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.

This is a working seminar, so that no details will be provided.

In this talk, we revisit an auxiliary space preconditioning method proposed by Xu [Computing 56, 1996], in which low-order nite element spaces are employed as auxiliary spaces for solving linear algebraic systems arising from high-order nite element discretizations. We provide a new convergence rate estimate and parallel implementation of the proposed algorithm. We show that this method is user-friendly and can play an important role in a variety of Poisson-based solvers for more challenging problems such as the Navier-Stokes equation. We investigate the performance of the proposed algorithm using the Poisson equation and the Stokes equation on 3D unstructured grids. Numerical results demonstrate the advantages of the proposed algorithm in terms of efficiency, robustness, and parallel scalability.

 In this talk, we present how to obtain Log-Lipschitz regularity of solutions to the 3D Navier-Stokes equations with initial data in $L^{r}cap L^{3}$, $1<r<frac{3}{2}$. As a corollary, we also obtain H"{o}lder regularity of the flow map.

 I present results obtained in collaboration with J. Froehlich and W. de Roeck on quantum Brownian motion. We consider a quantum particle on the lattice weakly coupled to a spatial array of independent non-interacting reservoirs in thermal states (heat baths). We prove that the motion of the particle is diffusive at large times. If in addition the particle is driven by a weak external force field, we show that the motion of the particle is diffusive around a mean ballistic motion with constant velocity proportional to the external force field. Moreover, we prove that the Einstein relation (or Green-Kubo formula) holds, linking the mobility of the particle with the diffusion constant at vanishing external force.

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)".

This is a working seminar. No details are provided.

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

I will discuss various aspects of the geometry of the extension graph.  I will discuss vertex link projections, the bounded geodesic image theorem, and a distance formula for right-angled Artin groups.  Joint with S. Kim.

I will discuss basics of extension graphs for right-angled Artin groups and the actions of right-angled Artin groups on their extension graphs.  The main result in this lecture will be a version of the Nielsen--Thurston classification for right-angled Artin groups.  Joint with S. Kim.

In the last several decades, conformal blocks have been studied by many algebraic geometers who are interested in the geometry of moduli spaces of vector bundles. Recently, people have begun to study the relation between conformal blocks and the birational geometry of the moduli space $bar{M}_{0,n}$ of stable pointed rational curves, because they give a huge family of base point free divisors on $bar{M}_{0,n}$. In this talk, I will discuss an example of interaction between three approaches to the birational geometry of $bar{M}_{0,n}$: stack theoretic viewpoint, GIT, and conformal blocks. This is based on a joint work with A. Gibney, D. Jensen and D. Swinarski.

Since Hassett and Hyeon initiated the study of birational geometry of moduli spaces of curves in a viewpoint of minimal model program, there have been a tremendous amount of results. In this talk, I will explain three different approaches toward this direction: stack theory, GIT and the minimal model program. After that, I am going to describe Mori's program for the moduli space of Deligne-Mumford pointed stable rational curves $bar{M}_{0,n}$. If the time is permitted, I will discuss some generalizations toward higher genera and relative cases.

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.