Invited Talk

In this short lecture series that is primarily for young graduate students, we aim to explain one of the first instances of the Langlands duality, namely, what the Langlands dual group means in a geometric manner. The bulk of the lectures will be devoted to developing some basic ideas that are useful for practitioners of number theory and representation theory, including the Tannakian formalism and Hecke algebras. 

1/30 강연 자료, 2/1 강연 자료

In this talk, we will consider the average value of the p-th Dirichlet coefficients of elliptic curves for a prime p in a fixed conductor range with given rank. Plotting this average yields a striking oscillating pattern, the details of which vary with the rank. Based on this observation, we can perform various data-scientific experiments. We will discuss some implications of the experimental results.

2022년 필즈상 수상자인 James Maynard의 연구 업적 중에서 소수의 간격에 대한 업적을 소개한다.

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Number Theory

In 1992, Claudia Spiro asked which set E determines an arithmetic function f uniquely in some set S of arithmetic functions under the condition f(a+b)=f(a)+f(b) for all a, b in E and she called E an additive uniqueness set for S. That is, she showed that the set of primes is an additive uniqueness set for multiplicative functions which do not vanish at some prime numbers. Since her intriguing result, many mathematicians have been solving enormously various problems related to Spiro’s paper. In this talk, we introduce several examples of additive uniqueness sets and related results.

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There is a well-known way of constructing a 1-dimensional rational distribution from circular units, called a circular distribution. We generalize this construction to a higher dimensional case, i.e. we construct an isomorphism between distributions on an n-dimensional rational vector space V and the n-th Milnor K-group of a circular algebra on V. Using this construction, we explicitly construct a GL_n-modular symbol with values in the n-th Milnor K-group-valued distributions on V, which we call the universal Dedekind-Eisenstein modular symbol. The logarithmic derivative of this modular symbol gives rise to an Eisenstein cocycle which contains key informations on periods of Eisenstein series and consequently the zeta values of totally real fields of degree n. The talk is based on the joint work with Glenn Stevens.

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We introduce Montgomery's 1974 paper on the pair correlation of zeros of the Riemann zeta function. This is the beginning of a new field of study, statistical understanding of the zero distribution via the random matrix theory.

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We study the distribution of eigenvalues of random matrices. In particular, we show how to compute n-level density for unitary matrices and compare it with n-level density for low-lying zeros of primitive Dirichlet L-functions.

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In this talk, we will review dynamics of SL2-action on hyperbolic spaces. We will then introduce Hurwitz complex continued fractions and dynamical properties of the complex Gauss map, in particular the finite range property. (Part of the talk is based on joint work with Jungwon Lee and Dohyeong Kim.)

We introduce the Riemann zeta function and proves its analytic properties. After that, we prove the Prime Number Theorem.

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We introduce Katz and Sarnak’s conjecture on low-lying zeros of L-functions in a nice family. After that, we explain what we have to calculate to compute one-level density. Lastly, we will review what the known results are.

As an application of low-lying zeros of L-function, we explain how to bound on the average analytic rank of elliptic curves.

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Oppenheim conjecture is the problem asking the condition of a quadratic form Q having arbitrarily small value Q(v), where v is chosen in the integral lattice. The conjecture was completely proved by Margulis showing that values of a quadratic form at integer points are dense in the real line if and only if the form is irrational under the condition that the form is non-degenerate, indefinite, and the rank is at least 3. In this talk, I will introduce some historical results related to this topic, so-called Oppenheim conjecture-typed problem, and how homogeneous dynamics contributes. We are particularly interested in studies about the distribution of joint values of integral vectors under a pair of a quadratic form and a linear form. This is joint work with Seonhee Lim and Keivan Mallahi-Karai.

Representation Theory

The Jantzen sum formula played an important role in calculating the decomposition number of symmetric groups. James and Mathas developed a q-analogue of Jantzen's sum formula and used it to characterize irreducible Weyl modules. In this talk, we will study the q-analogue of Jantzen's sum formula based on Mathas' book "Iwahori-Hecke algebras and Schur algebras of the symmetric group".

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In this talk, we review the relationship between RTT formalism and Drinfeld (current) presentation of the Yangian of type A. Using this connection, we interpret the isomorphism, introduced by Brundan-Kleshchev, from the Yangian of type A to finite rectangular W-algebra of type A. Also, we briefly review the homomorphism, constructed by Ueda, from the affine yangian to the universal enveloping algebra of the rectangular affine W-algebra and explain the relation between Brundan-Kleshchev’s map and Ueda’s map.

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First, we will construct dual canonical bases of symmetric and exterior powers of the natural representations of U_n=U_q(gl_n). And then, we will derive dual canonical bases of the irreducible polynomial representation of U_n by realizing the representation by means of a homomorphism between symmetric and exterior powers.

For a given vertex algebra, one can associate an associative algebra called a twisted Zhu algebra provided a proper gradation on the vertex algebra. As one of the most important example, a finite W-algebra is a twisted Zhu algebra of an affine W-algebra. In this talk, I will explain twisted Zhu algebras by showing some examples, including finite W-algebras, and review some representation theory behind them.

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Dirac reduction is the way to deal with Hamiltonian mechanics with constraints. In this talk, I explain the structures of classical W-algebras in terms of the modified Dirac reduction and why this modification is needed. If time allows, I will tell you that the results in our work which are extended to W-superalgebra with (or without) supersymmetric context. This is the joint work with Arim Song and Uhirinn Suh.

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The universal R-matrix is an element of a completion of quantum groups, which endows its module category a structure of a braided monoidal category. In this talk, we prove the uniqueness and existence of the universal R-matrix, its multiplicative formula, and discuss its applications to canonical bases.

The q-character is one of the most significant tools for finite-dimensional representations of quantum affine algebras. On one hand, one can regard the q-character as a kind of formal character and study it. On the other hand, the construction of it is largely motivated from a method of integrable systems, which is designed to solve the spectral problem of the Hamiltonian. The goal of this talk is to understand in this vein the corresponding (1+1)D quantum integrable model in terms of representations of quantum affine algebra. As always the key role is played by the R-matrix, which will be demonstrated in the toy example: spin 1/2 XXX chain.

Let g be a finite-dimensional simple Lie algebra over C. When g is of simply-laced type, a t-deformation K_t(g) of the Grothendieck ring of a category of finite-dimensional modules over a quantum loop algebra U_q(Lg) has a rich structure, especially (I) a deep connection with the dual canonical basis of the quantum group U_q(g) and (II) quantum cluster algebra structure of skew-symmetric type. In this talk, I briefly introduce the quantum virtual Grothendieck ring K_q(g), which can be viewed as a non-trivial generalization of K_t(g) for simply-laced types. Focused on the viewpoint of (I), I mainly explain some bases of K_q(g). Those bases are closely related to the dual-canonical/upper-global basis of finite type. This talk is based on joint work with Kyu-Hwan Lee and Se-jin Oh.

Deformed W-algebras have been studied by many mathematicians because of their interesting relation to integrable models associated to quantum affine algebras. In this talk, I will explain the construction of deformed W-algebra suggested by E.Frenkel, N.Reshetikhin, M.A. Semenov-Tian-Shansky, which is an analogue of Drinfeld-Sokolov reduction for non-deformed W-algebras. If time permits, I will also explain its relation to quantum affine algebras by describing deformed Miura transformation.

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In this talk, I explain the blocks of Hecke algebra and q-Schur algebra, which is a building block of their module category. In addition, I'll give some results on the irreducibility of well-known modules, such as Weyl modules and Specht modules. This is a summary of Chapter 5 of the main reference: Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of symmetric groups (1999).

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