Walk-regular graph were introduced by Godsil and McKay to understand when the characteristic polynomial of a graph in which a vertex is deleted does not depend on which vertex you delete. This notion was generalized to m-walk-regular graphs by Fiol and Garriga in order to understand how close you can come to a distance-regular graph. We observed that for many results on distance-regular graphs they also hold for 2-walk-regular. In this talk I will give an overview of which results can be generalized to 2-walk-regular graphs, and I also will give many examples of 2,3,4,5,-walk-regular graphs which are not distance-regular. At this moment all 6-walk-regular graphs known are distance-regular.

Amenability is one of those properties of group that has many different characterizations. I will discuss what it means in terms of invariant means, random walks and C* algebras. If time permits, I will also describe some related notions such as property rapid decay in the C* algebra setting.

 In this talk, I will review the recent progress on the flocking analysis of the Cucker-Smale flocking model introduced by Cucker and Smale in 2007, and will discuss seveal possible improvements to incorporate the collision avoidance and singular communication weights. 

In this talk, we will discuss uniqueness of positive solutions for the subelliptic heat equation on a manifold, which satisfies the generalized curvature dimension inequality (2009, F.Baudoin and N.Garofalo). This comes via another results; the global Poincaré inequalities and Sobolev inequalities on balls. Our results apply in particular to CR Sasakian manifolds with Tanaka-Webster-Ricci curvature bounded from below and Carnot groups of step two.

These lectures will use classi cation of surfaces in P4 of low
degree as a motivating storyline to discuss important
techniques in the study of projective surfaces. The main
topics will be: adjunction theory, liaison, multisecant lines,
special linear systems in the plane, vector bundle techniques
and Heisenberg-invariant varieties.

Further details
• Schedule: 2.00 pm - 3.30 pm
Monday (05/13) Room 2412
Wednesday (05/15) Room 2412
Monday (05/20) Room 2412
Wednesday (05/22) Room 3433 (n.b.)
Friday (05/24) Room 2412
• http://mathsci.kaist.ac.kr/andreash/spring2013/ranestad.html
For more information please contact Andreas Holmsen
(, Ext:7300).

A brief introduction of the most celebrated financial mathematical development with an emphasis on stock price option pricing will be presented.

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

These lectures will use classi cation of surfaces in P4 of low
degree as a motivating storyline to discuss important
techniques in the study of projective surfaces. The main
topics will be: adjunction theory, liaison, multisecant lines,
special linear systems in the plane, vector bundle techniques
and Heisenberg-invariant varieties.

Further details
• Schedule: 2.00 pm - 3.30 pm
Monday (05/13) Room 2412
Wednesday (05/15) Room 2412
Monday (05/20) Room 2412
Wednesday (05/22) Room 3433 (n.b.)
Friday (05/24) Room 2412
• http://mathsci.kaist.ac.kr/andreash/spring2013/ranestad.html
For more information please contact Andreas Holmsen
(, Ext:7300).

In the coming era of individualized custom medicine, all personal data including genetic background, diet habit, environmental exposure, and others will be used to make medical decisions such as which therapy should be used over an alternative. As all the strong genetic and nongenetic factors are being discovered for each common disease, all interactions between them will need to be explored as well. This is a doable but formidable task.

These lectures will use classi cation of surfaces in P4 of low
degree as a motivating storyline to discuss important
techniques in the study of projective surfaces. The main
topics will be: adjunction theory, liaison, multisecant lines,
special linear systems in the plane, vector bundle techniques
and Heisenberg-invariant varieties.

Further details
• Schedule: 2.00 pm - 3.30 pm
Monday (05/13) Room 2412
Wednesday (05/15) Room 2412
Monday (05/20) Room 2412
Wednesday (05/22) Room 3433 (n.b.)
Friday (05/24) Room 2412
• http://mathsci.kaist.ac.kr/andreash/spring2013/ranestad.html
For more information please contact Andreas Holmsen
(, Ext:7300).

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.

A symmetric matrix with complex entries may be diagonalized, so the corresponding quadratic form may be written as a sum of squares. There is a large variety of distinct sum of squares decompositions of the quadratic form. I shall present a compactification of this variety, and discuss and present old and new results on powersum decompositions for forms of higher degree.

KMRS Chair Professor Inaugural Lecture Series: Lecture 3

Algebraic geometry is the study of solutions sets to polynomial equations. Solutions that depend on an infinitesimal parameter are studied combinatorially by tropical geometry. Tropicalization works especially well for varieties that are parametrized by monomials in linear forms. Many classical moduli spaces (for curves of low genus and few points in the plane) admit such a representation, and we here explore their tropical geometry. Examples to be discussed include the Segre cubic, the Igusa quartic, the Burkhardt quartic, and moduli of marked del Pezzo surfaces. Matroids, hyperplane arrangements, and Weyl groups play a prominent role. Our favorites are E6, E7 and G32.

These lectures will use classi cation of surfaces in P4 of low
degree as a motivating storyline to discuss important
techniques in the study of projective surfaces. The main
topics will be: adjunction theory, liaison, multisecant lines,
special linear systems in the plane, vector bundle techniques
and Heisenberg-invariant varieties.

Further details
• Schedule: 2.00 pm - 3.30 pm
Monday (05/13) Room 2412
Wednesday (05/15) Room 2412
Monday (05/20) Room 2412
Wednesday (05/22) Room 3433 (n.b.)
Friday (05/24) Room 2412
• http://mathsci.kaist.ac.kr/andreash/spring2013/ranestad.html
For more information please contact Andreas Holmsen
(, Ext:7300).

KMRS Chair Professor Inaugural Lecture Series: Lecture 2

Maximum likelihood estimation is a fundamental computational task in statistics. We discuss this problem for manifolds of low rank matrices. These represent mixtures of independent distributions of two discrete random variables. This non-convex optimization problems leads to some beautiful geometry, topology, and combinatorics. We explain how numerical algebraic geometry is used to find the global maximum of the likelihood function, and we present a remarkable duality theorem due to Draisma and Rodriguez.

KMRS Chair Professor Inaugural Lecture Series: Lecture 1

 Interior point methods in linear programming travel along the central curve. We determine the degree, genus, and defining equations of this algebraic curve. These invariants, as well as the total curvature of the curve, are expressed in the combinatorial language of matroid theory. This is joint work with Jesus De Loera and Cynthia Vinzant.

Geometric Chevalley-Warning conjecture of Brown, Schnetz, and Esnault states that a projective hypersurface of degree d le n in Pn defines 1 modulo the class of A1 in the Grothendieck ring of varieties. I will construct virtually smooth quartic threefolds which are not stably rational over the field of complex numbers. This disproves the conjecture over any field of characteristic zero.

We will discuss connections between three notions in 3-dimensional topology that are, roughly speaking, algebraic, topological, and analytic. These are: the left-orderability of the fundamental group of a 3-manifold M, the existence of certain codimension 1 foliations on M, and the Heegaard Floer homology of M.

최근 반복되는 금융위기를 겪으면서 금융규제의 효과와 적절성에 대한 논의가 활발합니다. 특히 금융기관들의 건전성을 확보하기 위해 도입된 여러 규제들이 입안자들의 의도대로 작동하지 않는 경우도 많이 관측되고 있습니다. 본 발표에서는 대표적인 금융 규제들인 Stress Test와 위험자산 가중치의 설계에대해 고려해 봅니다. Stress Test에서 가장 중요한 요소인 Scenario선택을 손실의 Tail분포를 고려하여 관측자료로부터 수행하는 방법을 간략하게 살펴봅니다. 그리고, 은행에 대한 중요 규제인 자산들의 위험가중치를 어떻게 선택할 것인지에 대한 문제를 최적화모형을 통해 유도하고, 유도된 여러 성질들에 관해 발표합니다. 본 연구는 Paul Glasserman교수와 강철민씨와 함께 수행한 연구입니다.

The lectures will be an introduction to Dehn surgery. This is a construction, going back to Dehn in 1910, for producing closed 3-manifolds from knots. A natural generalization is Dehn filling, in which some torus boundary component $T$ of a 3-manifold $M$ is capped off with a solid torus $V$. If $alpha$ is the isotopy class of the loop on $T$ that bounds a disk in $V$, the resulting filled manifold is denoted by $M(alpha)$. Generically, the topological and geometric properties of $M$ persist in $M(alpha)$; in particular if $M$ is hyperbolic then $M(alpha)$ is usually also hyperbolic. If this fails then the filling is said to be {it exceptional}. We will outline a program to classify the triples $(M;alpha,beta)$ with $M(alpha)$ and $M(beta)$ exceptional, describing what is known in this direction and what remains to be done.

The lectures will be an introduction to Dehn surgery. This is a construction, going back to Dehn in 1910, for producing closed 3-manifolds from knots. A natural generalization is Dehn filling, in which some torus boundary component $T$ of a 3-manifold $M$ is capped off with a solid torus $V$. If $alpha$ is the isotopy class of the loop on $T$ that bounds a disk in $V$, the resulting filled manifold is denoted by $M(alpha)$. Generically, the topological and geometric properties of $M$ persist in $M(alpha)$; in particular if $M$ is hyperbolic then $M(alpha)$ is usually also hyperbolic. If this fails then the filling is said to be {it exceptional}. We will outline a program to classify the triples $(M;alpha,beta)$ with $M(alpha)$ and $M(beta)$ exceptional, describing what is known in this direction and what remains to be done.

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.

The lectures will be an introduction to Dehn surgery. This is a construction, going back to Dehn in 1910, for producing closed 3-manifolds from knots. A natural generalization is Dehn filling, in which some torus boundary component $T$ of a 3-manifold $M$ is capped off with a solid torus $V$. If $alpha$ is the isotopy class of the loop on $T$ that bounds a disk in $V$, the resulting filled manifold is denoted by $M(alpha)$. Generically, the topological and geometric properties of $M$ persist in $M(alpha)$; in particular if $M$ is hyperbolic then $M(alpha)$ is usually also hyperbolic. If this fails then the filling is said to be {it exceptional}. We will outline a program to classify the triples $(M;alpha,beta)$ with $M(alpha)$ and $M(beta)$ exceptional, describing what is known in this direction and what remains to be done.

 In this talk, I will describe construction and estimates for Green's function for elliptic and parabolic systems of second order in divergence form subject to various boundary conditions.
Here, we assume minimal regularity assumptions on the coefficients and domains.

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.