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.

We introduce a discrete dynamical system on the set of partial orientations of a graph, which generalizes Gioan’s cycle-cocycle reversal system. We explain how this setup allows for a new interpretation of the linear equivalence of divisors on graphs (chip-firing), and a new proof of Baker and Norine’s combinatorial Riemann-Roch formula. Fundamental connections to the max-flow min-cut theorem will be highlighted.

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)".

▶ 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 Algebraic Geometry, Nash Equilibria, Chemical Reaction Networks, Tropical Algebra

 I will explain the basic concepts of moduli and how moduli spaces can be constructed in algebraic geometry. Exploring the moduli spaces and issues arising from their construction lead to interesting interplay of geometry, algebra and computation.

▶ 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 Algebraic Geometry, Nash Equilibria, Chemical Reaction Networks, Tropical Algebra

Date: 2014. 6. 25(wed)

Time: 16: 30~17: 30

Place: E6-1 Room 1409

Abstract: Hilbert's Tenth Problem asks whether there is a general algorithm to
determine, given any polynomial in several variables, whether there exists a zero
with all coordinates in Z. It was proved in the negative by Yu. Matiyasevich in
1970. In the 70's J. R. Buchi attempted to prove a similar statement for a system
of quadric equations, and he was able to relate it to the following Diophantine
problem~

The mixed method for elasticity with weakly symmetric stress is a successful application of the finite element exterior calculus. In this talk, we first exploit the elasticity complex approach for the problem by Arnold, Falk, Winther, and survey its follow-up research. Then we introduce an abstract framework for unified error analysis of the method. Through examples, we will show that the framework covers most previously known mixed methods and also provides new mixed methods for the problem.

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)".

 First, we briefly review the derivation of nonlinear Schrodinger equation (NLS) from N-body linear Schrodinger equation via the cubic Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear equations. Such a derivation was established by the seminal works of Erdos-Schlein-Yau. In the derivation, the most involved part is the proof of unconditional uniqueness of solutions to GP hierarchy. Recently, Chen-Hainzl-Pavlovic-Seiringer gave a simpler alternative proof of uniqueness via the quantum de Finetti theorem. Adapting this new approach, we established the unconditional uniqueness of solutions to the GP hierarchy in a low regularity Sobolev type space. Precisely, we reduce the regularity requirement down to the currently known regularity requirement for unconditional uniqueness of solutions to NLS. This is a joint work with Kenneth Taliaferro and Zhihui Xie at UT Austin.

▶ 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 Algebraic Geometry, Nash Equilibria, Chemical Reaction Networks, Tropical Algebra

 Amoebas and coamoebas are the images of varieties of the complex algebraic torus under coordinatewise logarithm and argument maps, respectively. As shadows of the original variety, they retain some of its structure. When the variety is a hypersurface, the connected components of the complements of both the amoeba and coamoeba are convex. Henriques introduced a homological generalization of convexity and proved that complements of amoebas satisfy a weak form of this higher convexity.
 In this talk, I will explain these notions and describe some of the structure of coamoebas, namely their phase limit sets and shells, and then sketch how to use this structure to show that complements of coamoebas have this higher convexity of Henriques. This is joint work with Mounir Nisse.

 A matroid is a combinatorial notion that is a generalization of a spanning set of a vector space. To any loopless matroid, there correspond at least 3 kinds of convex polytopes: independent set polytope, base polytope, and spanning set polytope. In algebraic geometry context, base polytopes are preferred to the other two since base polytopes are closed under involution operation, and recovering the other two is easier. Moreover, in my recent research work, it turned out that base polytopes have a very special gluing property: when they glue through their codimension 2 common face, there are only finitely many cases! In this talk, I will first explain basics of matroids and base polytopes. After stating the gluing property (with a sketch of the proof), we will see how this gluing property plays its role concerning the classification of generic tropical planes of mathbb{TP}^5.

After a brief introduction to  the Waring ranks and cactus ranks of polynomials, we verify additive property of ranks and cactus  ranks of polynomials which are sums of particaular types of polynomials. This work is a natural generalization of the result of Carlini, Catalisano and Geramitta concerning sum of coprime monomials.

We discuss the notion of point scatterers, which is a renormalization of formal delta potentials for the Schrödinger equation in low-dimensional spaces. In particular, we will discuss the decomposition of periodic point scatterers which corresponds to Bloch's theorem of solid state physics.

 In Abo and Wan's study of Waring's problem for systems of skew-symmetric forms several defective systems were identified.  The most interesting cases occur when a certain secant variety of a Segre-Grassmann variety does not fill its natural ambient space as expected, but is a hypersurface instead. 

▶ 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 Algebraic Geometry, Nash Equilibria, Chemical Reaction Networks, Tropical Algebra

An r-dynamic proper k-coloring of a graph G is a proper k-coloring of G such that every vertex in V(G) has neighbors in at least min{d(v),r} different color classes. The r-dynamic chromatic number of a graph G, written χr(G) , is the least k such that G has such a coloring. By a greedy coloring algorithm, χr(G)≤rΔ(G)+1 and the equality holds if and only if G is r-regular with diameter 2 and girth 5. We improve the bound to χr(G)≤Δ(G)+2r when δ(G)≥2rlnn . In terms of the chromatic number, we prove χr(G)≤rχ(G) when G is k-regular with k≥(3+o(1))rlnr and show that χr(G) may exceed r1.377χ(G) when k=r. We prove χ2(G)≤χ(G)+2 when G has diameter 2, with equality only for complete bipartite graphs and the 5-cycle. Also, χ2(G)≤3χ(G) when G has diameter 3, which is sharp. This is joint work with SogolJahanbekam, Suil O, and Douglas B. West.

▶ 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 Algebraic Geometry, Nash Equilibria, Chemical Reaction Networks, Tropical Algebra

We describe how to approximate, in quasi-polynomial time, the largest independent set of polygons, in a given set of polygons. Our algorithm works by extending the result of Adamaszek and Wiese [AW13, AW14] to polygons of arbitrary complexity. Surprisingly, the algorithm also works for computing the largest subset of the given set of polygons that has some sparsity condition. For example, we show that one can approximate the largest subset of polygons, such that the intersection graph of the subset does not contain a cycle of length 4 (i.e., K2,2). To appear in SoCG 2014.

In recent year methods based on nonparametric estimation detection is more popular in signal processing community for estimating detecting the signal function from noisy degraded measurement. This is due to localized estimation. Recent approaches to processing and restoration of images and video brought together several powerful data-adaptive methods from different field of work. Examples include Moving Least Square (from computer graphics), the Bilateral Filter and Anisotropic Diffusion (from computer vision), Functional Gradient Decent, Kernel Regression and Iterative scaling (from Statistics).

In this talk we discussed basic of nonparametric estimation of density and distribution function followed by the class of robust nonparametric methods which are ideally suited for the reconstruction of signals and images ( in general function) form noise –- corrupted and sparse or irregularly sampled data. As the framework of nonparametric the methods do not depend on strong assumption about noise; and it is applicable to a wide variety of problems. In this talk, we consider image denoising and deblurring in nonparametric framework.