Beijing Institute of Mathematical Sciences and Applications

In this talk, we will explain the basic notions of smooth stratified spaces which arise in many situations, including orbit spaces, Hamiltonian reductions, and others. We will discuss the particular structure of Poisson bivectors on stratified spaces. Additionally, we will mention the quantization of a symplectic stratified space.

In this talk, we will first give a quick review of cWorkshopWorkshoplassic Calogero-Moser systems, including the physical background, historical developments and different generalizations. We then introduce the Calogero-Moser-Sutherland (CMS) models and spin CMS models. We will review the systems via Hamiltonian reductions and discuss their (super)integrability. An example from $SU(3)$ will be provided with details as a toy model of the spin CMS systems.

In this talk, I will focus on $5d N = 1 Sp(N)$ gauge theory with $Nf (≤ 2N + 3)$ flavors based on 5-brane web with $O5$-plane. Based on 5-brane web with $O5$-plane corresponding to non-toric geometry, we compute the Nekrasov partition function based on the topological vertex formalism with O5-plane. Rewriting it in terms of profile functions, we obtain the saddle point equation for the profile function after taking thermodynamic limit. By introducing the resolvent, we derive the Seiberg-Witten geometry and its boundary conditions as well as its relation to the prepotential in terms of the cycle integrals. They coincide with those directly obtained from the dual graph of the 5-brane web with $O5$-plane. This agreement gives further evidence for mirror symmetry which relates Nekrasov partition function with Seiberg-Witten curve in the case with orientifold plane. This talk is based on joint work with Futoshi Yagi.

Differential graded (dg for short) manifolds (a.k.a. Q-manifolds) emerged from a number of areas of mathematics and theoretical physics such as string theory, Hamiltonian mechanics, and derived geometry. Hamiltonian systems in symplectic graded manifolds encodes many interesting geometrical structures. For example, Courant algebroids, introduced by Liu, Weinstein and Xu, can be realized as Hamiltonian systems in symplectic graded manifolds of degree 2. Meanwhile, cohomology of Courant algebroids, defined via dg geometry, plays an important role in the AKSZ\'s construction of 3D topological Courant sigma models. For each regular Courant algebroid, we construct a minimal model and a Hodge-to-de Rham type spectral sequence to compute its cohomology. This is a joint work with X. Cai and Z. Chen.

Some important classes of infinite-dimensional Hamiltonian operators of Dirac structures are described in terms of differential geometry, theory of Lie algebras and group representation theory, corresponding integrable systems are considered. The reference is Irene Dorfman’s book in 1996.

Brace algebra structure is important in the solutions of Deligne’s conjecture on the Hochschild cochain complex of an associative algebra, and the studies of formality in deformation quantizations. In this talk, we will discuss a construction of homotopy brace algebras from Lie algebroid pairs.

Boundary groupoids can be used to model many analysis problems on singular spaces.In this talk, we first show that the $\eta$-term vanishes for elliptic differential operators on renormalizable boundary groupoids which is based on the method of renormalized trace similar to that of Moroianu and Nistor. Next we introduce the notion of {\em deformation from the pair groupoid}. Under the assumption that a deformation from the pair groupoid $M \times M$ exists for Lie groupoid $\mathcal{G}\rightrightarrows M$, we construct explicitly a deformation index map relating the analytic index on $\mathcal G$ and the index on the pair groupoid. We apply this map to boundary groupoids of the form $\mathcal{G}= M_0 \times M_0 \sqcup G \times M_1 \times M_1 \rightrightarrows M=M_0 \sqcup M_1$, where $G$ is an exponential Lie group, to obtain index formulae for (fully) elliptic (pseudo)-differential operators on $\mathcal{G}$, with the aid of the index formula by M. J. Pflaum, H. Posthuma, and X. Tang. It is joint work with Bing Kwan So (Jilin University).

The primary objective of this talk is to derive explicit formulas for rank one and rank two Drinfeld modules over a specific domain denoted by $\mathcal{A}$. This domain corresponds to the projective line associated with an infinite place of degree two. To achieve the goals, we construct a pair of standard Drinfeld modules whose coefficients are in the Hilbert class field of $\mathcal{A}$. We demonstrate that the period lattice of the exponential functions corresponding to both modules behaves similarly to the period lattice of the Carlitz module, the standard rank one Drinfeld module defined over rational function field. Moreover, we employ Anderson’s $t$-motive to obtain the complete family of rank two Drinfeld modules. This family is parameterized by the invariant $J = \lambda^{q^2+1}$ which effectively serves as the counterpart of the $j$-invariant for elliptic curves. Building upon the concepts introduced by van~der~Heiden, particularly with regard to rank two Drinfeld modules, we are able to reformulate the Weil pairing of Drinfeld modules of any rank using a specialized polynomial in multiple variables known as the Weil operator. As an illustrative example, we provide a detailed examination of a more explicit formula for the Weil pairing and the Weil operator of rank two Drinfeld modules over the domain $\mathcal{A}$. This is a joint work with Hu Chuangqiang.