CMC Algebra Seminar

2024 Seminars

2024. 12. 13. (Fri) 10:30-12:00  Room 1424
Haseo Ki (Yonsei University)

On Bogomolny-Schmit Conjecture for Maass forms

ABSTRACT. I will introduce Bogomolny-Schmit conjecture for Maass forms.

2024. 12. 2. (Mon) 10:30-12:00  Room 1424
Jaeseong Oh (Korea Institute for Advanced Study)

On Hikita's Proof of the Stanley-Stembridge Conjecture

ABSTRACT. The Stanley-Stembridge conjecture, a longstanding open problem in algebraic combinatorics, predicts that certain chromatic symmetric functions are e-positive. Recently, Hikita achieved a remarkable breakthrough by proving this conjecture through a probabilistic interpretation. In this talk, I will provide an overview of Hikita's approach, trying to understand the motivation of the key ideas with numerous examples. Additionally, I will discuss related open problems where Hikita's techniques may offer valuable insights.

2024. 11. 22. (Fri) 10:30-12:00  Room 8101
Se-jin Oh (Sungkyunkwan University)

Generalization of quantum Grothendieck rings

ABSTRACT. Grothendieck rings C_g of finite dimensional modules over quantum affine algebras have been intensively studied since they are related to various branches of mathematics and mathematical physics. The quantization of C_g introduced by Nakajima and Varanolo-Vasserot, called quantum Grothendieck ring, provides an Kazhdan-Lusztig algorithm for Jordan-Holder multiplicity of simple modules in standard modules for ADE cases. In this talk, I would like to introduce the generalization of the quantum Grothendieck ring and recent progress about it. If time permits, I would like to discuss the quantum positivity of KR-polynomials. This talk is based on the results with my collaborators, Fujita, Jang, Hernandez, Lee, Kashiwara, Kim, Park and Oya.

2024. 10. 31. (Thu) 09:30-10:30  Room online
Ryo Fujita (Research Institute for Mathematical Sciences)

Deformed Cartan matrices and generalized preprojective algebras, with application to quantum affine algebras

ABSTRACT. In their study of the deformed W-algebras associated with complex simple Lie algebras, E. Frenkel--Reshetikhin introduced certain two parameter deformations of the Cartan matrices. They are also important in the representation theory of quantum affine algebras as a key combinatorial ingredient. In this talk, we give a representation-theoretic interpretation of these deformed Cartan matrices (and their inverses) in terms of the generalized preprojective algebras recently introduced by Geiss--Leclerc--Schröer, and discuss its application to the quantum affine algebras in relation with the cluster theory. This talk is based on a joint work with Kota Murakami.

2024. 10. 11. (Fri) 10:00-11:00  Room online
Kento Osuga (University of Tokyo)

b-Hurwitz numbers from Whittaker vectors for W-algebras

ABSTRACT. G-weighted Hurwitz numbers can be interpreted as sum of ribbon graphs embedded into orientable surfaces with certain weight G, and G-weighted b-Hurwitz numbers are their non-orientable analogues with the notion of measure of non-orientability proposed by Chapuy-Dolega. It was further observed by Bomzon-Chapuy-Dolega that for (very) simple choices of G, their b-Hurwitz numbers satisfy Virasoro constraints. Then, one may ask: what if we choose G to be a more general form? Do they still satisfy Virasoro constraints or are they related to different algebras? In this talk I will show a nontrivial relation between b-Hurwitz numbers and so-called Whittaker vectors for W-algebras --- a generalisation of the Virasoro algebra. If time permits, I will also discuss how they are related to topological recursion as well as refined topological recursion. This talk is based on joint work with Nitin Chidambaram and Maciej Dolega, and the main reference is arXiv:2401.12814.

2024. 9. 24. (Tue) 10:30-12:00  Room 8101
Soojin Cho (Ajou University)

Permutation module decomposition of the cohomology of a regular semisimple Hessenberg variety

ABSTRACT. We consider the module structure of the cohomology of regular semisimple Hessenberg varieties (of type A) under the symmetric group action. We use BB(Bialynicki-Birula) basis of the cohomology space to construct permutation module decomposition of the cohomology for the Hessenberg varieties associated with lollipop graphs and the second degree cohomology for general cases. This talk is based on the works done with Jaehyun Hong and Eunjeong Lee, and the ongoing research with JiSun Huh and Seonjeong Park.

2024. 9. 20. (Fri) 10:30-12:00  Room 8101
Daeyeol Jeon (Kongju National University)

Construction of elliptic curves with prescribed torsion

ABSTRACT. In this talk, we aim to address the moduli problem associated with modular curves, which serve as the moduli space of elliptic curves. By solving this problem, we seek to construct elliptic curves prescribed torsion defined over specific number fields.

2024. 9. 12. (Thu) 10:00-11:00  Room online
Eugene Gorsky (University of California, Davis)

Delta Conjecture and affine Springer fibers

ABSTRACT. Delta Conjecture of Haglund, Remmel and Wilson is the identity describing the action of Macdonald operators on elementary symmetric functions. The conjecture was proved independently by Blasiak-Haiman-Morse-Pun-Seelinger, and D'Adderio-Mellit. In this talk, I will give a geometric model for Delta conjecture using affine Springer fibers. This is a joint work with Sean Griffin and Maria Gillespie.

2024. 9. 5. (Thu) 10:30-11:30  Room online
Brendon Rhoades (University of California, San Diego)

Harmonics and graded Ehrhart theory

ABSTRACT. Classical Ehrhart theory gives a family of results on counting lattice points in a lattice polytope P together with its integer dilations. The enumerative results of Ehrhart theory applied to P may be understood algebraically using the affine semigroup ring A_P attached to P. The orbit harmonics method gives a way to attach a graded ring to any finite point locus; we use this idea to define a canonical q-refinement of the Ehrhart series of P and make a number of conjectures about this object analogous to those in the classical setting. We attach a non-obvious bigraded ring H_P to P called the {\em harmonic algebra} of P. We conjecture that the harmonic algebra plays a role in the q-Ehrhart theory of P analogous to the role played by the affine semigroup ring in the classical Ehrhart theory of P. Joint with Vic Reiner.

2024. 8. 28. (Wed) 10:30-12:00  Room 8406
Sungsoon Kim (University of Picardy Jules-Verne)

Polynomial Representations of the Iwahori-Hecke algebras of type A and Beyond

ABSTRACT. It is a kind of survey talk on the polynomial representations of the Iwahori-Hecke algebras of the Coxeter group of type A_{n-1}, i.e. the symmetric group S_n, starting from Young's seminormal representation, works of Specht, and then generalisation to the Iwahori-Hecke algebra H(S_n). If time permits, there will be some discussions for other types like W(B_n) and/or W(D_n). The results for the case of complex reflection groups of type G(r,1,n) are still open.

2024. 8. 16. (Fri) 10:30-12:00  Room 8101
Yeansu Kim (Chonnam National University)

Classification of discrete series representations and its applications on the Langlands program

ABSTRACT. The main purpose of the talk is to introduce one main subject in the Langlands program, which is the classification of discrete series of p-adic groups. This is an important subject since it will provide a structure theory of refinement of local Langlands correspondence in the sense of filtration of admissible representations. It also has numerous applications in the Langlands program, such as local Langlands correspondence, Langlands functoriality conjectures, degenerate principal series, unitary duals, and so on. We first introduce our approach to the classification of discrete series, which is developed by Matić and myself. This generalizes the Mœglin-Tadić classification results for classical groups. If time permits, we will explain its application to the generic local Langlands correspondence via the Langlands-Shahidi method. This is a joint work with Ivan Matić, and its applications include work in progress with Krishnamurthy and Shahidi.

2024. 8. 9. (Fri) 10:30-12:00  Room 8101
Hae-Sang Sun (UNIST)

Distribution of Heilbronn variable and modular symbols

ABSTRACT. This is joint work with Hong Kwon. The Heilbronn variable is first introduced to analyze the first and second moment of the length of continued fraction on the rational numbers with a fixed denominator in the unit interval. In the talk, I will explain its history and our result on the distribution of Heilbronn variable. In the latter part, I will also explain how to extend our result to study the distribution of modular symbols.

2024. 7. 26. (Fri) 10:30-12:00  Room 8101
Jaeseong Heo (Hanyang University)

평균가능성과 톰슨 군 문제 (Amenability and Thompson group)

ABSTRACT. 평균가능성과 관련하여 Banach-Ruziewicz 문제, Banach-Tarski 역설을 소개하고, 폰노이만에 의해 정의된 (discrete) group의 평균가능성 (amenability), 폰노이만의 추측 및 그에 대한 결과들을 고찰한다. R. Thompson과 위상수학자들에 의해 독립적으로 발견된 Thompson 군의 여러가지 성질과 Thompson 군의 평균가능성 문제에 대하여 논의한다.

2024. 5. 23. (Thu) 10:30-12:00  Room 1424
Sungmun Cho (POSTECH)

Orbital integrals for classical Lie algebras and smoothening

ABSTRACT. I will introduce a new method, using smoothening of a certain scheme over DVR, to study stable orbital integrals for regular semisimple elements and for the unit element of the Hecke algebras in classical Lie algebras gl_n, u_n, sp_2n. As a byproduct, I will explain closed formulas for orbital integrals associated with an arbitrary element of the Hecke algebra in gl_3, and lower bounds/conjectures to estimate values of stable orbital integrals for the other cases. This is based on joint works with Taeyeoup Kang and/or Yuchan Lee.

Past Seminars

2023 Seminars

ORGANIZERS	Young-Hoon Kiem   -   Korea Institute for Advanced Study
	Chul-hee Lee   -   Korea Institute for Advanced Study
	Jaeseong Oh   -   Korea Institute for Advanced Study
	Sin-Myung Lee   -   Korea Institute for Advanced Study
	Byung-Hak Hwang   -   Korea Institute for Advanced Study