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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)
2006
January
12 Oh, ByeongKweon (Sejong University) On quadratic regular forms
Abstract:
In this talk, we present an overview of
recent results on positive definite integral quadratic forms which satisfy various
regularity properties. In particular, we show some finiteness results on
regular forms and almost regular forms, and we also show that the minimal
rank of nregular forms has an exponential lower bound.
January 26 Kim, Chang Heon
(Seoul Women's University) Twisted traces of
singular values
Abstract:
Let be a prime for which the congruence
group ( is of genus zero, and be the corresponding Hauptmodul.
We investigate the twisted traces of singular values of and construct infinite products
related to them.
February 9 Cho, Soojin (Ajou
University) Combinatorics of
LittlewoodRichardson numbers
Abstract:
There are well known reduction formulas
for the universal Schubert coefficients defined on Grassmannians. These coefficients
are also known as the LittlewoodRichardson coefficients in the theory of
symmetric functions. We restate the reduction formulas combinatorially and
provide a combinatorial proof for them.
February
16 Lee, Joongul (Hongik University) Refinement of a conjecture of Gross
February
16 Chae, HiJoon (Hongik University) Lfunctions and epsilon factors
March
9 Choi, Dohoon (POSTECH) The Space of Modular Forms Generated by Etaquotients
Abstract:
In this talk, we study the spaces of
modular forms on generated by etaquotients, where the
genus of is zero or is a prime. These partly give an answer to the following problem
suggested by Ono: Classify the spaces of modular forms which are generated by
etaquotients.
March
23 Kim, Ji Young (SNU) On almost
universal forms
Abstract: We present an overview of
results on positive definite integral quadratic forms which satisfy the universal property or the almost
nuniversal property. Furthermore, we classify all almost 2universal
positive definite integral diagonal forms of rank 6 up to isometry. These
results imply that there exist infinitely many such forms, up to isometry.
April
13 Choi, YounSeo (KIAS)
Is it possible to interpret the mock theta
function with the theory of partitions?
Abstract: In this talk, a few elementary
theories of partitions are introduced and also the definition of mock theta
functions is provided. Then we will discuss that the mock theta functions can
be understood by the theory of partitions.
April
27 Choi, So Young (KAIST) The genus
of arithmetic curves
Abstract:
For any element with a positive integer, we find the genus
of the arithmetic curve which are independent of , where is the Fricke involution.
We obtain that the genus of is zero if and only if or . As an application, we determine a
generator of the function field related to the groups of genus zero which will be used to
generate appropriate ray class fields over imaginary quadratic fields, and
show that the fixed point of in H is a
Weierstrass point of for all but finitely many .
May
11 Kang, SoonYi (KIAS)
Generalizations of two fundamental product
identities and their applications
Abstract:
Using some standard transformations in
the theory of basic hypergeometric series, we first derive a generalization
of a reciprocity theorem for a certain qseries found in Ramanujan's
lost notebook and utilize it to derive four variable generalizations of
Jacobi's triple and quintuple product identities. Then we present some
applications of the generalized product identities, including new
representations for generating functions for sums of six squares and those
for overpartitions.
May
25 Kang, SoonYi (KIAS) The
theory of elliptic functions in the signature 3
Abstract: We first briefly survey the
history of elliptic function theory which originated several branches of
mathematics such as complex function theory and Galois theory. Then we
discuss the main theme of Ramanujan's theory of elliptic functions, including
the classical inversion formula which shows the intertwined relations among
elliptic functions, theta functions, and hypergeometric functions. Nex, we
introduce his discovery of the theory of elliptic functions in alternative
bases. In this talk, we focus on the cubic analogue of the theory of elliptic
functions and present the recent progress on this topic.
June
8 Takashi Ono (Johns Hopkins University) Arithmetic of Hopf Maps I
June 9 Takashi Ono (Johns
Hopkins University) Arithmetic of Hopf Maps II
June 30 Oh, Hee (California Institute
of Technology) Counting rational points of
bounded height
Abstract:
A fundamental problem in modern
arithmetic geometry is to describe the set of rational points of a projective
variety in terms of geometric invariants. One of the main conjectures in this
area formulated by Manin in late eighties predicts the asymptotic of the
number of rational points of bounded height for Fano varieties. I will
discuss a recent proof of Manin's conjecture for the case of the wonderful
compactication of semisimple algebraic groups, based on the mixing property
of adelic groups (joint work with Gorodnik and Maucourant).
June 30 Andreas Schweizer (University
of Exeter)
On the mrank of the class groups of quadratic
function fields
Abstract: Fix a finite field of odd characteristic and an odd integer that is not divisible by . We are interested in the rank of the divisor class groups of
quadratic extensions of the rational function field . Although an analogue of the
CohenLenstra heuristics (due to Friedman and Washington) suggests that for
each positive integer a positive proportion of all should have rank at least , there are only few results for.
First we show: If there is one with rank at least , then there are infinitely many with rank at least . Then for each that is congruent to 1 modulo 4 we
explicitly construct an with rank at least 4. Similar
constructions give slightly weaker results if mod 4 and also for the ideal class group of the integral closure
of in .
July 11 Michael Schlosser
(University of Vienna) Elliptic enumeration
of nonintersecting lattice paths
Abstract:
We enumerate lattice paths in the planar integer lattice consisting of
positively directed unit vertical and horizontal steps with respect to a
specific elliptic weight function. The elliptic generating function of paths
from a given starting point to a given end point evaluates to an elliptic
generalization of the binomial coefficient. Convolution gives an identity
equivalent to Frenkel and Turaev's summation. This appears to be the first
combinatorial proof of the latter, and at the same time of some important
degenerate cases including Jackson's and Dougall's _{ }summation. By considering nonintersecting lattice paths we are
led to a multivariate extension of the summation which turns out to be a
special case of an identity originally conjectured by Warnaar, later proved
by Rosengren. We conclude with discussing some future perspectives. (A
preprint corresponding to this talk is available at http://www.arxiv.org/abs/math.CO/0602260.)'
July 27 Choi Dohoon (KIAS)
Congruence Properties for Fourier Coefficients of
Modular Forms of Half Integral Weight
Abstract: In this talk, we discuss congruence properties related to
Fourier coefficients of modular forms of half integral weight and their applications.
August 10 Andreas Bender (KIAS)
Developments on the Schinzel Hypothesis for
Function Fields
Abstract:
This will be a talk about work in
progress: I shall review the basic issues, explain what the main theorem says
and discuss the picture as it is emerging.
September 14 Lee, GeonNo (Yonsei
University)
Upper bound of degrees of CMfields with relative
class number one
Abstract:
In 1974, Stark showed that for any fixed integers m and h
with m > 2, there exist only finitely many CMfields of degree
2m with class number h. In 1975, Odlyzko proved that
there are only finitely many normal CMfields of a given class number. In
1979, Hoffstein showed that every normal CMfield of degree greater than 434
has relative class number greater than one. In 2003, Bessassi improved upon
the Hoffstein's bound: he proved that every normal CMfield of degree greater
than 266 has relative class number greater than one and that if the
Generalized Riemann Hypothesis is true then the normal CMfields with
relative class number one are of degrees less than or equal to 164. Now we
give the sharper upper bounds for the degrees of CMfields of relative class
number one. We show that the normal CMfields with relative class number one
are of degrees less than or equal to 96 assuming the Generalized Riemann
Hypothesis, the CMfields (not necessarily normal) with relative class number
one are of degrees less than or equal to 104 assuming the Generalized Riemann
Hypothesis, and the normal CMfields with relative class number one are of
degrees less than or equal to 216 without assuming the Generalized Riemann
Hypothesis. By many authors all normal CMfields of degrees less than or
equal to 96 with class number one are known except for the possible fields of
degree 64 or 96. Consequently class number one problem for normal CMfields
is solved under the Generalized Riemann Hypothesis except for these two
cases.
September 27 Yuichiro Taguchi (Kyushu
University) On extensions of truncated
discrete valuation rings
Abstract:
Let be a complete discrete valuation field whose residue field may
be imperfect, and let be an integer . We show that the category of finite
separable extensions of with "ramification bounded by " in the sense of
AbbesSaito depends only on the "length truncation" of the integer ring of .
October 12 Kim, Wook (Yonsei
University) On the standard module
conjecture
Abstract:Let be a quasisplit algebraic group over a adic field, and let be a parabolic subgroup of . For a tempered generic
representation of and a parameter , an induced module is called standard if is in the positive Weyl chamber. A
standard module has a unique irreducible quotient , the Langlands quotient. The standard
module conjecture states that is irreducible if and only if the
Langlands quotient is generic. The standard module conjecture together with
Shahidi's conjecture on the holomorphy on Lfunctions implies that
normalized intertwining operators associated to a globally generic cuspidal
representation are holomorphic. It is called "Assumption A" which
has been used to establish functoriality for many cases.
In this talk we prove the conjecture for general spin groups via
the LanglandsShahidi method and Muic's idea, and put some comments on the
general case based on the recent result by Heiermann and Muic.
October 26 Lee, Dong Uk (KIAS) padic monodromy of Picard moduli scheme
Abstract:
Let be an imaginary quadratic number field
and let be distinct natural numbers. Suppose a
rational prime splits in .
We show that the naive adic monodromy of the ordinary locus
of the good reduction over of Picard moduli scheme of signature is "as large as
possible", i.e. equal to We discuss an application of this
result to the bad reduction of Picard moduli scheme with parahoric level
structure.
October 26 Joseph Hundley
(POSTECH) An Introductory Overview of
the RankinSelberg Method
Abstract:
Without going into all of the details, we describe the idea of a Langlands function and of a RankinSelberg
integral for one.
We then discuss why relationships are to be expected between
* poles of Langlands unctions
* the conjectural "functorial liftings"
* nonvanishing of certain "period integrals,"
and how RankinSelberg integrals can be used to prove these relationships in
some cases.
November 10 Im, BoHae (ChungAng
University) The rank of abelian varieties
over cyclic fields
Abstract:
Let be a field of characteristic such that every finite separable
extension of is cyclic.
Let be an abelian variety over . We show that if is infinite but not locally finite, the rank of over is infinite.
November 23 Seo, Seunghyun
(Seoul National University)
A generalization of Cayley's formula for trees
and parking functions
Abstract: A leader of a tree on is a vertex which has no smaller
descendants in . In 2005, Gessel and Seo showed
which is a generalization of Cayley formula, where is given by .
In this talk, using a variation of Prufer code which is called a
"reverse Prufer code", we give a simple bijective proof of Gessel
and Seo's formula. We also discuss some related properties of parking
functions and other combinatorial objects.
December
14 Kim, Sangjib (National University of Singapore)
LittlewoodRichardson
numbers and related combinatorics
Abstract: Given
finite dimensional irreducible representations of labeled by their highest weights , , and , LittlewoodRichardson number is the multiplicity of in the tensor product of and .
Many combinatorial objects have been introduced to encode these numbers.
After reviewing some of them, we will investigate their properties and
relations among them. If time permits, I will also present a new
proof for LittlewoodRichardson rules.
December 28 Yee, Ae Ja (The
Pennsylvania State University)
Lecture Hall Theorem and a generalization of Euler's
theorem
Abstract: Lecture hall partitions are partitions whose parts
satisfy a certain ratio condition. Their enumeration by BousquetMelow and
Eriksson gives a finite version of a theorem of Euler on strict partitions.
In this talk, we will discuss lecture hall theorem and a generalization of
Euler's theorem. This is join work with Carla Savage.
