GLSM) proposed by Witten nearly thirty years ago was used to explain the mirror symmetry phenomena. GLSM relates to many important conjectures, like Landau-Ginzburg A model mirror to Landau-Ginzburg B model, Calabi-Yau model to Landau-Ginzburg model correspondence. GLSM has also intimate relation with GIT and can also be formulated in category theory. This report gives a short survey on the recent progress on this topic.）
2023/11/08：邓嘉龙（清华大学），Positive Scalar Curvature: Existence and Rigidity （The scalar curvature of a Riemannian metric is interesting not only in analysis, geometry, and topology, but also in physics. Enlargeable Length-structures will be introduced and showed that it is a new obstruction to the existence of a Riemannian metric with positive scalar curvature (PSC-metric). Thus, the connected sum of a closed manifold with some of locally CAT(0)-manifolds carry not PSC-metrics. We will also show that almost non-positive curved manifolds carry no PSC-metrics. On the other hand, harmonic maps with condition C will be used to show the rigidity theorem about the scalar curvature.）
2023/11/08：黄永红（新疆大学）Moduli space of Higgs bundles on DM curves （In this talk, I will present the basic constructions about the moduli spaces of Higgs bundles on Deligne-Mumford curves. To conclude, I will explicitly discrible the moduli space of rank 2 Higgs bundles on the pillowcase orbifold curve.）
2023/12/15：顾杰（东南大学）Resurgent structure in topological string (Topological string theory has (spacetime) instanton sectors, which the resurgence theory predicts to be completely controlled by the perturbative free energy via Stokes transformations. Recent results also suggest the Stokes constants are related to BPS/DT invariants. To make this picture concrete, one needs to first solve the instanton amplitudes and then calculate the Stokes constants. We demonstrate that the first problem can be solved exactly and completely through a trans-series extension of the BCOV holomorphic anomaly equations. We also show that valuable information on BPS invariants can be obtained through the calculation of Stokes constants. We will demonstrate our results with the example of the famous quintic manifold.)
2023/12/29：丁祥茂（中国科学院数学与系统科学研究院）Langlands Dualities through Bethe/Gauge Correspondence for 3d Gauge Theories (For non-simple laced Lie algebras, B_N and C_N are Langlands dual to each other in mathematical. In this article, we give two kinds of uniform realizations of Bethe/Gauge correspondence between 3d (or 2d) classical Lie group supersymmetry gauge theory with closed and open XXZ (or XXX) spin chain, the role of the ADE Lie algebras are self-dual, and the non-simple laced Lie algebras BN and CN, their position is exchanged. From Bethe/Gauge correspondence point of view, the two types of the effective superpotentials are Langlands duality to each other. More fantastically, for BN-type Lie algebra, fixed spin site by a boundary through Bethe/Gauge.)
2024/01/05：王百灵（澳大利亚国立大学）Symplectic reductions and quantisations （In this talk, we will first review symplectic reductions, geometric quantisation in terms of index theory and K-theory (K-homology), and the classical [Q, R] =0 theorems. In the second part, we will explore the quantum [Q, R] =0 in symplectic topology with the Q-functor given by QH and QK from the gauge Gromov-Witten theory. This is based on the long ongoing projects with Bohui Chen and Jianxun Hu, and recent work with Bohui Chen and Hailong Her.）
2024/03/15: 曹子皇（国家天文台）CSST 密集星场天体测量参数解算及其进展 (高精度的自行对于银河系的研究具有极为重要的作用。目前绝大多数银河系的研究都集中在太阳临近和外围区域，而对于保留着银河系早期形成历史和演化轨迹重要证据的核球却缺乏研究。重要原因是由于银河系中心区域恒星密度极高、消光高且不均匀、来自于不同子结构的恒星成分混杂，导致直接获取核球高质量的观测数据十分困难。 由于不受大气的影响，空间观测对于密集星场自行研究优势不言而喻，以HST和Gaia为代表的美国、欧洲空间天文项目已经在相关领域取得了丰硕成果。 可以断定：未来兼顾HST空间高分辨率观测能力和盖亚全天覆盖能力的大口径空间测光巡天数据可以极大地拓展密集星场高精度自行的研究。其中，即将发射的中国2米载人空间站工程巡天空间望远镜（China Space Station Telescope，CSST）就提供了这样一个重要的契机。其光学特性和设计指标，可保证其空间分辨能力与国外同级别空间望远镜相匹敌。对于银心方向，CSST会打破HST数据多限于低消光窗口、样本类型较为单一和Gaia对该区域空间分辨率较低的限制；对于本星系团中的近邻星系/矮星系，CSST会打破HST观测区域不够完备和Gaia观测极限星等较亮的的窘境；从而全面获得这些区域高精度绝对自行信息。 本报告就CSST在密集星场天体测量解算的关键问题进行全面的概括和深入的分析，向学生和相关研究人员展示相关数据处理的前沿观点。)
2024/04/17：曲华迪（南方科技大学）Action and periodic orbits of area-preserving surface diffeomorphisms（We are interested in the periodic points of area-preserving diffeomorphisms on surfaces. The celebrated Poincare-Birkhoff fixed point theorem and subsequent generalizations revealed the existence and abundance of periodic solutions in Hamiltonian systems, but the good structure of the collection of periodic solutions remains unknown. Converting area-preserving surface diffeomorphisms to Reeb flow via open book decomposition provided a way to relate the dynamics of area-preserving surface diffeomorphisms and the dynamic of Reeb flows of 3-dimensional contact manifolds, and ECH (Embedded Contact Homology) theory proves to be powerful. Though this way ,we want to conduct a quantitative analysis about the distribution of periodic orbit , based on the positive developments by Hutchins in 2016 and Weiler in 2019, our work delves into a more precise examination of the dependency of quantitative results, further generalizing and unifying these two results. Our investigation involves two crucial invariants of surface diffeomorphisms: the mean action and the rotation vector defined on the invariant measure set. We reveals the intrinsic connection between these two invariants, leading to the formulation of a further conjecture that generalize the Poincare-Birkhoff theorem.）
2024/04/17：张建路（中科院数学所）Selection principle of Generalized Hamilton-Jacobi equations（In 1987, Lions firstly proposed the homogenization for Hamilton-Jacobi equations, which revealed the significance of effective Hamiltonian in controlling the large time behavior of solutions. He also pointed out a vanishing discount procedure which is equivalent in obtaining the effective Hamiltonian, yet the convergence of solutions in this procedure was unknown until recently. In a bunch of joint works, we verified this convergence by using dynamical techniques.）
2024/04/19：徐晓濛（北京大学）Explicit evaluation of Stokes matrices via the isomonodromic approach（The Stokes matrices of linear systems and the associated isomonodromy equation play important roles in the study of Frobenius manifolds, stability conditions, representation theory and so on. The isomonodromy equation is a multi-time dependent Hamiltonian system, with the linear system as a Lax representation. Similar to the inverse scattering method, this talk solves the isomonodromy equation in terms of the scattering data, i.e., the Stokes matrices. Equivalently, it gives explicit expression of the Stokes matrices via the boundary values of the isomonodromy equation. As a consequence, many global properties of the solutions of isomonodromy equations can be derived. ）
2024/05/08：廖灵敏（武汉大学）Rational dynamics on the projective line of the field of p-adic numbers（We study rational maps as dynamical systems on the projective line of the field of p-adic numbers. We usually divide the space into two invariant parts: Julia set and Fatou set. For a rational map without critical point, the subsystem on the Julia set is topologically conjugate to some subshift of finite type on an alphabet of finite symbols, i.e., finite states Markov shift, and the subsystem on the Fatou set is described by a decomposition of minimal subsystems. However, if the rational map admits critical points, its dynamical behavior becomes complicated. In general, we prove that for a geometrically finite rational map, i.e., every critical point has finite forward orbit, the dynamics on its Julia set is topologically conjugate to a countable states Markov shift. This is a joint work with Shilei Fan, Hongming Nie and Yuefei Wang.）
2024/06/27: 蒋云峰（堪萨斯大学）A theory of counting surfaces in projective varieties(The theory of enumerative invariants of counting curves (Riemann surfaces) in projective varieties has been an important theory in the last decades.The enumerative invariants were motivated by theoretical physics—string theory and gauge theory, and include Gromov-Witten theory, Donaldson-Thomas theory and more recently Vafa-Witten theory. It is hoped that there may exist a theory of counting algebraic surfaces in projective varieties. A theory of counting surface in a Calabi-Yau 4-fold has been constructed using Donaldson-Thomas theory of 4-folds.In this talk I will try to give evidences of a counting surface theory using stable maps, and explain why it is difficult to construct the counting surface invariants.)
2024/08/19-23: 王明浩（Boston University）短期课程 Feynman Graph Integrals and Configuration Spaces （In this mini-course, I will introduce Feynman graph integrals from topological and holomorphic quantum field theories. For applications, I will sketch a proof of the formality of the configuration space of Euclidean spaces. Finally, I will discuss some conjectures related to the formality problem for the structure sheaf cohomology of configuration spaces. No preliminary knowledge in physics is required for this mini-course.
Reference:
1.Pascal Lambrechts, Ismar Volic, Formality of the little N-disks operad, arxiv.0808.0457
2.Maxim Kontsevich, Deformation quantization of Poisson manifolds, arxiv.9709040
3.Minghao Wang, Brian R. Williams, Factorization algebras from topological-holomorphic field theories, arxiv.2407.08667）
2024/09/20: 周正一（中科院数学所）Kahler compactification of C^n and Reeb dynamics (We will present two results in complex geometry: (1) A Kahler compactification of C^n with a smooth divisor complement must be P^n, which confirms a conjecture of Brenton and Morrow (1978) under the Kahler assumption; (2) Any complete asymptotically conical Calabi-Yau metric on C^3 with a smooth link must be flat, confirming a modified version of Tian’s conjecture regarding the recognition of the flat metric among Calabi-Yau metrics in dimension 3. Both proofs rely on relating the minimal discrepancy number of a Fano cone singularity to its Reeb dynamics of the conic contact form. This is a joint work with Chi Li. )
2024/09/25: 宗正宇（清华大学数学系）Remodeling conjecture with descendant(Based on the work of Eynard-Orantin and Marino, the Remodeling Conjecture was proposed in the papers of Bouchard-Klemm-Marino-Pasquetti in 2007 and 2008. The Remodeling Conjecture can be viewed as an all genus open-closed mirror symmetry for toric Calabi-Yau 3-orbifolds. In this talk, I will explain an all genus mirror symmetry for descendant Gwomov-Witten nvariants of toric Calabi-Yau 3- orbifolds. The B-model is given by the Laplace transform of the Chekhov-Eynard-Orantin invariants of the mirror curve. This talk is based on ongoing joint work with Bohan Fang, Melissa Liu, and Song Yu.)
2024/11/20: 乔帅格（北京大学）Lie groupoid structures on regular Donaldson Moduli spaces(In general, moduli spaces in geometry have quotient singularities and the Lie groupoids have advantages in recording these structures in smooth categories. Motivated by this principle, in this talk, we will base on the Donaldson moduli spaces of instantons on 4-manifolds and explain the framework of the construction. This is the joint work with Bohui Chen and Bai-ling Wang.)
2024/12/04: 孟国武（香港科技大学）Tulczyjew’s approach for particles in gauge fields(Tulczyjew’s unified approach to Lagrangian and Hamiltonian descriptions of particle dynamics is quite appealing to geometry-oriented minds. An advantage of this geometric approach is its flexibility in the sense that it can be easily adapted to different settings, especially to systems with singular Lagrangians or subject to constraints. As one more demonstration of its flexibility, in this talk it will be reported that Tulczyjew’s approach also works for particle dynamics in which an electrically charged particle moves in the presence of an external non-abelian gauge field. As a consequence, a nontrivial generalization of the Lagrange equation is obtained.)