Number Theory Seminars 2007

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January 11 Song, Joung Min  (DePauw University)

Title: Smooth numbers and the Riemann Hypothesis

Abstract: Questions about properties of positive integers whose prime factors are relatively small arise naturally in research of various problems in additive number theory. 'Smooth numbers,' or sometimes termed as 'crumbly numbers,' may be viewed as the opposites of prime numbers in the sense that prime numbers are naturally defined as 'unbreakables'. Density of smooth numbers are closely related to some of the most celebrated problems in number theory, such as the Prime Number Theorem, and the Riemann Hypothesis.

January 18 Myung, Sung (Inha University)

Title: Algebraic K-theory and Number fields

Abstract: An introduction to lower algebraic K-theory of algebraic number fields and their integer rings. Mostly classical definitions and computations will be shown, but some of speaker's results may be mentioned in connection with cyclotomic units.

January 25 Yang, Jae-Hyun (Inha University)

Title: Invariant differential operators on the Minkowski-Euclid Space

Abstract: In this lecture, I will talk about invariant differential operators on the Minkowski-Euclid space and present some related important problems. Using these invariant differential operators, we discuss automorphic forms on the Minkowski-Euclid space generalizing automorphic forms on the general linear group GL(n,R).

February 15 Moon, Hyunsuk (Kyungpook National University)

Title: On the rationality of zeta functions of modular representations of a discrete group

Abstract: (joint work with Shinya Harada) We show that the generating function defined by the numbers of isomorphism classes of absolutely irreducible representations of a finitely generated group into the general linear group of degree d over various finite fields is a rational function.

February 22 Kim, Seon-Hong (Sokmyung Women's University)

Title: On a generalization of an upper bound for exponential function

Abstract: With the introduction of a new parameter n, Kim recently generalized an upper bound for exponential function that implies the inequality between the arithmetic and geometric means. In this talk, we answer some of Kim's conjectures about the inequalities between Kim's generalized upper bound and the original one. We also see the validity of Kim's generalization for some further negative values of x for certain cases. The range of its validity for negative x is investigated through the study of the zero distribution of a certain family of quadrinomials. This is partially joint work with Jaegug Bae.

March 15 Kim, Ho-il (Kyungpook National University)

Title: Arithmetic Noncommutative Geometry

Abstract: First we introduce nonncomutative geometry briefly and then we explain the influence of this on the diverse subjects in mathematics and physics.

March 22 Kang, Soon-Yi (KIAS)

Title: Partitions Weighted by the Parity of the Crank

Abstract: Ranks and cranks are partition statistics which give combinatorial interpretations of Ramanujan's famous partition congruences. Motivated by intriguing results on the number of partitions with even rank minus the number with odd rank, we study the number of partitions with even crank minus the number with odd crank. In this talk, we present an asymptotic formula, Ramanujan type congruences, q-series identities that this crank difference satisfies. We also determine the exact values of the crank difference in the case of partitions into distinct parts, which is an analogue of pentagonal number theorem. 'This is a joint work with D. Choi and J. Lovejoy.'

April 12 Yang, Jae-Hyun (Inha University)

Title: Remark on Harmonic Analysis on Siegel-Jacobi Space

Abstract: In this talk, we briefly discuss the spectral theory of the Laplacian on Siegel-Jacobi space and also the decomposition of the regular representation of the Jacobi group on the associated Hilbert -space. I introduce the notion of Maass-Jacobi forms that play an important role in the spectral theory. I also plan to present some new results related to this talk.

April 26 Jeremy Lovejoy (LIAFA, CNRS and Universit'e Denis Diderot)

Title: Rank and Congruences for Overpartition Pairs

Abstract: Overpartition pairs have recently played a key role in combinatorial studies of q-series identities, including families of identities of the Rogers-Ramanujan type. An important statistic for overpartition pairs is the rank, which is generalization of Dyson's classical rank of a partition. Here we investigate the role that this generalized rank plays in congruence properties for the number of overpartition pairs of n

May 10 Kim, Ji-Young (National Institute for Mathematical Sciences)

Title: Recent developments of universal and regular Hermitian lattices

Abstract: In this talk, we introduce an overview of recent development of Hermitian lattices over imaginary quadratic fields. First part of this talk, we will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields for all positive square-free integer . For each imaginary quadratic field , we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13, 14 and 15, then L is universal. We call this the fifteen theorem or universal Hermitian lattices}. Note that the difference between Conway-Schneeberger's fifteen theorem and ours is the number 13. Second part of this talk, we will introduce a method to get all regular binary Hermitian lattices over imaginary quadratic fields for all positive square-free integers , and provide a complete list of (normal) binary regular Hermitian lattices

June 14 Jun, Byungheup (Yonsei University)

Title: Moduli stack of polarized 1-motives and the Schottky problem around the infinity

Abstract: We introduce the moduli problem for 1 motives with polarization. Polarized 1 motives arise from the mixed Hodge structure with monodromy weight filtration of degenerating family of curves. We ask the Schottky problem comparing the moduli of polarized 1 motives and the moduli of polarized MHS of the same type. We present a case where the VPMHS is not coming from any family of polarized 1-motives, though the coarse moduli are the same around the infinity.

July 5 Shin, Sug Woo (Harvard University)

Title: Various aspects of Hecke

Abstract: In this expository talk, we will give several interpretations of the Hecke operators on the space of classical modular forms. For instance, we will seek for connection with Hecke algebras in representation theory and Hecke correspondences in algebraic geometry. Our aim is no more ambitious than to show the relationship among different viewpoints.

July 12 Sun, Hae-Sang (UCLA)

Title: Non-vanishing mod p of Dirichlet L-values

Abstract: In this talk, we will explain two different approaches to prove Washington's theorem which is a non-vanishing result about the Dirichlet L-values twisted by characters with prime power conductors. The geometric and homological variants of the Washington's and Sinnott's proofs will be discussed. We will also discuss their generalization to various special L-values.

July 12 Han, Lin (Harvard University)

Title: l-adic monodromy theorem and its application

Abstract: We have a lot of l-adic Galois representations coming from various sources in algebraic geometry so analyzing it should be of importance. In this talk, I would introduce how Deligne and Grothendieck have developed the tool for doing it called l-adic monodromy theorem and then try to present its detailed proof. The main point is lying on seperating the image of the inertia group from the image of whole local Galois group and then realizing that the remaining image of given l-adic Galois representation is finite.

July 26 Choi, So Young (KIAS)

Title: Recursion formulas for coefficients of the Drinfeld discriminant function

Abstract: We find recursion formulas for coefficients of the Drinfeld Discriminant Function .

August 14 Anna Cadoret (Universite Bordeaux 1)

Title: On uniform boundedness for p-torsion on abelian varieties

August 28 Lee, Dong Uk (KIAS)

Title: Shimura varieties and Hodge theory I, II

Abstract: This will be a survey of some aspects of Shimura varieties and consists of two lectures. In the first lecture, we explain how Hermitian symmetric domains can be view as moduli spaces of polarizable Hodge structures. In the second lecture, we explain the notion of Shimura varieties of Hodge type and the Andre-Oort conjecture.

October 11 Andreas Bender (KIAS)

Title: Some applications of mathematical logic to number theory

October 18 Chang, Seunghwan (KAIST)

Title: Introduction to (\Phi,\Gamma)-modules I, II

Abstract: Every l-adic local Galois representation is quasi-unipotent in the sense that passing to finite extension of the base field it becomes tame and the action of the inertia becomes unipotent. However, p-adic ones don't have such a simple description and have much more complicated structure in general. In this talk, I will talk about a tool introduced by Fontaine called (,)-modules, which classify all p-adic representations by linearizing the Galois action, and about the aspects related to crystalline representations.

November 29 Seo, Soogil (Yonsei University)

Title: Etale cohomology of an algebraic group

Abstract: We use etale cohomology to prove some explicit results on the Galois cohomology of ideal class groups.