A del Pezzo cone is a generalized affine cone over a del Pezzo surface with respect to a pluri-anticanonical divisor.
We define an alpha function and compute all this functions on a smooth del Pezzo surfaces.
As an important application, we show that del Pezzo cones with lower degree do not admit non-trivial G_a-actions.

Several classical results in convexity, like the theorems of Caratheodory, Helly, and Tverberg, have colourful versions.

In this talk I plan to explain how two methods, the octahedral construction and Sarkaria’s tensor trick, can be used to prove further extensions and generalizations of such colourful theorems.

This is joint work with Suh Hyun Choi. Let p be a prime number. Suppose we have two modular forms whose weights are congruent modulo p^r(p-1), and q-expansions are congruent modulo p^r. (For example, consider modular forms given by topologically close points on an eigencurve.) People who do Iwasawa Theory believe that their p-adic L-functions are also congruent modulo p^r. In fact, if we push this idea further, we can also imagine there is a big p-adic L-function over an eigencurve which is integral and smooth. This is known in the ordinary prime case (i.e. the case where the slope of modular form is a p-adic unit), and in this case, the big p-adic L-function over the eigencurve is called the Kitagawa-Mazur p-adic L-function. In the non-ordinary case, so far we know relatively little. In this presentation, we will prove that the (non-integral) p-adic L-functions that I constructed are congruent for the above-said congruent modular forms assuming that Hecke algebras are Gorenstein. (The same technique can be applied to different p-adic L-functions.) We believe that this is one step towards a big integral smooth p-adic L-function over an eigencurve for a non-ordinary prime.

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 article "Class fields over real quadratic fields and Hecke operators - G.Shimura(1972)".

In this talk, we will investigate the algebraic construction of the Jacobian of a hyperelliptic curves.

We prove that a combinatorial triangulation L of a sphere admits an acute geodesic triangulation if and only if L does not have a separating three- or four-cycle. The backward direction is an easy consequence of the Andreev–Thurston theorem on orthogonal circle packings. For the forward direction, we consider the Davis manifold M from L. The acuteness of L will provide M with a CAT(-1) (hence, hyperbolic) metric. As a non-trivial example, we show the non-existence of an acute realization for an abstract triangulation suggested by Oum; the degrees of the vertices in that triangulation are all larger than four. This approach generalizes to triangulations coming from more general Coxeter groups, and also to planar triangulations. (Joint work with Genevieve Walsh)

A network represents a way of interconnecting any pair of users or nodes by means of some meaningful links. Thus, it is quite natural that its structure can be represented, at least in a simplified form, by a connected graph whose vertices represent nodes and whose edges represent their links.

 As an efficient method to investigate dynamical phenomena on networks such as electrical flow on a circuits, chemical reaction between molecules, behavior of biological individuals in their societies and so on, in a systematic way, we introduce the theory of discrete partial differential equations on networks. In order to do this, the calculus on networks is introduced, at first, after defining the partial derivatives at each nodes. Being based on this calculus, we discuss the various types of partial differential equations on networks. In particular, the solvabilities of (nonlinear) elliptic PDE and parabolic PDE on networks will be discussed.

Let $f in S=C[x_0,...,x_n]$ be a homogeneous polynomial with complex coefficients and denote by $f_0,...,f_n$ the partial derivatives of $f$.
Let $V(f)$ be the projective hypersurface defined by $f=0$. Then it is known that $V(f)$ is smooth if and only if $f_0,...,f_n$ is a regular sequence in $S$, i.e. there are no nontrivial syzygies involving $f_0,...,f_n$. We will discuss the case when the hypersurface $V(f)$ is nodal and show that there are no low degree nontrivial syzygies involving $f_0,...,f_n$. We'll explain the relations of this algebraic question to the topology and the Hodge theory of the hypersurface $V(f)$.

In this talk, we consider the invisibility cloaking. The aim of the  invisibility cloaking is to hide an object from observation, and it has been actively studied since last decade. I will introduce a cloaking method based on the transformation optics and related research.

Let $f in S=C[x_0,...,x_n]$ be a homogeneous polynomial with complex coefficients and denote by $f_0,...,f_n$ the partial derivatives of $f$.
Let $V(f)$ be the projective hypersurface defined by $f=0$. Then it is known that $V(f)$ is smooth if and only if $f_0,...,f_n$ is a regular sequence in $S$, i.e. there are no nontrivial syzygies involving $f_0,...,f_n$. We will discuss the case when the hypersurface $V(f)$ is nodal and show that there are no low degree nontrivial syzygies involving $f_0,...,f_n$. We'll explain the relations of this algebraic question to the topology and the Hodge theory of the hypersurface $V(f)$.

 I'll define relatively quasiconvex subgroups, and talk about how to do Dehn filling while preserving quasiconvexity

Let $f in S=C[x_0,...,x_n]$ be a homogeneous polynomial with complex coefficients and denote by $f_0,...,f_n$ the partial derivatives of $f$.
Let $V(f)$ be the projective hypersurface defined by $f=0$. Then it is known that $V(f)$ is smooth if and only if $f_0,...,f_n$ is a regular sequence in $S$, i.e. there are no nontrivial syzygies involving $f_0,...,f_n$. We will discuss the case when the hypersurface $V(f)$ is nodal and show that there are no low degree nontrivial syzygies involving $f_0,...,f_n$. We'll explain the relations of this algebraic question to the topology and the Hodge theory of the hypersurface $V(f)$.

I'll state the main theorem and sketch a proof.

I'll define and give examples of relatively hyperbolic groups, and talk about what it means to do Dehn filling on a group pair.

In the first lecture, I'll describe an explicit construction of negatively curved metrics on closed 3-manifolds obtainod by Dehn filling of cusped hyperbolic manifolds.  I also plan to sketch an application by Cooper and Long to finding surface subgroups of 3-manifolds. I'll talk about how to extend the 2＼pi Theorem to cusped hyperbolic manifolds of dimension larger than 3.

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 "Modular forms and projective invariants - J.Igusa(1967)".

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 investigate the algebraic construction of the Jacobian of a hyperelliptic curves.

In this talk, I give an introduction to Nonparametric (NP) Bayesian statistical modeling with some applications. First, I describe some key components of Bayesian statistical inference. Then, I begin with some motivating examples for which parametric modeling may have limitations and introduce a NP Bayes methodology for more flexible modeling. Focuses are on NP Bayes approaches involving Dirichlet process (DP) and some DP-extended processes. Finally, I discuss computation-based inference procedure focusing on Markov Chain Monte Carlo (MCMC) and conclude with some remarks of future research directions.

 Understanding the parameter spaces of dynamical systems has long been the dream of the greatest mathematicians. Even Newton asked: what initial conditions(positions, velocities masses) lead to a stable solar system?
 There are exceedingly few cases where we can answer such questions: no one knows anything about the parameter space for the 3-body problem. But for the simplest nonlinear dynamical system, z 7! z2 + c with parameter c, we do understand the parameter space.
 The crucial object in parameter space is the Mandelbrot set: it features some very delicate combinatorics, which can be written exactly.
 In my lecture I will attempt to describe these combinatorial laws, and sketch where they come from.

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 "Modular forms and projective invariants - J.Igusa(1967)".

In this talk, we will investigate the algebraic construction of the Jacobian of a hyperelliptic curves.

The main theme of the three lectures is to explain how syzygystratifications can be used to describe the birational geometry of curves with or without a level structure. The first lecture will deal with the basics of Koszul cohomology and the statements of the Green respectively Green-Lazarsfeld secant conjectures. Voisin's solution to the generic Conjecture will be sketched. In the second lecture, I will describe the implications of Green's Conjecture to the moduli space of curves, and how using the moduli space, one can prove Green's Conjecture for curves on arbitary K3 surfaces. Finally in the third lecture I will present three wide-ranging generalizations to Green's Conjecture which have been
recently used to compute the Kodaira dimension of the moduli space of curves with a torsion point of order p in its Jacobian variety.

 Averaging methods for ordinary differential equations is an
old idea going back to Laplace.
Recently along with my collaborators a new approach to averaging based
on Young measures has been introduced which both simplifies some of
the older and tedious approaches and allows for new applications to problems
arising is the KdV equations, flocking models due to Cucker and Smale,
and the Kuramoto system of oscillators.

The main theme of the three lectures is to explain how syzygystratifications can be used to describe the birational geometry of curves with or without a level structure. The first lecture will deal with the basics of Koszul cohomology and the statements of the Green respectively Green-Lazarsfeld secant conjectures. Voisin's solution to the generic Conjecture will be sketched. In the second lecture, I will describe the implications of Green's Conjecture to the moduli space of curves, and how using the moduli space, one can prove Green's Conjecture for curves on arbitary K3 surfaces. Finally in the third lecture I will present three wide-ranging generalizations to Green's Conjecture which have been
recently used to compute the Kodaira dimension of the moduli space of curves with a torsion point of order p in its Jacobian variety.

KMRS 집중 강연

일시: 2013년 3월 5일, 7일, 12일, 14일, 19일(화,목) 14시30분~15시 45분

 In 1956 John Nash established the existence of global smooth embedding of an n-dimensional Riemannian manifold M^n into m-dimensional Euclidean space for m <= n(3n + 11)/2 if M^n is compact, m <= n(n + 1)(3n + 11)/2 if M^n is non-compact.(See John Nash, Annals of Math. 65, 1956) In Nash's case the system of PDE's is under determined with many more unknowns than equations. In the determined case when m = n(n+1)=2 Nash's (1956) theorem does not apply and we can search for smooth local embeddings. These lectures will discuss this problem from the view of an applied mathematician, i.e.,

The main theme of the three lectures is to explain how syzygystratifications can be used to describe the birational geometry of curves with or without a level structure. The first lecture will deal with the basics of Koszul cohomology and the statements of the Green respectively Green-Lazarsfeld secant conjectures. Voisin's solution to the generic Conjecture will be sketched. In the second lecture, I will describe the implications of Green's Conjecture to the moduli space of curves, and how using the moduli space, one can prove Green's Conjecture for curves on arbitary K3 surfaces. Finally in the third lecture I will present three wide-ranging generalizations to Green's Conjecture which have been
recently used to compute the Kodaira dimension of the moduli space of curves with a torsion point of order p in its Jacobian variety.

KMRS 집중 강연

일시: 2013년 3월 5일, 7일, 12일, 14일, 19일(화,목) 14시30분~15시 45분

 In 1956 John Nash established the existence of global smooth embedding of an n-dimensional Riemannian manifold M^n into m-dimensional Euclidean space for m <= n(3n + 11)/2 if M^n is compact, m <= n(n + 1)(3n + 11)/2 if M^n is non-compact.(See John Nash, Annals of Math. 65, 1956) In Nash's case the system of PDE's is under determined with many more unknowns than equations. In the determined case when m = n(n+1)=2 Nash's (1956) theorem does not apply and we can search for smooth local embeddings. These lectures will discuss this problem from the view of an applied mathematician, i.e.,

The derived category of bounded complexes of coherent sheaves on an algebraic variety is an interesting invariant of the algebraic variety. There is more symmetry than the varieties themselves in the sense that there are different varieties with equivalent derived categories. There is a surprising parallelism between the minimal model program and the semi-orthogonal decompositions of derived categories. I will review some old and new results in this direction.

 KMRS 집중 강연

일시: 2013년 3월 5일, 7일, 12일, 14일, 19일(화,목) 14시30분~15시 45분

 In 1956 John Nash established the existence of global smooth embedding of an n-dimensional Riemannian manifold M^n into m-dimensional Euclidean space for m <= n(3n + 11)/2 if M^n is compact, m <= n(n + 1)(3n + 11)/2 if M^n is non-compact.(See John Nash, Annals of Math. 65, 1956) In Nash's case the system of PDE's is under determined with many more unknowns than equations. In the determined case when m = n(n+1)=2 Nash's (1956) theorem does not apply and we can search for smooth local embeddings. These lectures will discuss this problem from the view of an applied mathematician, i.e.,

 I will show some easy calculations concerning examples on the topics of the colloquium talk, e.g., Fourier-Mukai transform, grade restriction windows for GIT quotients, etc.

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.

 KMRS 집중 강연

일시: 2013년 3월 5일, 7일, 12일, 14일, 19일(화,목) 14시30분~15시 45분

 In 1956 John Nash established the existence of global smooth embedding of an n-dimensional Riemannian manifold M^n into m-dimensional Euclidean space for m <= n(3n + 11)/2 if M^n is compact, m <= n(n + 1)(3n + 11)/2 if M^n is non-compact.(See John Nash, Annals of Math. 65, 1956) In Nash's case the system of PDE's is under determined with many more unknowns than equations. In the determined case when m = n(n+1)=2 Nash's (1956) theorem does not apply and we can search for smooth local embeddings. These lectures will discuss this problem from the view of an applied mathematician, i.e.,

We discuss about a result of Littlewood  on the horizontal distribution of the zeros of the Riemann zeta-function ζ(s) in the critical strip and further we discuss about the progress made on the zeros of  ζ(s) locally in the neighbourhood of the critical line. (An old work of mine jointly done with Professor K. Ramachandra).

Let V be a projective hypersurface of fixed degree and dimension which has only isolated singular points. We show that, if the sum of the Milnor numbers at the singular points of V is large, then V cannot have a point of large multiplicity, unless V is a cone. As an application, we give an affirmative answer to a conjecture of Dimca and Papadima.

Let V be a projective hypersurface of fixed degree and dimension which has only isolated singular points. We show that, if the sum of the Milnor numbers at the singular points of V is large, then V cannot have a point of large multiplicity, unless V is a cone. As an application, we give an affirmative answer to a conjecture of Dimca and Papadima.

 KMRS 집중 강연

일시: 2013년 3월 5일, 7일, 12일, 14일, 19일(화,목) 14시30분~15시 45분

 In 1956 John Nash established the existence of global smooth embedding of an n-dimensional Riemannian manifold M^n into m-dimensional Euclidean space for m <= n(3n + 11)/2 if M^n is compact, m <= n(n + 1)(3n + 11)/2 if M^n is non-compact.(See John Nash, Annals of Math. 65, 1956) In Nash's case the system of PDE's is under determined with many more unknowns than equations. In the determined case when m = n(n+1)=2 Nash's (1956) theorem does not apply and we can search for smooth local embeddings. These lectures will discuss this problem from the view of an applied mathematician, i.e.,

Given a degeneracy locus arising from a map of vector bundles, the Thom-Porteous formula allows to express its Chow ring fundamental class as a polynomial in the Chern classes of the two bundles. In this talk I will present the geometry involved in the universal case which represents the core of one of the proofs of the formula. Moreover, I will explain how it is possible to generalize this result to other oriented cohomology theories as, for instance, the graded Grothendieck ring of vector bundles, connective K-theory and algebraic cobordism.