World experts and early-career researchers will come together to exchange ideas and to build research networks.
Representation theory is a prominent branch of mathematics which studies abstract algebraic structures by relating them to well-understood ones, traditionally linear maps on a vector space. It connects to many other branches (category theory, combinatorics, harmonic analysis, geometry, topology, number theory, ...) and as such plays a major role in unified frameworks such as the Langlands program.
One of the important applications of representation theory and indeed one of its orgins is the study of symmetries in physical models. Classical or quantum integrable systems can be regarded as dynamical systems with “maximal” symmetry. In classical mechanics, integrability means a sufficient number of independent integrals of motion, and quantum integrability has a similar meaning. Sometimes integrable systems, especially infinite-dimensional ones, make it possible to study effects that are usually masked by complicated chaotic dynamics in non-integrable systems. Quantum integrable systems also inspired the discovery of new algebraic structures, such as quantum groups.
Together, representation theory and integrable systems form a modern and powerful area of science at the interface of pure mathematics and applications.
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Tuesday,Wednesday,Thursday,Friday | 09:00 - 18:00 | A6-101 | ZOOM 13 | 637 734 0280 | BIMSA |
| Time\Date | Jun 2 Tue |
|---|---|
| 16:30-17:30 | Oksana Yakimova |
*All time in this webpage refers to Beijing Time (GMT+8).
16:30-17:30 Oksana Yakimova
Near-derivations and their applications to Lie algebras
We extend E.B. Vinberg’s theory of quasi-derivations of algebras to a broader framework of near-derivations. This deepens connections between Poisson geometry and Lie theory. While basic results apply to arbitrary algebras, our main focus lies on the Poisson algebra $(S(\mathfrak{q}),\{\, ,\ \})$ associated to a Lie algebra $\mathfrak{q}$. It will be shown that a near-derivation $D$ of $(S(\mathfrak{q}),\{\, ,\ \})$ naturally gives rise to a pencil of compatible Poisson structures on the dual of $\mathfrak{q}$. Moreover, using $D$ one may naturally construct a Poisson-commutative subalgebra of $S(\mathfrak{q})$.<br><br>Special emphasis is placed on near-derivations arising from $\mathfrak{q}$ itself, which lead to both classical and novel families of compatible Poisson brackets. At the end of the talk, we will compare properties of near-derivations of $\mathfrak{q}$ with Nijenhuis operators in $\mathfrak{gl}(\mathfrak{q})$, highlighting parallels between these two frameworks.
- Jiakang BAO (University of Tokyo, Tokyo, Japan)
Quiver (BPS) Algebras and Crystal Representations - Nikita BELOUSOV (BIMSA, Beijing, China)
Eigenfunctions of BC Toda chain - Siqi CHEN (Qilu University of Technology, Jinan, China)
Painlevé transcendents and recurrence relations for orthogonal polynomials with discontinuous Gaussian-type weights - Andrei GRIGOREV (NRU HSE, Moscow, Russia)
Monodromy-free Shrodinger operators and master functions for affine $\mathfrak{sl}_{2}$ - Yuan MIAO (Kavli IPMU, University of Tokyo, Tokyo, Japan)
Hidden Onsager symmetry in XXZ model at root of unity $\mathfrak{sl}_{2}$ - Maria ONUFRIENKO (Lomonosov Moscow State University, Moscow, Russia)
Semiglobal classification of corank-1 singularities in integrable Hamiltonian systems with three degrees of freedom - Rahul SINGH (YMSC, Tsinghua University, Beijing, China)
$q$–opers and Bethe Ansatz for open spin chains - Jinfeng SONG (The Hong Kong University of Science and Technology, Hong Kong, China)
Braid group symmetries on Poisson homogeneous spaces - Pengyu SUN (Shanghai University, Shanghai, China)
Soliton solutions of the cross-ratio equation on the half-plane - Xiaolu YUE (City University of Hong Kong, Hong Kong, China)
$(1,m)$-type biorthogonal polynomials and discrete Painlevé-type equations - Siyao YIN (Sobolev Institute of Mathematics, Novosibirsk, Russia)
Integrable Birkhoff Billiards inside Cones - Weinan ZHANG (The Hong Kong University, Hong Kong, China)
Quantum symmetric pairs at roots of unity