图的同调类中圈的渐近展开

Lusztig 层、特征循环与 Nakajima 箭图簇的 Borel-Moore 同调

Lusztig sheaves, characteristic cycles and the Borel-Moore homology of Nakajima's quiver varieties

本文利用特征循环构建量子群典范基到 Nakajima 箭图簇 Borel-Moore 同调的态射，证明整数实现并给出张量积簇同构猜想的新证明。

Odd characteristic classes in entire cyclic homology and equivariant loop space homology

Given a compact manifold $M$ and $g\in C^{\infty}(M,U(l;\mathbb{C}))$ we construct a Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part of the equivariant (entire) cyclic Chen-normalized bar complex $\underline{\mathscr{C}}(Ω_{\mathbb{T}}(M\times \mathbb{T}))$ of $M$, and which is mapped to the odd Bismut-Chern character under the equivariant Chen integral map. It is also shown that the assignment $g\mapsto \mathrm{Ch}^-(g)$ induces a well-defined group homomorphism from the $K^{-1}$ theory of $M$ to the odd homology group of $\underline{\mathscr{C}}(Ω_{\mathbb{T}}(M\times \mathbb{…

局部有限同调中$\mathrm{GL}_n(\mathbb{Z})$的轮类，规范积分与ζ值

The wheel classes in the locally finite homology of $\mathrm{GL}_n(\mathbb{Z})$, canonical integrals and zeta values

本文研究了轮类在局部有限同调和热带几何中的表现，证明了规范积分与奇ζ值的比例关系，并揭示了其在不同数学结构中的非零类存在。

The Geometry of integrable systems. Tau functions and homology of Spectral curves. Perturbative definition

In this series of lectures, we (re)view the "geometric method" that reconstructs, from a geometric object: the "spectral curve", an integrable system, and in particular its Tau function, Baker-Akhiezer functions and "current amplitudes", all having an interpretation as CFT conformal blocks. The construction identifies Hamiltonians with cycles on the curve, and times with periods (integrals of forms over cycles). All the integrable structure is formulated in terms of homology of contours, the phase space is a space of cycles where the symplectic form is the intersection, the Hirota operator is …

同调、上同调与示性类的新颖编码

Novel Encodings of Homology, Cohomology, and Characteristic Classes

本文通过扩展环面码并利用其错误的拓扑结构，首次在拓扑量子纠错码中直接编码纤维丛的阻碍类，从而实现了陈类、庞特里亚金类等示性类的新颖编码，并明确构造了 $S^2$ 的欧拉类实例。