The Siegel-Ramachandra invariants, as special values of Siegel functions of one variable, generate ray class fields over imaginary quadratic fields. Generalizing these invariants we shall introduce ray class invariants of certain CM-fields obtained from classical theta constants of multi-variables. And we will determine the action of the Galois group on these invariants in a concrete way by making use of Shimura's reciprocity law.

This is a joint work with Koo and Shin.

In this talk, we develop an equivalent condition for a primitive Fricke family of level $N$ to be totally primitive when $N$ is different from $4$. Furthermore, we present generators of the function field of the modular curve of level $N$ in terms of Fricke and Siegel functions. By using the functions belonging to Fricke families, we shall construct generators of the ray class fields over imaginary quadratic fields as an application of class field theory.

This is a joint work with Koo and Shin.

The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function ω(q) (resp. ν(-q)). Similar results for partitions with the corresponding restriction on each even part are also obtained, one of which involves the third order mock theta function φ(q). Congruences for the smallest parts functions associated to such partitions are obtained. Two analogues of the partition-theoretic interpretation of Euler’s pentagonal theorem are also obtained. This is joint work with George Andrews and Atul Dixit.

We will highlight two examples of the interplay of combinatorics and orthogonal polynomials by considering two recent one-parameter extensions of Hermite polynomials: a curious q-analog in connection with q-Weyl algebra and the 2D-Hermite polynomials. As application we derive a generalization of Touchard-Riordan formula for crossings of chords joining pairs of 2n points on a circle and a new Kibble-Slepian type formula for the 2D-Hermite polynomials, which extends the Poisson kernel for these polynomials.

The Riemann zeta-function, which encodes information about the integers and the prime numbers, has been studied extensively. Its values at 2,4,... are well-known, but much less is known about its values at 3,5,... . This difference can be explained to an extent by the different behaviour of certain groups (algebraic K-groups) of the rationals.
In this talk, we discuss some basic examples of such K-groups, and some links between them and arithmetic.

Approximate random k-colouring of a graph G=(V,E) is a very well

 Combining information from different source is an important practical problem. Using hierarchical area level models, we establish a frequentist framework for combining information from different source to get improved prediction for small or large area estimation. The best prediction is obtained by the conditional expectation of the observable latent variable given all available observation. The model parameters are estimated by two-level EM algorithm. Estimation of the mean squared prediction error is discussed.

  Sponsored by National Agricultural Statistical Agency (NASS) of US department of Agriculture, the proposed method was applied to the crop acreage prediction problem combining information from three sources: The first source is the June Area Survey (JAS), which is obtained by the probability sampling. The second source is from the Farm Service Agency (FSA) data, which is obtained from a voluntary participation of certain programs. The third source is from the classification of the satellite image data, called Cropland Data Layer (CDL).

 We deal with special Hurwitz' schemes of curves, that is the coverings of the Riemann sphere having only odd ramification points. We will discuss the relation, considered firstly by Serre and Fried, with theta characteristic. In the line of a joint research project with G. Farkas and J. Naranjo we present some recent existence results in the particular case of elliptic and hyperelliptic curves.

 The Neumann-Poincare (NP) operator is a boundary integral operator which arises naturally when solving boundary value problems using layer potentials. It is not self-adjoint with the usual inner product. But it can symmetrized by introducing a new inner product on H^{-1/2} spaces using Plemelj's symmetrization principle. Recently many interesting spectral properties of the NP operator have been discovered. I will discuss about this development and various applications including solvability of PDEs with complex coefficients and plasmonic resonance.

Chemotaxis models are based on spatial or temporal gradient measurements
by individual organisms. The key contribution of Keller and Segel (J Theor
Biol 30:225–234, 1971a; J Theor Biol 30:235–248, 1971b) is showing that erratic
measurements of individuals may result in an accurate chemotaxis phenomenon as a
group. In this paper we provide another option to understand chemotactic behavior
when individuals do not sense the gradient of chemical concentration by any means.
We show that, if individuals increase their dispersal rate to find food when there is
not enough food, an accurate chemotactic behavior may be obtained without sensing
the gradient. Such a dispersal has been suggested by Cho and Kim (Bull Math Biol
75:845–870, 2013) and was called starvation driven diffusion. This model is surprisingly
similar to the original Keller–Segel model. A comprehensive picture of traveling
bands and fronts is provided.

 Let C be a smooth curve which is complete intersection of a quadric and a degree k>2 surface in the 3 dimensional projective space. Let C(2) be its second symmetric power. of C. We study the finite generation of the extended canonical ring R(Δ,K):=⨁(a,b)H^0(C(2),aΔ+bK), where Δ is the image of the diagonal and K is the canonical divisor. We show that R(Δ,K) is finitely generated if and only if the difference of the two linear series defined on C by the rulings of the quadric is a torsion non-trivial line bundle. Then we show that this holds on an analytically dense locus of the moduli space of such curves. The results have been obtained in a joint work with Antonio La Face and Michela Artebani.

The subject of this talk is wave equations that arise from geometric considerations. Prime examples include the wave map equation and the Yang-Mills equation on the Minkowski space. On the one hand, these are fundamental field theories arising in physics; on the other hand, they may be thought of as the hyperbolic analogues of the harmonic map and the elliptic Yang-Mills equations, which are interesting geometric PDEs on their own.

Our main concern will be global well-posedness for large data of these PDEs in dimensions where the conserved energy is critical with respect to the scaling symmetry of the equations. I will first explain the ‘threshold conjecture’ for wave maps and its resolution by Sterbenz-Tataru (cf. related work by Krieger-Schlag and Tao), as well as its latest refinement in my work with A. Lawrie. I will also describe my recent work with D. Tataru on the global well-posedness of the energy critical Maxwell-Klein-Gordon system, which shares many similarities with the Yang-Mills equation.

The subject of this talk is wave equations that arise from geometric considerations. Prime examples include the wave map equation and the Yang-Mills equation on the Minkowski space. On the one hand, these are fundamental field theories arising in physics; on the other hand, they may be thought of as the hyperbolic analogues of the harmonic map and the elliptic Yang-Mills equations, which are interesting geometric PDEs on their own.

Our main concern will be global well-posedness for large data of these PDEs in dimensions where the conserved energy is critical with respect to the scaling symmetry of the equations. I will first explain the ‘threshold conjecture’ for wave maps and its resolution by Sterbenz-Tataru (cf. related work by Krieger-Schlag and Tao), as well as its latest refinement in my work with A. Lawrie. I will also describe my recent work with D. Tataru on the global well-posedness of the energy critical Maxwell-Klein-Gordon system, which shares many similarities with the Yang-Mills equation.

We first investigate the Kronecker Ugendtraum (= Hilberet 12th Problem) which initiates the study of algebraic number theory. We also briefly review the class number one problem in terms of modular functions without using L-function arguments. And, over cyclotomic fields and imaginary biquadratic fields we show how to construct class fields(= abelian extensions) by making use of high dimensional modular functions.

We consider the conductivity problem in the presence of adjacent circular inclusions with constant conductivities. When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution can be arbitrary large. In this paper we derive an asymptotic formula of the solution, which characterizes the gradient blow-up of the solution in terms of conductivities of inclusions as well as the distance between inclusions. The asymptotic formula is expressed in bipolar coordinates in terms of the Lerch transcendent function, and it is valid for inclusions with arbitrary constant conductivities. We illustrate our results with numerical calculations.

If one were to write up a list of keywords that describe recent development in algebraic geometry, it would be hard to miss the words like "derived category" or "categorification" on the top part. One basic problem in algebraic geometry is to study how a variety can be embedded in other varieties. In 2011, Bondal categorified the embedding problem and raised the following question.

Question. (Fano visitor problem) Let Y be a smooth projective variety. Is there a Fano variety X equipped with a fully faithful embedding of the derived category of Y into that of X?
If there is such an X, then Y is called a Fano visitor and X a Fano host of Y. In this talk, I will talk about a joint work with In-Kyun Kim, Hwayoung Lee and Kyoung-Seog Lee in which we proved that every complete intersection is a Fano visitor. I will also discuss related questions and problems.

I will explain the basic structure theory and representation theory of reductive and semisimple algebraic groups, and illustrate an application to invariant theory. A connected reductive group is naturally a central extension of a connected semisimple group by a torus, which enables one to reduce problems about reductive groups to problems about semisimple groups and tori. I will apply this principle to the study of weight decompositions of representations and obtain a precise formula relating the states of reductive group actions and the states of their derived group actions.

To be rational, or not to be, that is the internal conflict a cubic faces; its linear destiny provides a way to rationalize its life in a rich world and its whimsical action shows a way for a cubic to refuse its rational life in an impoverished field.

Given a vector bundle E over a smooth scheme X, a classical result of Kempf-Laksov describes the Schubert classes of grassmann bundles Gr(d,E) by means of a Jacobi-Trudi determinant whose entries are polynomials in the Chern classes of E and the universal bundle U_d. More recently, using a similar geometric framework, Kazarian was able to obtain a Pfaffian formula describing the Schubert classes of the Lagrangian grassmann bundle. In this talk I will present how these determinantal and Pfaffian formulas can be generalized to connective K-theory, an oriented cohomology theory which can be specialized to both the Chow ring and the Grothendieck ring of vector bundles. This is a joint work with T. Ikeda, T. Matsumura, H. Naruse.

In his death bed letter to Hardy, Ramanujan introduced mock theta functions, which are now prototypes of mock modular forms. The coefficients of mock modular forms encode the number of certain combinatorial objects and we will discuss how mock modularity works to investigate arithmetic properties for these counting functions. On the other hand, generating functions for certain unimodal sequences are now becoming prototypes of quantum modular forms. We will discuss how they are related and what we expect from quantum modularity.

수학에서 형식적인 정의를 보면 전혀 관계가 없어 보이는 것이 어떤 경우 서로 밀접히 연관된 경우가 많다. 이런 관계성을 정확히 파악하고 증명하는 것이 수학의 중요한 일면일 것이다. PDE를 공부하다 보면 계산이 너무 많아 수학에서 추구하는 미적인 성향이 약하다고 생각하는 경향이 많다. 본 강연에서는 PDE분야에서 아름다운 결과 중 하나인 Symmetry에 관하여 별로 관련성이 없어 보이는 Maximum Principle을 통하여 보이는 방법에 대하여 소개하고자 한다.