학과 세미나 및 콜로퀴엄
In this talk, I will present deformations of compact holomorphic Poisson manifolds on the basis of Kodaira-Spencer's analytic deformation theory and extend their "Theorem of existence" for deformations of complex structures in the context of holomorphic Poisson deformations. I will discuss infinitesimal version of deformations of compact holomorphic Poisson manifolds and describe a differential graded Lie algebra governing holomorphic Poisson deformations of a compact holomorphic Poisson manifold in the language of "functor of Artin rings".
A complex normal variety $X$ is called a symplectic variety if it admits a holomorphic symplectic 2-form $omega$ on the regular part $X_{reg}$ and $omega$ extends to a holomorphic 2-form on a resolution $Y$ of $X$. Compared with the compact case, there are a lot of examples of affine symplectic varieties. They are not only interesting objects in algebraic geometry, but also play important roles in geometric representation theory.
The aim of this talk is to characterize the nilpotent variety of a complex semisimple Lie algebra among affine symplectic varieties. The main result is that if $(X, omega)$ is an affine singular symplectic variety embedded in an affine space as a complete intersection of homogeneous polynomials and $omega$ is homogeneous, then $(X, omega)$ coincides with the nilpotent variety
$N$ of a complex semisimple Lie algebra together with the Kostant-Kirillov 2-form $omega_{KK}$.
The proof of the main result uses the theory of Poisson deformation, holomorphic contact geometry, Mori theory and some elementary representation theory.
A complex normal variety $X$ is called a symplectic variety if it admits a holomorphic symplectic 2-form $omega$ on the regular part $X_{reg}$ and $omega$ extends to a holomorphic 2-form on a resolution $Y$ of $X$. Compared with the compact case, there are a lot of examples of affine symplectic varieties. They are not only interesting objects in algebraic geometry, but also play important roles in geometric representation theory.
The aim of this talk is to characterize the nilpotent variety of a complex semisimple Lie algebra among affine symplectic varieties. The main result is that if $(X, omega)$ is an affine singular symplectic variety embedded in an affine space as a complete intersection of homogeneous polynomials and $omega$ is homogeneous, then $(X, omega)$ coincides with the nilpotent variety
$N$ of a complex semisimple Lie algebra together with the Kostant-Kirillov 2-form $omega_{KK}$.
The proof of the main result uses the theory of Poisson deformation, holomorphic contact geometry, Mori theory and some elementary representation theory.
A complex normal variety $X$ is called a symplectic variety if it admits a holomorphic symplectic 2-form $omega$ on the regular part $X_{reg}$ and $omega$ extends to a holomorphic 2-form on a resolution $Y$ of $X$. Compared with the compact case, there are a lot of examples of affine symplectic varieties. They are not only interesting objects in algebraic geometry, but also play important roles in geometric representation theory.
The aim of this talk is to characterize the nilpotent variety of a complex semisimple Lie algebra among affine symplectic varieties. The main result is that if $(X, omega)$ is an affine singular symplectic variety embedded in an affine space as a complete intersection of homogeneous polynomials and $omega$ is homogeneous, then $(X, omega)$ coincides with the nilpotent variety
$N$ of a complex semisimple Lie algebra together with the Kostant-Kirillov 2-form $omega_{KK}$.
The proof of the main result uses the theory of Poisson deformation, holomorphic contact geometry, Mori theory and some elementary representation theory.
자연과학동(E6-1) Room 1409
Discrete Math
Hiu-Fai Law (University of Hamburg Germany)
F-Matchings in a tree
자연과학동(E6-1) Room 1409
Discrete Math
Given trees F and T, a F-matching is a collection of disjoint copies F in T. Generalizing results from Wagner (2009), Alon et al (2011) proved that the number of F-matchings for fixed F in a random tree T whose size tends to infinity is a.a.s. a multiple of any given integer m>0. We will discuss inverse and extremal problems on the number of F-matchings.
In this talk, we discuss the cuspidal group of $J_0(pq)$ and the rational torsion points of $J_0(pq)$.
We prove the following statement. If a prime l does not divide 6pq*gcd(p-1, q-1)*gcd(p-1,q+1)*(q-1,p+1), then the l-primary part of the rational torsion subgroup of $J_0(pq)$ is isomorphic to the l-primary subgroup of the cuspidal group.
We define and study an extended hyperbolic space which contains the hyperbolic space and de Sitter space as subspaces, and which is obtained as an analytic continuation of the hyperbolic space. The construction of the extended hyperbolic space gives rise to a complex valued geometry consistent with both the hyperbolic and de Sitter space. Such a construction inspires a new concrete insight for the study of the hyperbolic geometry and Lorentzian geometry as a unified object. We also discuss the advantage of this new geometric model as well as some of its applications.
Volume conjecture was first introduced by Kashaev late 90's and has arisen as a very important research problem in low dimensional topology since it involves many mathematical areas such as knot theory, hyperbolic geometry and quantum representation as well as theoretical physics.
In this talk I would like to give an introductory description of the conjecture for the general audience, especially with the figure eight knot complement. And then I will discuss an optimistic limit technique to obtain a new approach to knot/hyperbolic invariant of link complement.
Symbolic dynamics, a part of discrete dynamical systems, is the study of spaces consisting of infinite arrays defined by certain constraints on finite subsets. A rich theory has been developed on 1-dimensional symbolic dynamics in the last several decades, but the higher-dimensional case is more problematic and many problems occurring in multidimensional symbolic dynamics involve computability issues. In this talk, an introduction to the theory of multidimensional symbolic dynamics and the basic notion of the theory of computation will be presented. In the second part of the talk, we will see why computability problems are crucial for the development of the theory in the multidimensional actions.
E6-1, ROOM 1409
Discrete Math
Andrew Goodall (Charles University, Prague)
Graph Polynomials from simple graph sequences
E6-1, ROOM 1409
Discrete Math
The chromatic polynomial evaluated at a positive integer n is equal to the number of homomorphisms to the complete graph on n vertices. Many important graph polynomials are likewise determined by counting homomorphisms to a sequence of (multi) graphs, such as the Tutte polynomial and the independence polynomial. We give a powerful construction that produces graph polynomials in this way from sequences of simple graphs. It uses just two fundamental operations (disjoint union and interpretations of relational structures) and starts from a simple set of building blocks (coloured transitive tournaments). This will be illustrated by many examples, some of whose combinatorial properties have been well explored (such as that of the chromatic polynomial), but most of which are new. This is joint work with Jarik Nesetril and Patrice Ossona de Mendez.
We present a critical threshold phenomenon on the $L^1$-asymptotic completeness for the nonlinear Vlasov equation with a self-consistent force. For a long-ranged self-consistent force, we show that the nonlinear Vlasov equation has no $L^1$-asymptotic completeness, which means that the nonlinear Vlasov flow cannot be approximated by the corresponding free flow in $L^1$-norm time-asymptotically. In contrast, for a short-ranged force, the nonlinear Vlasov flow can be approximated by the free flow time-asymptotically. Our result corresponds to the kinetic analogue of scattering results to the Schrödinger-type equations in quantum mechanics.
Propensity score adjustment is a popular technique for handling unit nonresponse in samplesurveys. If the response probability depends on the study variable that is subject to missingness, estimating the response probability often relies on additional distributional assumptions about the study variable. Instead of making fully parametric assumptions about the population distribution of the study variable and the response mechanism, we propose a new approach of maximum likelihood estimation that is based on the distributional assumptions of the observed part of the sample. Since the model for the observed part of the sample can be verified from the data, the proposed method is less sensitive to failure of the assumed model of the outcomes. Generalized method of moments can be used to improve the efficiency of the proposed estimator. Results from a limited simulation study are presented to compare the performance of the proposed method with the existing methods. We also present an application of the proposed method to the exit poll for the 19th legislative election in Korea.
자연과학동(E6-1) Room 1409
Discrete Math
Tomoo Matsumura (KAIST)
Pfaffian Sum formula of the double Schubert polynomials for the symplectic Grassmannians
자연과학동(E6-1) Room 1409
Discrete Math
It is well-known that the ring of symmetric polynomials has a special basis, consisting of Schur polynomials and being indexed by Young diagrams. There underlies the representation theory of general linear groups and the combinatorics of Young diagrams. The geometry behind it is the geometry of Schubert varieties of Grassmannians of subspaces in a complex vector space. If we change the geometric setup to the case of Grassmannians of isotropic subspaces of a symplectic vector space, we have analogous functions called Schur Q-functions and their slight generalization. I will explain an explicit closed formula to express those functions as sums of Pfaffians and also the geometry and combinatorics behind it. If time allows, I will mention the technique of how we obtained the formula, that involves the so-called divided difference operators. This is a joint work with Takeshi Ikeda.
In this talk, we want to introduce famous geometers in Histroy of Mathe-
matics from C.F. Gauss, and B. Riemann, to S.S.Chern by way of E. Cartan.
Related to these great mathematicians some remarks on Fields medals and
Math. prizes in ICM will be remarked and some explanations about Poincare
conjecture and S.S. Chern conjecture will be given in detail.
Every minimal hypersurface in Sn+1(1) with constant scalar curvature
has the property that the squared norm of the second fundamental tensor h2
is constant.
From such a view point S.S.Chern conjectured the following problem:
Chern's conjecture For n-dimensional compact minimal hypersurfaces
in Sn+1(1) with constant scalar curvature, the value h2 of the squared norm
of the second fundamental forms should be discrete.
Finally, related to this conjecture we will report some solved problems
until now.
The higher Chow group is introduced by S. Bloch, which satisfies localization long exact sequence extending the classical Chow group. It is also related to arithmetic questions such as the special value of L-functions. It is interesting question to find varieties with big higher Chow group. In this talk, we construct surfaces over the formal Laurent series field over C, with big higher Chow group. We use the etale cycle map c(X) and the monodromy weight spectral sequence to compute the lower bound of dim(Im(c(X))).
Computational social choice is a growing discipline at the interface of social choice theory and computer science. It is concerned with the application of computational techniques to the study of social choice mechanisms, and with the integration of social choice paradigms into computing. Several problems have been investigated in the framework of parameterized complexity. This talk will describe the election scheme of Charles Dodgson. In 1876 the mathematician Charles Dodgson (better known as Lewis Carroll) formulated a rule that defines the winner of an election even if there is no Condorcet winner. A candidate who beats every other candidate in pairwise comparison is said to be a Condorcet winner. Unfortunately, an election may have a cyclic structure (candidate a beats b, candidate b beats c and c beats a), and therefore no candidate who beats all others in pairwise comparison. The Dodgson Score measures how close a candidate is to being a Condorcet winner. Candidates with a minimum Dodgson Score are the election winners. As are many election methods, Dodgson Score is NP-hard. This talk discusses the complexity of Dodgson Score in the parameterized framework. We give a reduction from Multi-Colored Clique to show that Dodgson Score, parameterized by the number of votes, is W[1]-hard. When parameterized by the number of swaps, Dodgson Score is FPT, but we show by polynomial parameter transformation that it has no polynomial kernel.
We now have rather satisfactory answer to many questions about orbifolds CC^n/G, where n = 2 or 3 and G is a finite subgroup of SL(n,CC). For example, the G-Hilbert scheme provides a standard crepant resolution of singularities, every projective crepant resolution represents an appropriate moduli functor involving G-equivariant sheaves on CC^n, and the derived category, K theory, homology or cohomology of a crepant resolution can be treated in terms of G-equivariant structures on CC^n. However, not much is known about n >= 4, or about finite subgroups of GL(3,CC). The talk will describe some ongoing work on the case of the terminal 3-fold points 1/r(1,a,r-a), mainly due to JUNG Seung-Jo (Warwick), and on some cases in dimension >= 4 that are almost tractable.
We study the convex hull of the symmetric moment curve Uk(t)=(cost, sint, cos3t, sin3t, …., cos(2k-1)t, sin(2k-1)t) in R2k and provide deterministic constructions of centrally symmetric polytopes with a record high number faces. In particular, we prove the local neighborliness of the symmetric moment curve, meaning that as long as k distinct points t1, …, tk lie in an arc of a certain length φk > π/2, the points Ut1, …, Utk span a face of the convex hull of Uk(t). In this talk, I will use the local neighborliness of the symmetric moment curve to construct d-dimensional centrally symmetric 2-neighborly polytopes with approximately 3d/2 vertices.
This is joint work with Alexander Barvinok and Isabella Novik.
In the theory of elliptic P.D.E.'s, an overdetermined problem is one where both the Dirichlet and Neumann boundary values are prescribed. This puts strong geometric constraints on the domain. A famous result of J. Serrin asserts that if is a bounded domain D in R^n which admits a function u solution of ꠑ Delta u =-1 in D with zero Dirichlet boundary value and constant Neumann boundary values, then D is a ball. The boundary of D is then a sphere, a constant mean curvature. I will present other similar results and in particular the existence of a 1-to-1 correspondence between harmonic functions which solve an overdetermined problem and a certain type of minimal surfaces.