Analytic and combinatorial aspects of representation theory

表示论是数学的一个领域，它交织了几个学科的思想，包括代数，分析，组合学和数学物理，以研究参数化对称性的抽象数学对象的实现（也称为表示）。在动力学系统和物理学中出现的许多问题中，人们会遇到连续的对称群，它们被称为李群。李群是属于微分几何的对象，但对它们表示的研究基本上依赖于称为李代数的某些代数对象。李群和李代数的表示论在调和分析、数论、代数组合学和理论物理中起着杰出的作用。** 在我的研究在未来五年，我的目标是回答问题的李群和李代数的表示理论，和李代数的推广，这是所谓的李超代数。这些问题与对称多项式理论密切相关。对称多项式，如杰克和麦克唐纳多项式及其变形，经常出现在表示论中。这导致了显着的相互作用之间的表示论和代数组合，和洞察力提供的每一个这些分支的数学丰富了其他一个实质性的。我提出的研究目标是通过关注李超代数的情况来揭示这些相互作用，并与其他代数对象（如量子群）建立新的联系。从李代数到李超代数的过渡导致了许多新的挑战和技术困难，解决它们需要新颖的想法。我还计划继续我的研究酉表示李群使用工具从分析。这些表示是在希尔伯特空间上实现的，通常底层的李代数并不作用于整个表示空间。特别是，带有李代数作用的规范稠密子空间，如光滑向量空间，在理论中起着至关重要的作用。在我的研究中，我将研究有限维和无限维李群的表示。有限维的情况包括真实的和p-adic半单李群，因此我的研究的影响将在自守形式的理论。在无限维的情况下，所研究的群包括loop群和Virasoro群。后者群的最重要的一类表示是正能量的酉表示类。因此，在无限维的情况下，我的研究的影响将是在数学物理。

Representation theory is an area of mathematics that interweaves ideas from several disciplines, including algebra, analysis, combinatorics, and mathematical physics, to study realizations (also known as representations) of abstract mathematical objects that parametrize symmetries. In many problems that emerge in dynamical systems and physics, one encounters continuous groups of symmetries, which are called Lie groups. Lie groups are objects that belong to differential geometry, but the study of their representations relies substantially on certain algebraic objects called Lie algebras. Representation theory of Lie groups and Lie algebras plays a distinguished role in harmonic analysis, number theory, algebraic combinatorics, and theoretical physics. ******In my research over the next five years, I aim to answer questions in representation theory of Lie groups and Lie algebras, and a generalization of Lie algebras which are called Lie superalgebras. These questions are closely related to the theory of symmetric polynomials. Symmetric polynomials, such as Jack and Macdonald polynomials and their deformations, occur frequently in representation theory. This has lead to remarkable interactions between representation theory and algebraic combinatorics, and the insight provided by each of these branches of mathematics has enriched the other one substantially. The goal of my proposed research is to shed more light on these interactions by focusing on the case of Lie superalgebras, and to establish new connections with other algebraic objects such as quantum groups. The passage from Lie algebras to Lie superalgebras results in many new challenges and technical difficulties, and tackling them requires novel ideas.******I also plan to continue my study of unitary representations of Lie groups using tools from analysis. These representations are realized on Hilbert spaces, and typically the underlying Lie algebra does not act on the entire representation space. In particular, canonical dense subspaces which carry an action of the Lie algebra, such as the space of smooth vectors, play a crucial role in the theory. In my research I will study representations of both finite and infinite dimensional Lie groups. The finite dimensional case includes real and p-adic semisimple Lie groups, and therefore the impact of my research will be in the theory of automorphic forms. In the infinite dimensional case, the groups under investigation include loop groups and the Virasoro group. One of the most important classes of representations of the latter groups is the class of unitary representations of positive energy. Therefore in the infinite dimensional case the impact of my research will be in mathematical physics.