정수론 mini-workshop

일시: 2015. 2. 27일(금)

           14:00-14:50 최소영(동국대)

           15:00-15:50 전병흡 (연세대)     

         16:00-16:50 이정연 (이화여대)

Title and Abstract

Rational period funcions and cycle integrals in hinger level cases (최 소영 교수, 동국대)

abstract : Generalizing the results of Duke, Imamoglu and Toth we give an effective basis for the space of period polynomials in higer level case.

From Euler-Maclaurin formula to the rationality and integrality of zeta values (전 병흠 박사, 연세대)

abstract : By using the Euler-Maclaurin summation formula and asymptotic expansion of Shintani generating function, we express the zeta values. From this expression, we derive the result of Klingen-Siegel concerning the zeta values of totally real number fields. We also discuss the method which can derive the integrality by using the related homological properties.

Indivisibility of class numbers of real quadratic function fields (이 정연 박사, 이화여대)

abstract : In this paper we work on indivisibility of the class numbers of real quadratic

function fields. We find an explicit expression for a lower bound of the density of real quadratic function fields (with constant field whose class numbers are not divisible by a given prime . We point out that the explicit lower bound of such a density we found only depends on the prime , the degrees of the discriminants of real quadratic function fields, and the condition: either or not.

 In this talk we discuss characterizations of Burniat surfaces constructed by bidouble covers. Mendes Lopes and Pardini dealt with a characterization of a Burniat surface with K^2=6. They showed that a minimal surface S of general type with p_g=0, K^2=6 and the degree 4 of the bicanonical map of S is a Burniat surface with K^2=6. Zhang considered the surface S with K^2=5. He proved that the surface S with K^2=5 is a Burniat surface with K^2=5 when the image of the bicanonical map of S is smooth. We consider that a minimal surface S of general type with p_g=0, K^2=4 and the degree 4 of the bicanonical morphism of S is a Burniat surface with K^2=4 and of non nodal type when the image of the bicanonical morphism of S is smooth.

Let S be a complete intersection surface defined by a net N of quadrics in P^5. In this talk we analyze GIT stability of nets of quadrics in P^5 up to projective equivalence, and discuss some connections between a net of quadrics and the associated discriminant sextic curve. In particular, we prove that if S is normal and the discriminant of S is stable then N is stable. And we prove that if S has the reduced discriminant and the discriminant is stable then the N is stable. Moreover, we prove that if S has simple singularities then the associated discriminant has simple singularities.

 In 2007, Y. Lee and J. Park provided a new method to construct surfaces of general type via Q-Gorenstein smoothing. Using the same technique, we were able to attain an algebraic construction of some Dolgachev's surfaces, for which there was an analytic construction (using logarithmic transform), but nothing have been known on its algebraic construction. In this talk, we shortly introduce the technique of Y. Lee and J. Park, and discuss how we construct Dolgachev's surfaces using this technique. On the other hand, P. Hacking provided a way to construct an exceptional vector bundle associated to a degeneration of surfaces with p_g = q = 0. We explicitly provides how to yield such vector bundles on Dolgachev's surfaces, and discuss what can be studied with these bundles. 

 

Finding a criterion of when a q-hypergeometric series can have modularity is an interesting open problem in number theory. Nahm's conjecture relates this question to the Bloch group in algebraic K-theory. I will give an introduction to the conjecture and explain its close relationship with various objects such as the dilogarithm function, Y-systems and Q-systems.

In this lecture series I will explore several problems of analytic number theory in the context of function fields over a finite field. Some of the problems can be approached by methods different that those of traditional analytic number theory and the resulting theorems can be used to check existing conjectures over the integers, and to generate new ones. Among the problems discussed are: counting primes in short intervals and in arithmetic progressions; Chowla's conjecture on the autocorrelation of the Möbius function, the additive divisor problem, moments of L-functions, and statistics of zeros of L-functions and connections with random matrix theory.

In automorphic representation theory, the Gross-Prasad(GP) conjecture has generated much attention in recent years. In this talk, we will explain the conjecture and introduce my relevant work

In this talk, we will survey the book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

In this talk, we will survey the article "Modular forms and projective invariants - J.Igusa(1967)".

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

In this talk, we will survey the book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

In this talk, we will survey the book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

In this talk, we will survey the article "Modular forms and projective invariants - J.Igusa(1967)".

We consider the one-dimensional cubic fractional nonlinear Schrödinger equation. Due to non-locality of the fractional Laplacian, the equation does not have any Galilean-type invariance. Despite of lack of this symmetry, we can still construct a new class of traveling soliton solutions by a rather involved variational argument. 

By coarse classification theorem of tight contact structures, it is known that every closed, atoroidal 3-manifolds attains at most finite tight contact structures up to contact isotopy. However, the explicit number, even the existence, of the tight contact structures remains a mystery. In this talk we introduce basic theories for classification problem and investigate the number of tight contact structures of certain hyperbolic 3-manifolds up to contact isotopy. 

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

In this talk, we will survey the article "Class fields over real quadratic fields and Hecke operators - G.Shimura(1972)".

In this talk, we will survey the book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

In this talk, we will survey the article "Modular forms and projective invariants - J.Igusa(1967)".