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 book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

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)".

A Halin graph is constructed from a plane embedding of a tree whose non-leaf vertices have degree at least 3 by adding a cycle through its leaves in the natural order determined by the embedding. In this talk, we prove that every 3-connected {K_{1,3},P_5} -free graph has a spanning Halin subgraph. This result is best possible in the sense that the statement fails if K_{1,3} is replaced by K_{1,4} or P_5 is replaced by P_6. This is a joint work with Guantao Chen, Jie Han, Songling Shan, and Shoichi Tsuchiya.

Assume K is a convex body in R^d and X is a (large) finite subset of K. How many convex polytopes are there whose vertices belong toX? Is there a typical shape of such polytopes? How well does the maximal such polytope (which is actually the convex hull of X) approximate K? In this lecture I will talk about these questions mainly in two cases. The first is when X is a random sample of n uniform, independent points from K. In this case motivation comes from Sylvester's famous four-point problem and from the theory of random polytopes. The second case is when X is the set of lattice points contained in K and the questions come from integer programming and geometry of numbers. Surprisingly (or not so surprisingly), the answers in the two cases are rather similar. The methods are, however, very different.

Assume K is a convex body in R^d and X is a (large) finite subset of K. How many convex polytopes are there whose vertices belong toX? Is there a typical shape of such polytopes? How well does the maximal such polytope (which is actually the convex hull of X) approximate K? In this lecture I will talk about these questions mainly in two cases. The first is when X is a random sample of n uniform, independent points from K. In this case motivation comes from Sylvester's famous four-point problem and from the theory of random polytopes. The second case is when X is the set of lattice points contained in K and the questions come from integer programming and geometry of numbers. Surprisingly (or not so surprisingly), the answers in the two cases are rather similar. The methods are, however, very different.

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 book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

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 introductary talk, we define fractional Laplace operator and investigate its basic properties mostly in "Drichlet to Neuman map" point of view. And then we study elliptic equations involving fractional Laplacians and compare them with local equations.

In this introductary talk, we define fractional Laplace operator and investigate its basic properties mostly in "Drichlet to Neuman map" point of view. And then we study elliptic equations involving fractional Laplacians and compare them with local equations.

The theory of complex multiplication allows one to construct explicit class fields and cryptographic curves of genus g=1 or g=2. Essential to this is are special values of the j-invariant (g=1) or absolute Igusa invariants (g=2). Class invariants are special values of arbitrary modular functions that lie in the same field as the aforementioned values. Such class invariants can replace the j-invariant or Igusa invariants in applications, which speeds op these applications when the class invariants have small height. Schertz gave a systematic way of creating class invariants using modular functions for the group Gamma_0(N) in the case g=1. I will show how to generalize Schertz's method to modular functions on higher-dimensional moduli spaces. This is joint work with Andreas Enge.

 The functions of living cells are regulated by the complex biochemical network, which consists of stochastic interactions among genes and proteins. However, due to the complexity of biochemical networks and the limit of experimental techniques, identifying entire biochemical interaction network is still far from complete. On the other hand, output of the networks, timecourses of genes and proteins can be easily acquired with advances in technology. I will describe how to use oscillating timecourse data to reveal biochemical network structure by using a fixed-point criteria. Moreover, I will describe how mathematical modeling can be used to understand the dynamics and functions of complex biochemical networks with an example of circadian clock. Finally, in biochemical networks, reactions occur on disparate timescale. This timescale separation has been used to project deterministic models of biochemical networks onto lower-dimensional slow manifolds with quasi-steady state approximation (QSSA). I will discuss whether this reduction technique for deterministic systems can be used for stochastic systems. Specifically, I will show when macroscopic rate functions derived with QSSA (e.g. Hill functions) can be used to derive the propensity functions for microscopic rates.

Abstract:

In this talk, we consider the Carlitz multiple polylogarithms (CMPLs) at algebraic  points. We show that they form a graded algebra defined over the base rational function field. We further show that any multiple zeta value (MZV) defined by Thakur can be expressed as a linear combination of CMPLs at algebraic points, which is a generalization of the work of Anderson-Thakur on the depth one case.
As a conseqence, we obtain a function field version of Goncharov's conjecture
for MZVs.

The Lojasiewicz exponent is an analytic invariant of hypersurfaces with isolated singularities. It is an open question whether it is a topological invariant as well: so far this has been proven to be the case for weighted homogeneous isolated singularities of curves and surfaces. In this talk I will focus on the Lojasiewicz exponent of normal rational singularities of complex surfaces in comparison to its counterpart associated to the ideals of the local ring. Via this comparison I will give a bound for the case of ADE singularities.  This is a joint work with Meral Tosun and Gülay Kaya.

Title: Numerical Computing with Chebfun

 Chebfun is a Matlab-based system for numerical computing with functions as opposed to just numbers. This talk will describe some of the algorithms behind Chebfun and demonstrate it in action, including the extension to two dimensions known as Chebfun2.

Wiring diagrams are widely used combinatorial objects that are mainly used to describe reduced words of a permutation. In this talk, I will mention a fun property I recently found about those diagrams, and then introduce other results and problems related to this property.

TBA

 We use the results of the previous talk. Developing a mass formula for the space of binary quartic forms, and using a squarefree sieve, we show that the average size of the 2-Selmer groups of elliptic curves is 3. This yields an upper bound of 1.5 on the average rank of elliptic curves. This is joint work with ManjulBhargava

Lecture 3

▶ Date: 11:00, July 29, 2014

▶ Place: E2, Room 3221

▶ Speaker: Inwon Kim(UCLA)

▶ Title: Quasi-static evolution and congested crowd motion

▶ Abstract: In this talk we investigate the relationship between Hele-Shaw evolution with a drift and a transport equation with a drift potential, where the density is transported with a constraint on its maximum. The latter model, in a simplified setting, describes the congested crowd motion with a density constraint. When the drift potential is convex, the crowd density is likely to aggregate, and thus if the initial density starts as a patch (i.e. if it is a characteristic function of some set) then it is expected that the density evolves as a patch. We show that the evolving patch satisfies a Hele-Shaw type equation. This is joint work with Damon Alexander and Yao Yao.

 ♦ Title:  Beyond Endoscopy and the Trace Formula

♦ Date : July 29, 2014 (Tuesday) / July 30, 2014 (Wednesday) / 

            August 1, 2014 (Friday) 15:00 ~ 16:00

♦ Room : 자연과학동(E6-1) Room 1409

♦ Speaker:  S. Ali Altug (Columbia University)

♦ Abstract:

In his 2004 paper, "Beyond Endoscopy", Langlands proposed an approach to (ultimately) attack the general functoriality conjectures by means of the trace formula. For a (reductive algebraic) group G over a global field F and a representation r : L G →GL(V ), the strategy, among other things, aims at detecting those automorphicrepresentations of G for which the L-function, L(s,   π, r), has a pole at s = 1. Langlands‘ suggestion is to use the the trace formula together with an averaging process to Capture these poles via “techniques", which may or may not be available, of analytic number theory.

In these lectures I will start by going over the aforementioned paper focusing on G =GL(2) over ℚ,and describe the limiting procedure (and some of the dicultiesthat come with it).

I will then move on to some results (which there are not many) related to the problem, and discuss the current state of matters.

Robertson and Seymour proved that graphs are well-quasi-ordered by the minor relation and the weak immersion relation. In other words, given infinitely many graphs, one graph contains another as a minor (or a weak immersion, respectively). Unlike the relation of minor and weak immersion, the topological minor relation does not well-quasi-order graphs in general. However, Robertson conjectured in the late 1980's that for every positive integer k, the topological minor relation well-quasi-orders graphs that do not contain a topological minor isomorphic to the path of length k with each edge duplicated. We will sketch the idea of our recent proof of this conjecture. In addition, we will give a structure theorem for excluding a fixed graph as a topological minor. Such structure theorem were previously obtained by Grohe and Marx and by Dvorak, but we push one of the bounds in their theorems to the optimal value. This improvement is needed for our proof of Robertson's conjecture. This work is joint with Robin Thomas.

In this talk, we adapt Bhargava's geometry-of-numbers-methods to determine the number of GL(2,Z)-orbits on integral binary quartic forms. We use this result, along with a parametrizationdue to Birch and Swinnerton-Dyer, to prove that the average rank of elliptic curves is finite.

This is joint work with Manjul Bhargava.

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 book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

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)".

 Let X be a complex projective variety of dimension n equipped with a very ample line bundle L and a choice of valuation ν on its homogeneous coordinate ring R = R(L). Given this data, we can associate to(X, R, ν) a convex body of (real) dimension n, called the Okounkov body ∆ = ∆(X, R, ν). In many cases ∆is in fact a rational polytope; indeed, in the case when X is a nonsingular projective toric variety, the ringR and valuation ν may be chosen so that ∆ is the Newton polytope of X. It has been proved (Anderson, Kaveh) that, in many cases of interest (such as those arising in representation theory and Schubert calculus), the Okounkov body gives rise to a toric degeneration of X; in particular, this construction simultaneously generalize many toric degenerations given in the literature (e.g. Alexeev-Brion, Caldero, Kogan-Miller).

Lecture 1

▶ Date: 15:00~15:50, July 24, 2014

▶ Place: E2, Room 3221

▶ Speaker: Soojung Kim(NIMS)

▶ Title:Harnack inequality for nondivergent parabolic operators on Riemannian manifolds

▶ Abstract: In this talk, I will discuss the Krylov-Safonov theory which is the analogue of the De Giorgi-Nash-Moser theory. In particular, I will explain the Krylov-Safonov Harnack inequality for parabolic operators on certain Riemannian manifolds. This result gives a new nondivergent proof for the Li-Yau Harnack inequality of the heat equation on manifolds with nonnegative Ricci curvature. This talk is based on a joint work with Seick Kim and Ki-Ahm Lee.

Lecture 2

▶ Date: 16:00~16:50, July 24, 2014

▶ Place: E2, Room 3221

▶ Speaker: Minha Yoo(NIMS)

▶ Title: A drift approximation for nonlinear parabolic PDEs with oblique boundary data

Lecture 1

▶ Date: 15:00~15:50, July 24, 2014

▶ Place: E2, Room 3221

▶ Speaker: Soojung Kim(NIMS)

▶ Title:Harnack inequality for nondivergent parabolic operators on Riemannian manifolds

▶ Abstract: In this talk, I will discuss the Krylov-Safonov theory which is the analogue of the De Giorgi-Nash-Moser theory. In particular, I will explain the Krylov-Safonov Harnack inequality for parabolic operators on certain Riemannian manifolds. This result gives a new nondivergent proof for the Li-Yau Harnack inequality of the heat equation on manifolds with nonnegative Ricci curvature. This talk is based on a joint work with Seick Kim and Ki-Ahm Lee.

Lecture 2

▶ Date: 16:00~16:50, July 24, 2014

▶ Place: E2, Room 3221

▶ Speaker: Minha Yoo(NIMS)

▶ Title: A drift approximation for nonlinear parabolic PDEs with oblique boundary data

This lecture is independent of the previous three lectures. Matching many mass distributions (measures) in an optimal way is an important mathematical problem and has natural applications, e.g. in economics and physics. Focusing on mathematical aspects, we will explain some of the key concepts and results. A key notion is the Monge-Kantorovich barycenter, which is itself a measure and a geometric barycenter with respect to the Monge-Kantorovich distance on the space of probability measures.

We consider the unconditional uniqueness (UU) of solutions to the Cauchy problem for certain nonlinear dispersive equations on the torus. Our proof of UU is based on successive time-averaging arguments (integration by parts with respect to time variable). This approach was taken by Babin, Ilyin, and Titi (2011) for the periodic KdV equation, and has been applied to other equations such as the modified KdV equation and higher-order KdV-type equations. Recently, Guo, Kwon, and Oh (2013) obtained the optimal UU result for one-dimensional cubic NLS equation. We note that they needed to apply integration by parts infinitely many times, while for the KdV and the modified KdV cases the optimal results were obtained by finitely many applications of integration by parts. In this talk we prove UU for general NLS equations in higher dimensions and of higher (odd) degree nonlinearities, one-dimensional cubic derivative NLS, and the modified Benjamin-Ono equations by this method.

Let X be a complex projective variety of dimension n equipped with a very ample line bundle L and a choice of valuation ν on its homogeneous coordinate ring R = R(L). Given this data, we can associate to(X, R, ν) a convex body of (real) dimension n, called the Okounkov body ∆ = ∆(X, R, ν). In many cases ∆is in fact a rational polytope; indeed, in the case when X is a nonsingular projective toric variety, the ringR and valuation ν may be chosen so that ∆ is the Newton polytope of X. It has been proved (Anderson, Kaveh) that, in many cases of interest (such as those arising in representation theory and Schubert calculus), the Okounkov body gives rise to a toric degeneration of X; in particular, this construction simultaneously generalize many toric degenerations given in the literature (e.g. Alexeev-Brion, Caldero, Kogan-Miller).

 Let X be a complex projective variety of dimension n equipped with a very ample line bundle L and a choice of valuation ν on its homogeneous coordinate ring R = R(L). Given this data, we can associate to(X, R, ν) a convex body of (real) dimension n, called the Okounkov body ∆ = ∆(X, R, ν). In many cases ∆is in fact a rational polytope; indeed, in the case when X is a nonsingular projective toric variety, the ringR and valuation ν may be chosen so that ∆ is the Newton polytope of X. It has been proved (Anderson, Kaveh) that, in many cases of interest (such as those arising in representation theory and Schubert calculus), the Okounkov body gives rise to a toric degeneration of X; in particular, this construction simultaneously generalize many toric degenerations given in the literature (e.g. Alexeev-Brion, Caldero, Kogan-Miller).

 Let X be a complex projective variety of dimension n equipped with a very ample line bundle L and a choice of valuation ν on its homogeneous coordinate ring R = R(L). Given this data, we can associate to(X, R, ν) a convex body of (real) dimension n, called the Okounkov body ∆ = ∆(X, R, ν). In many cases ∆is in fact a rational polytope; indeed, in the case when X is a nonsingular projective toric variety, the ringR and valuation ν may be chosen so that ∆ is the Newton polytope of X. It has been proved (Anderson, Kaveh) that, in many cases of interest (such as those arising in representation theory and Schubert calculus), the Okounkov body gives rise to a toric degeneration of X; in particular, this construction simultaneously generalize many toric degenerations given in the literature (e.g. Alexeev-Brion, Caldero, Kogan-Miller).

 Current techniques for proving cases of Langlands reciprocity rely (in part) on understanding the geometry of certain local deformation spaces of Galois representations. In this talk, we will discuss a way to construct (the irreducible components of) semi-stable deformation rings in small Hodge-Tate weights of irreducible mod p representations of the absolute Galois group of Q_p.

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 book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

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)".

The choosability $chi_ell(G)$ of a graph $G$ is the minimum $k$ such that having $k$ colors available at each vertex guarantees a proper coloring.
Given a toroidal graph $G$, it is known that $chi_ell(G)leq 7$, and $chi_ell(G)=7$ if and only if $G$ contains $K_7$.
Cai, Wang, and Zhu proved that a toroidal graph $G$ without $7$-cycles is $6$-choosable, and $chi_ell(G)=6$ if and only if $G$ contains $K_6$.
They also prove that a toroidal graph $G$ without $6$-cycles is $5$-choosable, and conjecture that $chi_ell(G)=5$ if and only if $G$ contains $K_5$.
We disprove this conjecture by constructing an infinite family of non-$4$-colorable toroidal graphs with neither $K_5$ nor cycles of length at least $6$; moreover, this family of graphs is embeddable on every surface except the plane and the projective plane.
Instead, we prove the following slightly weaker statement suggested by Zhu: toroidal graphs containing neither $K^-_5$ (a $K_5$ missing one edge) nor $6$-cycles are $4$-choosable.
This is sharp in the sense that forbidding only one of the two structures does not ensure that the graph is $4$-choosable.

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.

▶ Date: May 15 ~ July 3

▶ Time: Thur. & Fri., 10:00-12:00 (Exercise session: 15:00-17:00)

▶ Description:

Many models in the sciences and engineering can be described by non-linear polynomial equations. This course offers an introduction to both theoretical and computational methods for working with such models. It is aimed at graduate students from across the mathematical sciences (Mathematics, EECS, Statistics, Physics, etc).

▶ Syllabus:

Each week of the semester is about a different topic in non-linear algebra, according to the schedule below. Auditors interested in a particular topic are welcome to attend just that week. Enrolled students will attend all weeks.

- Gröbner Basics, Elimination, Decomposing Varieties, Sparse Polynomial Systems, Semidefinite Programming, Moments and Sums of Squares,Representations and Invariants, Tensors and their Rank, Orbitopes, Maximum Likelihood, Numerical

As an application of the methods explained in the previous two lectures, we explain a few recent results on optimal transport problems, regarding the nature of the singular set where the solution is not smooth.

We explain some of the key concepts and techniques for regularity theory of the Monge-Ampere equation and optimal transport maps. This is the second lecture.

Optimal transportation theory studies phenomena where mass distributions are matched in an efficient way, with respect to a given transportation cost. In the most standard case, optimal transport maps are given by the gradient of convex functions that solve the Monge-Ampere equation. We explain some of the most basic concepts and techniques for regularity theory of the Monge-Ampere equation.