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Lecture Video Playlist : https://www.youtube.com/playlist?list=PLSgu8su7kN2_OPd-Wg5799ZCM3vsPycwC
Dylan G.L. Allegretti (Yau Mathematical Sciences Center, Tsinghua University)
[ Talk 1 ]
Title: Skein algebras and quantized Coulomb branches, I
Abstract: In the first talk, I will introduce the Kauffman bracket skein algebra as a quantization of the SL(2,C)-character variety of a surface and describe this quantization explicitly for some simple surfaces.
[ Talk 2 ]
Title: Skein algebras and quantized Coulomb branches, II
Abstract: In the second talk, I will review the notion of a quantized K-theoretic Coulomb branch introduced by Braverman, Finkelberg, and Nakajima. I will describe this quantized Coulomb branch explicitly in the case of a quiver gauge theory.
[ Talk 3 ]
Title: Skein algebras and quantized Coulomb branches, III
Abstract: In the third talk, I will associate a quantized Coulomb branch to any compact surface of genus at most one with boundary. I will describe recent work with Peng Shan relating this quantized Coulomb branch to the skein algebra in some cases.
* If you would like to access lecture videos of "Skein algebras and quantized Coulomb branches," please contact Professor Hyunkyu Kim or Professor Dylan Allegretti.
Daniel C. Douglas (Max Planck Institute for Mathematics in the Sciences)
Title: Quantum higher Teichmüller theory
Abstract: We survey some (semi-)recent work and open problems at the intersection of higher Teichmüller theory and quantum topology. The guiding philosophy is that of Fock-Goncharov duality.
John F.R. Duncan (Institute of Mathematics, Academia Sinica)
Title: Rational Triangles, Elliptic Curves and Sporadic Groups
Abstract: Monstrous Moonshine was initiated by an observation relating the largest sporadic finite simple group to supersingular elliptic curves. We will explain how a closer look at the geometric side of this coincidence leads to a role for the smallest sporadic finite simple group in the congruent number problem of antiquity.
Ryo Fujita (Research Institute for Mathematical Sciences, Kyoto University)
[ Talk 1 ]
Title: Quantum affine algebras, graded quiver varieties and generalized Schur-Weyl duality, I
Abstract: In this series of talks, I explain a geometric construction of finite-dimensional representations of quantum affine algebras (of simply-laced type) using graded quiver varieties due to Nakajima, and some of its applications. In the first talk, I will review some basics of finite-dimensional representations of quantum affine algebras and Nakajima's geometric construction.
[ Talk 2 ]
Title: Quantum affine algebras, graded quiver varieties and generalized Schur-Weyl duality, II
Abstract: In the second talk, I will explain a relationship between the graded quiver varieties and the derived category of quiver representations due to Keller--Scherotzke. Via Nakajima's construction, it gives us some important information about the R-matrices between the fundamental modules.
[ Talk 3 ]
Title: Quantum affine algebras, graded quiver varieties and generalized Schur-Weyl duality, III
Abstract: In the third talk, I will talk about the generalized Schur-Weyl duality between quantum affine algebras and quiver Hecke algebras due to Kang--Kashiwara--Kim, and its geometric interpretation via graded quiver varieties. If time permits, I would mention very recent progress in the case of non-simply-laced types.
Ivan Chi-Ho Ip (Hong Kong University of Science and Technology)
Title: Regular Positive Representations
Abstract: In the study of positive representations of split real quantum groups, it is natural to consider an associated embedding of .png){kind=small}
to certain quantum cluster algebra related to the moduli space of framed G-local systems. Over the past few years, various new variations of such representations are realized as certain reductions of the original one, and share some common features which we call "regular". We will explain the current status of the classification of these regular positive representations.
Ryosuke Kodera (Chiba University)
Title: Affine Yangians and W-algebras in AGT correspondence
Abstract: Around 2009, Alday-Gaiotto-Tachikawa proposed a correspondence between 4-dimensional gauge theories and 2-dimensional conformal field theories. It is now called AGT correspondence. One goal is to establish a statement of the form "An algebra related to a conformal field theory (Virasoro algebras, affine Lie algebras, and W-algebras, which are their generalizations) acts on the cohomology group of the moduli space of instantons associated with a gauge theory''. Furthermore, it is expected that such actions may be obtained through a kind of quantum groups called Yangians.
In my talk, I will give an introduction to AGT correspondence and explain the speaker's result that connects the affine Yangians and the W-algebras associated with rectangular nilpotent elements (joint work with Mamoru Ueda, arXiv:2107.00780).
Zhihao Wang (University of Groningen and Nanyang Technological University)
Title: The Frobenius map for stated SL(n)-skein modules.
Abstract: In this talk, I will introduce the stated SL(n)-skein modules, the classical limit of stated SL(n)-skein modules, and the explicit construction of the Frobenius map for SL(n). I will also introduce some relations I calculated for stated SL(n)-skein modules, which show the well-definedness of the Frobenius map and the transparency (or root of unity transparency) of the image of the Frobenius map.
Tao Yu (Southern University of Science and Technology)
Title: Quantum trace maps for SL(n)-skein algebras
Abstract: The skein algebra of a surface is a quantization of the character variety, and the quantum trace map is a quantization of the classical trace map that sends a curve on the surface to the coordinate expression of the trace function. We review the n=2 case constructed by Bonahon-Wong and describe the generalization to all n.