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Introduction
(article in ¡®KIAS News Letter¡¯ 2005 Autumn *in Korean*)
Past
Seminars: 2007  2006 
2005
Previous
Organizers:
SoonYi Kang (March 2005 – August 2005)
Daeyeol Jeon (September 2005 – February 2006)
Andreas Bender (March 2006 – August 2006)
SeokMin Lee
(September 2006 – February 2007)
Dohoon Choi
(March 2007 – August 2007)
2005
March 10 Kwon,
SounHi (Korea University) Class numbers of
number fields
Abstract: We will discuss the class number one problem for normal
totally imaginary number fields.
March 15 Andreas
Bender (Kyoto University) The Schinzel hypothesis
in the function field case
Abstract: The Schinzel hypothesis is a farreaching generalization
of the twin prime conjecture, with applications in diophantine geometry. We
shall look at a very weak form of this conjecture which can be proved if the
ground ring is rather than .
March 24 Pierre
Matsumi (KIAS)
Introduction to a cohomological hasse principle
for arithmetic varieties
Abstract: In 20th century, Hasse, Noether... proved the famous localglobal
short exact sequence for Brauer groups over number fields. This was, from the
cohomological viewpoint, reinterpreted by Kato, who generalized the above
short exact sequence to a beautiful complex for arithmetic varieties called
Katocomplex. In this talk, I will explain the brief history on it until now
and talk on my progress during my KIAS fellowship.
April 14 Seo,
Soogil (KIAS) Cohomology groups of the
class groups
Abstract: We compute cohomologies of the class groups in terms of
special units. The motivation starts from comparison between the group
structures of the class groups and special units of Euler systems. The method
to compute cohomology of these for pparts goes back to Iwasawa, Gold,
Kim and etc. More recently Schoof made an interesting way to compute it for
nonp parts over . We will
introduce his method and extend it to extension using spectral sequence of
HochchildSerre and results of Iwasawa, Gold, Kim.
April 19 David Burns
(King's College London)
Congruences
between derivatives of abelian Lfunctions
April 21 David Burns
(King's College London)
Congruences
between derivatives of abelian Lfunctions II
April 28 Ki, Haseo (Yonsei
Univ.) Zeros of
approximations of Zeta functions
May 12 Lee, JoungYeon
(Seoul National University)
Class number one
problem of RichaudDegert Type
Abstract: A certain
type of real quadratic fields is RichaudDegert (, is not 5, positive square free and such
that divides and ). When or , this is called
narrowRichaudDegert. Originally, Chowla conjectured that for . And Yokoi
conjectured that if . Both
conjectures were proved by Biro. The rest part was conjectured for , by Mollin. We
prove this conjecture by calculating the special values of some Zeta
functions. This completes the class number one problem for real quadratic fields
of narrowRichaudDegert type.
May 17 Solomon Friedberg (Boston
College) Sums of
Lfunctions and applications to number theory
May 17 Solomon Friedberg
(Boston College) Theta functions on Odd Orthogonal Groups
Abstract: Given an homomorphism , the Langlands
functoriality conjecture predicts the existence of a map from the automorphic
representations of to those of . In the case that
the adelic points of the algebraic groups , are replaced by their covers, one may
specify an analogue of the group
(depending on the cover), and then one may still expect the existence of a
correspondence. In this talk we describe joint work with Profs. D. Bump
(Stanford) and D. Ginzburg (TelAviv University) in which we construct such a
map for the double cover of the split special orthogonal groups, raising the
genuine automorphic representations to those of . To do so we
use as integral kernel a of rather exotic theta representation, exotic since,
in contrast to the classical theta correspondence, this representation is not
minimal in the sense of corresponding to a minimal coadjoint orbit.
May 26 Kim, JonLark
(University of Nebraska at Lincoln)
CapacityApproaching
LowDensity ParityCheck Codes
Abstract: Suppose we
want to send a message to a receiver over a noisy channel, for example, a
pair of wires, a band of radio frequencies, a beam of light, and generally,
magnetic storage devices, compact discs, etc. The fundamental problem in
communication is to determine what message was sent on the basis of the
received message. In 1948 Claude Shannon introduced a number called the
capacity of the channel and showed in a nonconstructive way, that arbitrary
reliable communication is possible at any rate below the channel capacity.
The design of codes with efficient encoding and decoding algorithm which
approach the capacity of the channel is one of main areas in coding theory.
Coding theorists have been led to search for codes that have considerable
algebraic or combinatorial structures.
Lowdensity
paritycheck (LDPC) codes, originally introduced by Gallager in 1962, have
been one of the hottest topics in coding theory since MacKay and Neal in 1996
demonstrated that some long LDPC codes approach near Shannon limit under the
sumproduct algorithm. In this talk, we overview LDPC codes, Tanner graphs,
and discuss our results on the two constructions of LDPC codes. One is based
on finite groups, generalizing the idea of Lubotzky, et al and independently
Margulis' construction of Ramanujan graphs. The other is based on
algebraically defined qregular bipartite graphs suggested by Lazebnik
and Ustimenko.
June 9 Lee,
HyangSook (Ewha Womans University)
Pairing based
Cryptosystem and Vector Decomposition Problem
Abstract: Pairings
were first used in cryptography as a cryptanalytic tool for reducing the
discrete log problem on some elliptic curves to the discrete log problem in a
finite field. Positive cryptographic applications based on pairings arose
from the work of Joux, who gave a simple one round tripartite DiffieHellman
protocol on supersingular curves. Many cryptographic schemes based on the
pairings have been developed recently, such as identity based encryption,
identity based signature schemes and identity based authenticated key
agreements and identity based signcryption. We discuss some pairingbased
cryptographic schemes in this talk. More recently the cryptography based on the
vector decomposition problem(VDP) was suggested and it was proved that the
VDP in a two dimensional vector space is at least hard as the computational
DiffieHellman problem in a one dimensional subspace under some condition.
However we present the VDP can be solved for a certain basis although the
conditions are satisfied.
June 23 Shim, Kyungah
(Ewha Womans University)
A New Class of
Problems for Cryptographic Schemes
Abstract: We
introduce a new class of problems which can be considered as a
composition of a computation problem and a decision problem. We provide a new
hard problem which belongs to the new class. As the gap problems which deal
with the gap of difficulty between computation problems and decision problems,
we define the gap problems based on the gap of difficulty between composition
problems and computation problems. Finally, we discuss their
applications to cryptography.
July 14 Im, BoHae
(University of Utah)
The rank of
elliptic curves and infinite multiplicity of roots of unity on elliptic
curves
Abstract: Let be a number field, an algebraic closure of and an elliptic curve defined over . Let be the absolute Galois group of over . We prove that
there is a subset of Haar measure 1 such that for every , the spectrum
of in the natural representation of consists of all roots of unity, each of
infinite multiplicity. In particular, we discuss the infinite multiplicity of
the eigenvalue 1 case which shows the infinite rank of a given elliptic curve
and the openness condition of such a subset for each root of unity.
July 28 Yang, JaeHyun
(Inha University) Harmonic Analysis on Homogeneous Spaces I
Abstract: I give a
survey talk on recent progress in harmonic analysis on semisimple or
reductive symmetric spaces reviewing classical harmonic analysis.
July 28 Andreas Bender
(KIAS) The Schinzel hypothesis in the function field case
Abstract: The
Schinzel hypothesis is a farreaching generalization of the twin prime
conjecture, with applications in diophantine geometry. We shall look at a
very weak form of this conjecture which can be proved if the ground ring
is rather than
August 4 Yang,
JaeHyun (Inha University) Harmonic Analysis on Homogeneous Spaces II
Abstract: I give a
lecture of harmonic analysis on homogeneous spaces of nonreductive type that
are important arithmetically and geometrically. I present some new results
and compare these results with those obtained in the case of reductive
symmetric spaces.
August 4 Lee, KyuHwan
(University of Toronto)
IwahoriHecke algebras
of SL_{2} over 2dimensional local fields
August 11 Shin, Sug Woo
(Harvard University) Shimura varieties and Langlands Correspondence
Abstract: Shimura
varieties often arise as moduli spaces of abelian varieties with additional structures.
Some examples are elliptic modular curves and Hilbert modular varieties. On
the other hand, mathematicians believe that there is Global Langlands
Correspondence, namely a correspondence between Galois representations and
automorphic representations. Local Langlands Correspondence can also be
formulated and are believed to be compatible with global correspondence.
Shimura varieties are especially interesting because Langlands Correspondence
is presumably realized in their cohomology spaces. This idea is encoded in
the recent proof of Local Langlands Conjecture by Harris and Taylor.
With these generalities in mind, I will focus
on easy and primitive examples of Shimura varieties as cyclotomic fields,
imaginary quadratic fields and elliptic modular curves. We will see how
Langlands Correspondence can be seen in their cohomologies. Adelic point of
view of these objects will be emphasized on the way.
August 11 Lee, KyuHwan
(University of Toronto)
Spherical Hecke
algebras of GL_{n} over 2dimensional local fields
Abstract: After
considering construction of an invariant measure on GL_{n}
over 2dimensional local fields, we will define spherical Hecke algebras of
these groups. Then we will discuss Satake isomorphism.
August 17 Henry Kim
(University of Toronto) Langlands functoriality conjecture
August 18 Yang, JaeHyun
(Inha University) Siegel modular forms and Jacobi forms
Abstract: I give a
talk about the theory of Siegel modular forms and Jacobi forms in the aspects
of arithmetic, geometry and representation theory.
August 25 Kim, TaeKyun
(Kongju University) On Iwasawa padic Lfunctions
September 8 Kim,
Donggyun (Korea University)
Elementary
introduction to local Langlands correspondence for GL(2)
September 22 Park,
PooSung (Seoul National Univ.) 2Universal Hermitian Forms
Abstract: A positive
definite hermitian lattice is said to be 2universal if it represents all
positive definite binary hermitian lattices. We find all ternary and
quaternary 2universal hermitian lattices over imaginary quadratic fields and
provide the 15theorem type of criteria for 2universality of hermitian
lattices. We also investigate asymptotic behavior of minimal ranks of
2universal hermitian lattices over imaginary quadratic fields. As an
application we discuss the solvability of certain types of Diophantine
equations.
October 13 Kim, Hoil
(Kyungbook National University) Quantum Theta Functions
Abstract: We want to
describe the quantum abelian varieties by using two lattice structures
representing complex structures and noncommutative structures. Then we
analyze the symmetry on them, leading to the orbifolds of them.
October 27 Park, Jinsung
(KIAS) On the zeta regularized determinant and the Selberg zeta function
Abstract: In this
talk, I will explain the relation of the zeta regularized determinant of the
Laplacian and the Selberg zeta function for the congruence subgroup of .
November 10 Jeong, Kyeonghoon
(Seoul National Univ.)
Some Lie algebra
applications to number theory
November 17 Sohn, Jaebum
(Yonsei University)
Equivalent continued
fractions and acceleration of its convergence
Abstract: The
BauerMuir transformation is useful to prove equivalence of continued
fractions. With appropriate choice of modifying factors , it gives
equivalent continued fraction up to modify sense. In this talk, we first
examine equivalent continued fractions up to modified convergence of several
Ramanujan type continued fractions that include the RogersRamanujan
continued fraction and the Ramanujan's cubic continued fraction by using this
transformation. And then we comment about the acceleration of convergence of
these continued fractions.
November 17 Kim, SeonHong
(Chosun University)
Densely algebraic
bounds for the exponential function
Abstract: An upper
bound for that implies the inequality between the
arithmetic and geometric means is generalized with the introduction of a new
parameter . The new upper
bound is smoothly and densely algebraic in , and valid for for arbitrarily large positive provided that is sufficiently close to . The range of
its validity for negative is investigated through the study of
the zero distribution of a certain family of quadrinomials.
December 8 Kim, Daeyeoul (Sogang
University) Algebraic integrers for basic Appell series
Abstract: In this
talk, we state properties for basic Appell series. And then we prove
algebraic properties for them.
December 22 Lee, Yoonjin
(Simon Fraser University)
The structure of
the class groups of global function fields
Abstract: The problem
of determining the structure of the class group dates back to Gauss. In this
talk we discuss the structure of the class groups of global function fields.
Let be a finite field and a transcendental element over .
We show an explicit method of constructing,
for positive integers and with , infinitely
many global function fields K of degree m over such that has a given unit rank and the ideal class group of contains a subgroup isomorphic to .
