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.

L-functions are very interesting tools that number theorists have been using
since 18th century. Those also appear in the local Langlands conjecture. Brie
y,
the local Langlands conjecture asserts that there exists a `natural' bijection between
two di erent sets of objects: Arithmetic (Galois or Weil-Deligne) side and analytic
(representation theoretic) side. In each side, we can de ne the L-functions of those
objects. The L-functions from analytic side are de ned by Shahidi (Langlands-
Shahidi method) and the L-functions from arithmetic side are Artin L-functions.
The natural question is whether two L-functions are equal through the local Lang-
lands correspondence. If it is, we can use the properties of the L-functions from
arithmetic side to study L-packet, the object in the analytic side, which is the set of
irreducible admissible representations of quasi split group G over p-adic eld. The
equality of L-functions has an interesting application in proving the generic Arthur
L-packet conjecture. The generic Arthur L-packet conjecture states that if the L-
packet attached to Arthur parameter has a generic member, then it is tempered.
(Remark that this conjecture relates to the generalized Ramanujan conjecture). In
this talk, I will explain those in the case of split GSpin groups. Furthermore, I
will explain the classi cation of strongly positive discrete series representations of
GSpin groups over p-adic eld which is one of the main tools in the proof of the
equality of L-functions.

 In this talk, I will discuss the existence and uniqueness of singular solutions for some semilinear elliptic equations in radial settings. I am interested in the monotonicity of singular solutions.  For supercritical case, the asymptotic behavior of the singular solution turns out to be self-similar at infinity.
For the critical Sobolev exponent, we specify what is the unique positive solution with self-similarity at infinity.
 Then, all other solutions with slow decay are of Delaunay-Fowler type.

In this talk, we discuss lower semicontinuity and lower bounds for a Chen-Lubensky energy describing nematic/smectic liquid crystals with physically realistic boundary conditions. The Chen-Lubensky energy captures stable phases of the liquid crystal material, ranging from purely nematic or smectic states to coexisting nematic/smectic states. By including appropriate additional terms, the model
includes the effects of applied electric or magnetic fields, and/or electrical self-interactions in the case of polarized liquid crystals. As a consequence of our results, we establish existence of minimizers with weak or strong anchoring of the director field (describing molecular orientation) at the boundary, and Dirichlet or Neumann boundary conditions on the smectic order parameter for the liquid crystal material. This is a joint work with P.Bauman and D. Phillips at Purdue University, USA.

 In this talk, we will prove finite energy global well-posedness of the Chern-Simons-Higgs equations on the (2+1)-dimensional Minkowski space for general compact non-abelian gauge groups. The case of abelian gauge groups was recently established by [Selberg-Tesfahun, DCDS-A 2013] using the Lorenz gauge; in the non-abelian case, however, conventional gauges (such as Lorenz or Coulomb) become troublesome for large initial data. To address this difficulty, we will utilize the caloric-temporal gauge, introduced in [Oh, arXiv 2012] for the Yang-Mills equations, which is constructed using the Yang-Mills heat flow. This is despite the apparent lack of a naturally associated heat flow for the Chern-Simon-Higgs equations

 I will discuss an optimal transport problem arising when there are many mass distributions to match together in a cost efficient way, explaining a joint work with Brendan Pass. This talk will be introductory.

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 balancing domain decomposition by constraints (BDDC) preconditioner is defined by a coarse component, expressed in terms of primal constraints across the interface between the subdomains, and local components given in terms of Schur complements of local subdomain problems. A BDDC method for vector field problems discretized with Raviart-Thomas finite elements is introduced. Our method is based on a new type of weighted average developed to deal with more than one variable coefficient. A bound on the condition number of the preconditioned linear system is also provided which is independent of the values and jumps of the coefficients across the interface and has a polylogarithmic condition number bound in terms of the number of degrees of freedom of the individual subdomains. Numerical experiments are also presented, which support the theory and show the effectiveness of our algorithm. This is joint work with Olof Widlund and Clark Dohrmann.

As a systems scientist, biology looks all full of mysteries that are not understandable. A cell, the basic unit of life, consists of numerous molecules that highly interact with each other. Such interaction between molecules often results in paradoxical observations in many biological experiments. I was intrigued whether there exists any evolutionary design principle behind the puzzling dynamics of living systems. To unravel such a hidden design principle underlying complex phenomena, we need a systems biological approach by combining mathematical simulation and biochemical experimentation. In this talk, I will present the state space analysis of a molecular interaction network that is critical for cell fate determination and further discuss how to control such a network to change the cell fate as we want. The proposed state space analysis demonstrates that implementation of an attractor landscape to analyze a biological network is useful for gaining a better understanding of the complex network dynamics and the resulting cell fate determination.

Since the seminal work of Floer and Weinstein in 1986, the Lyapunov-Schmidt reduction method has been regarded as one of powerful methods in constructing solutions for a variety of elltipic equations such as the nonlinear Schrödinger equation, the Allen-Cahn equation, the prescribed scalar curvature problem and so on. In this talk, I will briefly review the reduction method and show how it can be applied to construct solutions for the Lane-Emden-Fowler equation on certain symmetric domains having thin toroidal holes.

Abstract:
We will review some old and new results and tools in anayltic number
theory. Especially, we will be interested in constructing their function
field analogs. Necessary tools from analytic number theory and
probabilistic language will be reviewed. Topics will include:

1. Elementary tools in anayltic number theory and probability theory
2. Prime number races and Chebyshev's bias
3. Distribution of the values of Mobius function
4. Prime number races in elliptic curves

강연일자 : 6/11, 12, 13, 18, 19, 21, 27, 28

시간: 16:00-17:30

장소: E6-1 #1409

We will discuss the connection between the circular-orderability of the fundamental group of a 3-manifold M and the existence of certain codimension-1 foliations on M via Thurston's universal circle theory. This theory provides a motivation to study group actions on the circle with dense invariant laminations. As an one lower dimensional example, we will give a complete characterization of Fuchsian groups in terms of its (topological) invariant laminations.

 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.