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Information Center for Mathematical Science

세미나

Information Center for Mathematical Science

세미나

연세대학교 수학과 세미나

연세대학교 수학과 세미나

연세대학교 수학과 세미나

연세대학교 수학과 세미나
  • Title : Fock-Sobolev spaces
  • Speaker : 조홍래
  • Data : 2019-01-07 15:00:00
  • Host : 연세대학교
  • Place :

연세대학교 수학과 세미나

고려대학교 수학과 세미나

The main purpose of this talk is to provide an effective version of a result due to Einsiedler, Mozes, Shah and Shapira, on the equidistributio​n of rational points on expanding horospheres in the space of unimodular lattices in at least 3 dimensions. Their proof uses techniques from homogeneous dynamics and relies in particular on measure-classif​ication theorems due to Ratner. Instead, we pursue an alternative strategy based on spectral theory, Fourier analysis and Weil’s bound for Kloosterman sums which yields an effective estimate on the rate of convergence in the space of (d + 1)-dimensional Euclidean lattices with d > 1. This extends my work with J. Marklof, on the 3-dimensional case (2017). This is a joint work with D. El-Baz and B. Huang.

고려대학교 수학과 세미나

Vertical Sato-Tatetheore​m for holomorphic modular forms concerns the distribution of eigenvaluesof a fixed Heckeoperator $T_p$acting on the space of weight $k$ and level $N$ modular forms, as $k+N\to \infty$. Itwas proven by Serre (and independently by Sarnak)that there exists a limiting measure $\mu_p$, which depends only on $p$, such thatthe eigenvalues become equidistributed​ relatively to $\mu_p$. Fix $N$ for simplicity. Then this can berestated in terms of the discrepancy between two measures: a probabilitymeas​ure $\mu_{p,k}$ supported on the eigenvalues of the Heckeoperator, and $\mu_p$, i.e., it is equivalent to $D(\mu_{p,k}, \mu_p) \to0$. Regarding the rate of convergence, in the context of arithmetic quantumchaos, it was suggested both by speculation and numerical test that \[ D(\mu_{p​,k}, \mu_p) =O(k^{-1/2+\eps​ilon}). \] In this talk, I'm going to disprove thisby showing that \[ D(\mu_{p,​k}, \mu_p) =\Omega(k^{-1/3​}\log^2 k). \] This is a joint work with NaserTalebizade​hSardariand Simon Marshall. Texas A&M

2019 서울대학교 수리과학부 세미나

Macdonald polynomials are associated with an irreducible affine root system, and are of two types: symmetric and nonsymmetric. The former are orthogonal polynomials with rational function coefficients in q,t, which are invariant under the corresponding finite Weyl group; upon setting q=t=0, they specialize to the irreducible characters of semisimple Lie algebras, in particular to Schur polynomials in type A. Macdonald polynomials have deep connections with: double affine Hecke algebras (DAHA), p-adic groups, integrable systems, conformal field theory, statistical mechanics, Hilbert schemes etc. This series of lectures will explore two closely related sides of the story of Macdonald polynomials: their central role in the representation theory of affine Lie algebras, and combinatorial constructions. I will start with the definition of Macdonald polynomials, and their construction in terms of the DAHA. I will continue with two classes of combinatorial formulas for Macdonald polynomials and the connection between them: the type-independent Ram-Yip formula, based on the so-called alcove model, and tableau formulas in classical types. Then I will discuss the way in which various specializations of Macdonald polynomials occur in representation theory, particularly as graded characters of certain modules for affine Lie algebras (Demazure modules, Kirillov-Reshetikhin modules, and several variations of them). The mentioned alcove model leads to a combinatorial model for the corresponding Kashiwara crystals; these are colored directed graphs encoding representations of quantum algebras in the limit of the quantum parameter going to 0. I will conclude with several recent developments in the area. The lectures contain joint work with my collaborators: Satoshi Naito, Daisuke Sagaki, Anne Schilling, Travis Scrimshaw, and Mark Shimozono, as well as my students Arthur Lubovsky and Adam Schultze. They will be largely self-contained, and only basic knowledge of the representation theory of Lie algebras is assumed.

2019 서울대학교 수리과학부 세미나

Macdonald polynomials are associated with an irreducible affine root system, and are of two types: symmetric and nonsymmetric. The former are orthogonal polynomials with rational function coefficients in q,t, which are invariant under the corresponding finite Weyl group; upon setting q=t=0, they specialize to the irreducible characters of semisimple Lie algebras, in particular to Schur polynomials in type A. Macdonald polynomials have deep connections with: double affine Hecke algebras (DAHA), p-adic groups, integrable systems, conformal field theory, statistical mechanics, Hilbert schemes etc. This series of lectures will explore two closely related sides of the story of Macdonald polynomials: their central role in the representation theory of affine Lie algebras, and combinatorial constructions. I will start with the definition of Macdonald polynomials, and their construction in terms of the DAHA. I will continue with two classes of combinatorial formulas for Macdonald polynomials and the connection between them: the type-independent Ram-Yip formula, based on the so-called alcove model, and tableau formulas in classical types. Then I will discuss the way in which various specializations of Macdonald polynomials occur in representation theory, particularly as graded characters of certain modules for affine Lie algebras (Demazure modules, Kirillov-Reshetikhin modules, and several variations of them). The mentioned alcove model leads to a combinatorial model for the corresponding Kashiwara crystals; these are colored directed graphs encoding representations of quantum algebras in the limit of the quantum parameter going to 0. I will conclude with several recent developments in the area. The lectures contain joint work with my collaborators: Satoshi Naito, Daisuke Sagaki, Anne Schilling, Travis Scrimshaw, and Mark Shimozono, as well as my students Arthur Lubovsky and Adam Schultze. They will be largely self-contained, and only basic knowledge of the representation theory of Lie algebras is assumed.

2019 서울대학교 수리과학부 세미나

Macdonald polynomials are associated with an irreducible affine root system, and are of two types: symmetric and nonsymmetric. The former are orthogonal polynomials with rational function coefficients in q,t, which are invariant under the corresponding finite Weyl group; upon setting q=t=0, they specialize to the irreducible characters of semisimple Lie algebras, in particular to Schur polynomials in type A. Macdonald polynomials have deep connections with: double affine Hecke algebras (DAHA), p-adic groups, integrable systems, conformal field theory, statistical mechanics, Hilbert schemes etc. This series of lectures will explore two closely related sides of the story of Macdonald polynomials: their central role in the representation theory of affine Lie algebras, and combinatorial constructions. I will start with the definition of Macdonald polynomials, and their construction in terms of the DAHA. I will continue with two classes of combinatorial formulas for Macdonald polynomials and the connection between them: the type-independent Ram-Yip formula, based on the so-called alcove model, and tableau formulas in classical types. Then I will discuss the way in which various specializations of Macdonald polynomials occur in representation theory, particularly as graded characters of certain modules for affine Lie algebras (Demazure modules, Kirillov-Reshetikhin modules, and several variations of them). The mentioned alcove model leads to a combinatorial model for the corresponding Kashiwara crystals; these are colored directed graphs encoding representations of quantum algebras in the limit of the quantum parameter going to 0. I will conclude with several recent developments in the area. The lectures contain joint work with my collaborators: Satoshi Naito, Daisuke Sagaki, Anne Schilling, Travis Scrimshaw, and Mark Shimozono, as well as my students Arthur Lubovsky and Adam Schultze. They will be largely self-contained, and only basic knowledge of the representation theory of Lie algebras is assumed.

2019 서울대학교 수리과학부 세미나

I will give a emph{very} introductory, informal talk—no prerequisites beyond a first course in algebraic geometry—on work using moduli spaces of curves to explore general questions in birational geometry over the past 50 years. The first half of the talk will recall basic notions from birational geometry and then review what moduli spaces are, sticking to the case of curves. The second half will outline classical constructions of these spaces and illustrate the interplay between constructions of a moduli space

2019 서울대학교 수리과학부 세미나

I hope to convince you that questions about factorizations of polynomials with non-negative real coefficients---not necessarily into irreducibles and possibly with conditions on the degrees of the factors---are worth studying by discussing several examples that have interesting answers. The question in the title arises from an equivalent formulation that asks what we can say about a finite collection of dice from knowing the probability distribution of the totals that arise when the dice are rolled. No knowledge of the game of craps will be required, but you may enjoy preparing by streaming this classic scene

서울대학교 물리천문학부 세미나

한국고등과학원 세미나

We study the possibility of a deconfined quantum phase transition in the two dimensional Shastry-Sutherland spin model, using both numerical and field theoretic techniques. We argue that the quantum phase transition between a two fold degenerate plaquette valence bond solid (pVBS) order and Neel order may be described by a deconfined quantum critical point (DQCP) with emergent O(4) symmetry. Further, using the infinite density matrix renormalization group (iDMRG) numerical technique, we verify the emergence of an intermediate pVBS order, between the dimer and Neel ordered phases. By analyzing the correlation length spectrum for different orders, we provide evidence for deconfinement and emergent O(4) symmetry at the phase transition between the pVBS and N\'eel orders. Such a phase transition has been reported in the recent pressure tuned experiments in the Shastry-Sutherland lattice material SrCu2(BO3)2. The non-symmorphic lattice structure of the Shastry-Sutherland compound leads to extinction points in the scattering, at which we predict sharp signatures for DQCP in both phonon and magnon spectra associated to the spinon continuua. Our result would guide the experimental search for DQCP in quantum magnets.

한국고등과학원 세미나

Recently, a lot of work has been done on the SYK model and its tensor counterparts. These SYK-like tensor models are first introduced by Witten based on the work of Gurau et al. I start by introducing these SYK-like tensor models and the motivations to study them. After this,? based on our work, I will propose a systematic way to: a) Diagonalize the Hamiltonian of these tensor models analytically b) Identify the gauge invariant/singlet states among the eigenstates of the Hamiltonian I will then discuss a nontrivial example where we have identified the complete singlet spectrum analytically using the above method. I conclude my talk by listing some future directions.

한국고등과학원 세미나

In this talk, we shall discuss the notion of very weak solutions for quasilinear parabolic equations and study the existence theory for the homogeneous and nonhomogeneous problem. In both cases, we obtain existence of very weak solutions which requires obtaining improved and in some cases, sharp a priori estimates below the natural exponent that enables us to apply compactness arguments. Some of the estimates we obtain are new even for the heat equation on bounded domains. This is joint work with Sun-Sig Byun and Wontae Kim.

KAIST 수리과학과 세미나

A Verra fourfold is a smooth projective complex variety defined as a double cover of P^2x P^2 branched along a divisor of bidegree (2,2). A Verra fourfold is a smooth projective complex variety defined as a double cover of P^2x P^2 branched along a divisor of bidegree (2,2). These varieties are similar to cubic fourfolds in several ways (Hodge theory, relation to hyperkaehler fourfolds, derived categories). Inspired by these multiple analogies, I consider the Chow ring of a Verra fourfold. Among other things, I will show that the multiplicative structure of this Chow ring has a curious K3-like property.

KAIST 수리과학과 세미나

The generalized Franchetta conjecture as formulated by O’Grady is about algebraic cycles on the universal K3 surface. It is natural to consider a similar conjecture for algebraic cycles on universal families of hyperkaehler varieties. This has close ties to Beauville’s conjectural ``splitting property’’, and the Beauville-Voisin conjecture (stating that the Chow ring of a hyperkaehler variety has a certain subring injecting into cohomology). In my talk, I will attempt to give an overview of these conjectures, and present some cases where they can be proven. This is joint work with Lie Fu, Charles Vial and Mingmin Shen.

KAIST Discrete Math 세미나

Problems such as Vertex Cover and Multiway Cut have been well-studied in parameterized complexity. Cygan et al. 2011 drastically improved the running time of several problems including Multiway Cut and Almost 2SAT by employing LP-guided branching and aiming for FPT algorithms parameterized above LP lower bounds. Since then, LP-guided branching has been studied in depth and established as a powerful technique for parameterized algorithms design. In this talk, we make a brief overview of LP-guided branching technique and introduce the latest results whose parameterization is above even stronger lower bounds, namely μ(I)=2LP(I)-IP(dual-I). Here, LP(I) is the value of an optimal fractional solution and IP(dual-I) is the value of an optimal integral dual solution. Tutte-Berge formula for Maximum Matching (or equivalently Edmonds-Gallai decomposition) and its generalization Mader’s min-max formula are exploited to this end. As a result, we obtain an algorithm running in time 4^(k-μ(I)) for multiway cut and its generalizations, where k is the budget for a solution. This talk is based on a joint work with Yoichi Iwata and Yuichi Yoshida from NII.

KAIST 수리과학과 세미나

A flag Bott tower is a sequence of flag bundles such that each stage of which comes from the induced full-flag bundle of a sum of holomorphic line bundles. A flag Bott manifold is not toric variety but it has a torus action. In this talk, we consider the standard torus action on a flag Bott manifold and compute its equivariant cohomology ring.