歷史演講 PAST TALKS

摘要： In this talk, we introduce some recent results on the transience, recurrence and strong ergodicity of regime-switching diffusion processes. Both the regime-switching diffusion processes with switching in a finite state space and that in an infinite countable state space are considered. Some on-off type criteria on recurrence can be provided based on the M-matrix theory and the Perron-Frobenius theorem. We apply these criteria to some concrete examples such as Ornstein-Uhlenbeck process with switching.

摘要： In this talk, we will present some of our recent results on the study stochastic ecology and population models. Treating predator-prey systems, our effort focuses on asymptotic properties. We obtain sufficient and almost necessary conditions for permanence and ergodicity of the stochastic predator-prey models with Beddington-DeAngelis functional response. Both non-degenerate and degenerate diffusions are considered. We will also consider ecology models of two species described by Kolmogorov systems of competitive type with a randomly switching component. Here we aim to establish the existence of an invariant probability measure.

摘要： In this talk we present an extension of the classical price impact model of Almgren and Chriss to incorporate the uncertainty of order fills. The extended model can be recast as alternatives to uncertain impact models, stochastic liquidity models, and an approximation of models for liquidation with limit orders. Optimal strategies are determined by maximizing the expected final P&L and various P&L-risk tradeoffs including utility maximization. Closed form expressions for optimal strategies are obtained in linear cases. The results suggest a type of adaptive volume weighted average price (VWAP) and adaptive Almgren-Chriss strategies. Possible generalizations to transient impact models are also discussed.

摘要： The master equation is one of the popular tools to explore the relative arbitrage opportunities in the literature of the stochastic portfolio theory. However, the traditional master equation only depends on the state of the market portfolio. E.R. Fernholz(2002) extended the master equation in two aspects. One is also related to the time and the other needs the ranks of the market portfolio. This talk will introduce the two extended master equation and the related works.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： We consider two applications using Mean Field Games (MFG) technique. Specifically, in order to solve the optimal strategy with the common noise, we need to introduce the stochastic partial differential equation (SPDE). The explicit solutions are obtained in these two examples.

摘要： The stochastic portfolio theory, established by Fernholz (2002) and Fernholz (2008), develops many researches in the multi-dimensional equity markets. This talk gives an overview of the stochastic portfolio theory. Except for the fundamental setup, we also introduce the useful tool, master equation and propose the relative open problems.

Abstract: Dynamic programming is one of useful methods in control theory and it is used in optimization problems under time-evolving states not only in engineering but also in other wide areas. A conventional dynamic programming treats costs determined by current state but it is recently considered important to work with costs depending on past history of states. In this talk, we overview dynamic programming methods for path- dependent costs in discrete- and continuous-time control problems recently studied in control and stochastic communities. We will see that value functions with path- dependent costs satisfy dynamic programming principles. Under dynamic programming principle, we can expect value functions should satisfy dynamic programming partial differential equations at least formal levels. However we need to specify a notion of a derivative on the state variable because the state space is infinite-dimensional. We introduce and explain several notions on infinite-dimensional calculus which fit to dynamic programming for path-dependent costs: one notion is co-invariant derivative studied in deterministic control community and the other is horizontal and vertical derivatives studied in stochastic community. We will also mention an idempotent expansion method for a particular problem.

Abstract: Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by $\lambda_k(M,g)$ the $k$-th eigenvalue of the Laplace-Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M, g)\mapsto \lambda_k(M,g)$. We propose a computational method for finding the conformal spectrum $\Lambda^c_k(M,[g_0])$, which is defined by the eigenvalue optimization problem of maximizing $\lambda_k(M,g)$ for $k$ fixed as $g$ varies within a conformal class $[g_0]$ of fixed volume $\text{vol}(M,g) = 1$. We also propose a method for the problem where $M$ is additionally allowed to vary over surfaces with fixed genus, $\gamma$. This is known as the topological spectrum for genus $\gamma$ and denoted $\Lambda^t_k(\gamma)$. Our computations support a conjecture of N. Nadirashvili (2002) that $\Lambda^t_k(0) = 8 \pi k$, attained by a sequence of surfaces degenerating to a union of $k$ identical round spheres. Furthermore, based on our computations, we conjecture that $\Lambda^t_k(1) = \frac{8\pi^2}{\sqrt{3}} + 8\pi (k-1)$, attained by a sequence of surfaces degenerating into a union of an equilateral flat torus and $k-1$ identical round spheres. The values are compared to several surfaces where the Laplace-Beltrami eigenvalues are well-known, including spheres, flat tori, and embedded tori. In particular, we show that the flat torus of volume one with the largest $k$-th Laplace-Beltrami eigenvalue attains $\lambda_k = 4\pi^2 \left\lceil \frac{k}{2} \right\rceil^2 \left( \left\lceil \frac{k}{2} \right\rceil^2 - \frac{1}{4}\right)^{-\frac{1}{2}}$. Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.

Abstract: We shall prove that of the Hausdorff dimension of a planar set is $>\frac{8}{5}$, then the three dimensional Lebesgue measure of the set of non-congruent triangles determined by $E$ is positive. Applications to geometric incidence theorems will be given.

Abstract: We study the three dimensional active scalar equation called the magneto-geostropic equation which was proposed by Moffatt and Loper as a model for the geodynamo processes in the Earth's fluid core. When the viscosity of the fluid is not taken to be zero, the constitutive law that relates the drift velocity $u(x,t)$ and the scalar temperature $\theta(x,t)$ produces two orders of smoothing. We study the implications of this property. For example, we prove that in the case of the non-diffusive ($\varepsilon_\kappa=0$) active scalar equation, initial data $\theta_0\in L^3(\R^3)$ implies the existence of unique, global weak solutions. If $\theta_0\in W^{s,3}(\R^3)$ with $s>0$, then the solution $\theta(x,t)\in W^{s,3}(\R^3)$ for all time. In the case of positive diffusivity ($\varepsilon_\kappa>0$), even for singular initial data $\theta_0\in L^3(\R^3)$, the global solution is instantaneously $C^\infty$-smoothed and satisfies the drift-diffusion equation classically for all $t>0$.

Abstract : The free additive convolution is a binary operation on the set of Borel probability laws on R, which allows us to deduce the law of a sum of free self-adjoint random variables in a non-commutative probability space. In this talk, we will discuss some regularization properties about free additive convolutions. More precisely, we will talk about regularity questions about free additive convolution semigroups, free Levy processes and free infinitely divisible laws. Some recent progresses in free probability theory will be discussed as well.

Abstract: The famous inner-outer factorization plays a fundamental role in the theory of the Hardy space. The same factorization holds for the Dirichlet space but its role in the Dirichlet space is more subtle and not well understood. We will discuss its application, in particular, its relevance to the zero set problem and the cyclic vector problem.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： We consider two applications using Mean Field Games (MFG) technique. Specifically, in order to solve the optimal strategy with the common noise, we need to introduce the stochastic partial differential equation (SPDE). The explicit solutions are obtained in these two examples.

摘要： We illustrate what a systemic risk is by proposing an inter‐bank borrowing and lending model using a stochastic flocking system with jump‐diffusion processes. We use the Laplace transform approach and the inversion formula instead to calculate the systemic risk. We then integrate a game feature with jumps where each bank controls its rate of borrowing/lending to a central bank.

Abstract: ppOpen-HPC is an open source infrastructure for development and execution of optimized and reliable simulation code on post-peta-scale (pp) parallel computers based on many-core architectures, and it consists of various types of libraries, which cover general procedures for scientific computation. In ppOpen-HPC, we are focusing on five types of discretization methods for scientific computing, which are FEM, FDM, FVM, BEM, and DEM. ppOpenHPC provides a set of libraries covering various types of procedures for these five methods, such as parallel I/O of data-sets, assembling of coefficient matrix, linear-solvers with robust and scalable preconditioners, adaptive mesh refinement (AMR), and dynamic load-balancing. The target post-peta-scale system is the Post T2K System of the University of Tokyo and University of Tsukuba based on many-core architectures, such as Intel MIC/Xeon Phi. It will be installed in FY.2015-2016 and its peak performance is expected to be 20-30 PFLOPS.　Source codes of ppOpen-HPC with documents are available at http://ppopenhpc.cc.u-tokyo.ac.jp/ . In this presentation, recent progresses of the "ppOpen-HPC" project are overviewed. Especially, achievements in the development of ppOpen- MATH/MG, which is a geometric multigrid solver in ppOpen-HPC, are described in detail.

Abstract: We will explain some intimate connections between the symmetric groups of regular polyhedra and Dynkin diagrams of simple Lie algebras from both geometric and group theoretic viewpoints.

Abstract : Cayley polytopes are integral polytopes which are “small” in some sense. In this talk, I give an algebro-geometric characterization of Cayley polytopes by corresponding toric varieties. As a corollary, we determine when the Seshadri constants of toric varieties are one.

Abstract: The representation theory is a powerful tool that realizes problems in abstract algebra in the language of linear algebras. On one hand, it has ubiquitous applications, e.g., Fourier analysis, number theory, algebraic geometry, topology and mathematical physics. On the other hand, the approaches to representation theory are diverse, including methods from geometry, ring theory, combinatorics, topology and differential equations. In this talk I will cover the following topics: 1. Introduction to representation theory. 2. Classification theorem of simple Lie algebras. 3. Weyl groups, Linkage principles and Kazhdan-Lusztig theory. 4. Categorification. Prerequisites: 1. Linear algebra 2. Undergraduate algebra

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： We consider the stochastic fractional heat equations [SFHE]. We like to show that when the spatial variables are far apart, the solutions at those points are “independence.” This helps to find the right scale of the chaotic behaviors of the SFHE.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： In this seminar series, we plan to study the basic theory of stochastic differential equations. We will mainly follow the lecture note by Prevot-Rochner (2007). The approach is to consider SPDE as a partial differential equation with the noise which is a Wiener process taking values in a Hilbert space or Banach space. This approach leads to the study of martingale theory in infinite dimensional spaces (Hilbert spaces or Banach spaces), and also the stochastic integrals with respect to the Hilbert space valued martingale. We will also have chance to compare different approaches by Walsh (white noise and martingale measure) and also Da Prato-Zabczyk (in Hairer lecture note). Last year, Shang-Yuan Shiu discussed Walsh’s approach. We hope to find chance to come back to this again this year. References 1. R. C. Dalang, D. Khoshnevisan, C. Mueller, D. Nualart and Y. Xiao (2008), A Mini-course on Stochastic Partial Differential Equation, Lcture Note in Mathematics, 1962 2. C. Prevot and M. Rochner (2007), A Concise Course on Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1905 3. Martin Hairer (2009), An Introduction to Stochastic PDEs 4. J.B. Walsh (1986), An Introduction to Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1108

Abstract: It has been known from the 70’s that all closed orientable 3-manifolds admit CR structures. Among all these manifolds, we prove that only sphere and the lens spaces may admit non-trivial torsion-free pseudo-gradient CR Yamabe soliton structures. For a non-trivial complete soliton, which always admits a Riemannian metric with indefinite curvature sign, we show that it must be diffeomorphic to the Euclidean space if it is simply-connected, torsion-free and has certain bound on the Webster curvature. This is a joint work with Huai-Dong Cao and Shu-Cheng Chang.

摘要： In this seminar series, we plan to study the basic theory of stochastic differential equations. We will mainly follow the lecture note by Prevot-Rochner (2007). The approach is to consider SPDE as a partial differential equation with the noise which is a Wiener process taking values in a Hilbert space or Banach space. This approach leads to the study of martingale theory in infinite dimensional spaces (Hilbert spaces or Banach spaces), and also the stochastic integrals with respect to the Hilbert space valued martingale. We will also have chance to compare different approaches by Walsh (white noise and martingale measure) and also Da Prato-Zabczyk (in Hairer lecture note). Last year, Shang-Yuan Shiu discussed Walsh’s approach. We hope to find chance to come back to this again this year. References 1. R. C. Dalang, D. Khoshnevisan, C. Mueller, D. Nualart and Y. Xiao (2008), A Mini-course on Stochastic Partial Differential Equation, Lcture Note in Mathematics, 1962 2. C. Prevot and M. Rochner (2007), A Concise Course on Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1905 3. Martin Hairer (2009), An Introduction to Stochastic PDEs 4. J.B. Walsh (1986), An Introduction to Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1108

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： A random walk is basically looking at the time sequence in a binomial process. It is one of the most basic and well-studied topics in probability theory. Researchers who work with perturbations of random walks, or with particle systems and other models often use random walks as a basic ingredient. In this talk, I will present some interesting properties of one dimensional random walk.

Abstract: A random walk (RW) is a mathematical formalization of a path that consists of a succession of random steps. It was first introduced by George Polya around 1905. Random walks have been used in many fields: ecology, economics, psychology, computer science, physics, chemistry, and biology. A self-avoiding walk (SAW) is a sequence of moves on a lattice that does not visit the same point more than once. It was first introduced by Paul Flory in 1953. Very little is known rigorously about the self-avoiding walk from a mathematical perspective, although physicists have provided numerous conjectures that are believed to be true and are strongly supported by numerical simulations. In this talk, I will introduce some relation between RW and SAW.

摘要： In this seminar series, we plan to study the basic theory of stochastic differential equations. We will mainly follow the lecture note by Prevot-Rochner (2007). The approach is to consider SPDE as a partial differential equation with the noise which is a Wiener process taking values in a Hilbert space or Banach space. This approach leads to the study of martingale theory in infinite dimensional spaces (Hilbert spaces or Banach spaces), and also the stochastic integrals with respect to the Hilbert space valued martingale. We will also have chance to compare different approaches by Walsh (white noise and martingale measure) and also Da Prato-Zabczyk (in Hairer lecture note). Last year, Shang-Yuan Shiu discussed Walsh’s approach. We hope to find chance to come back to this again this year. References 1. R. C. Dalang, D. Khoshnevisan, C. Mueller, D. Nualart and Y. Xiao (2008), A Mini-course on Stochastic Partial Differential Equation, Lcture Note in Mathematics, 1962 2. C. Prevot and M. Rochner (2007), A Concise Course on Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1905 3. Martin Hairer (2009), An Introduction to Stochastic PDEs 4. J.B. Walsh (1986), An Introduction to Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1108

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： In this seminar series, we plan to study the basic theory of stochastic differential equations. We will mainly follow the lecture note by Prevot-Rochner (2007). The approach is to consider SPDE as a partial differential equation with the noise which is a Wiener process taking values in a Hilbert space or Banach space. This approach leads to the study of martingale theory in infinite dimensional spaces (Hilbert spaces or Banach spaces), and also the stochastic integrals with respect to the Hilbert space valued martingale. We will also have chance to compare different approaches by Walsh (white noise and martingale measure) and also Da Prato-Zabczyk (in Hairer lecture note). Last year, Shang-Yuan Shiu discussed Walsh’s approach. We hope to find chance to come back to this again this year. References 1. R. C. Dalang, D. Khoshnevisan, C. Mueller, D. Nualart and Y. Xiao (2008), A Mini-course on Stochastic Partial Differential Equation, Lcture Note in Mathematics, 1962 2. C. Prevot and M. Rochner (2007), A Concise Course on Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1905 3. Martin Hairer (2009), An Introduction to Stochastic PDEs 4. J.B. Walsh (1986), An Introduction to Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1108

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

We will introduce a non-standard braided tensor structure on the category of finite dimensional representations of a complex simple Lie algebra, first defined by Drinfeld by means of the Knizhnik-Zamolodchikov equations in 1990. This category is also closely related to the representation theory of the corresponding affine Lie algebra. In particular, following Kazhdan and Lusztig's remarkable result in 1994, we should really think of Drinfeld's category as a certain representation category of the affine Lie algebra, equipped with a nontrivial tensor structure. In this talk, we will try to explain how the Knizhnik-Zamolodchikov equations and affine Lie algebras naturally appear in this story.

We study a class of evolution equations within the framework of the so-called Euler- Poincaré differential (EPDiff) equations. The EPDiff equations arise in studies related to the Riemannian geometry of groups of diffeomorphisms, as well as the fluid mechanics of nonlinear waves. The evolution of the dependent variable of an EPDiff system is coupled to an auxiliary field defined by the action of an invertible elliptic operator onto the base field. We study the finite-dimensional particle formulation of the EPDiff equations that is in a manner reminiscent of the streamfunction-vorticity formulation of the Euler equations in hydrodynamics. With the help of the Lagrangian formulation of the EPDiff evolution equations, we develop particle algorithms for the multidimensional EPDIFF system in both its dispersionless and dispersive forms. Our focus of the particle algorithms is on the class of non-smooth kernels, and in particular the “conon” case, for which the motion of particles are restricted in a one-dimensional channel. The one-channel reduction reduces the particle system to the “completely integrable” one-dimensional case, even though the “single channel” solution retains its dependence on two spatial dimensions. We study collision cases of these non-smooth kernels in a one-dimensional channel and bring forth the effects, due to the lack of Lipshitz continuity in the associated systems of ordinary differential equations. We also demonstrate that potentially the particle algorithms can be used as a numerical algorithm for solving the EPDIFF equations. We compare the numerical solutions between the particle algorithm and a splitting pseudospectral method developed in this paper. We find that the two results are virtually indistinguishable, provided a sufficiently large number of particles is used for the particle algorithm. Finally we link the N-particle algorithm to the application of template matching. Template matching is commonly used in problems of image registration and pattern recognition. Template matching can be formulated as finding the shortest or least expensive path of continuous deformation of one geometric object (reference) into another one (target). In this context, the time-dependent deformation process is the so-called geodesic evolution, and the derivation of the geodesic evolution equations falls into the Euler-Poincaré theory, which produces the EPDiff motion equations. Therefore, mathematically the EPDiff equations describe geodesic motion on the diffeomorphism group, and have the same form as the template matching equations. We propose a template matching algorithm based on the extended Kalman filter.

摘要： In this seminar series, we plan to study the basic theory of stochastic differential equations. We will mainly follow the lecture note by Prevot-Rochner (2007). The approach is to consider SPDE as a partial differential equation with the noise which is a Wiener process taking values in a Hilbert space or Banach space. This approach leads to the study of martingale theory in infinite dimensional spaces (Hilbert spaces or Banach spaces), and also the stochastic integrals with respect to the Hilbert space valued martingale. We will also have chance to compare different approaches by Walsh (white noise and martingale measure) and also Da Prato-Zabczyk (in Hairer lecture note). Last year, Shang-Yuan Shiu discussed Walsh’s approach. We hope to find chance to come back to this again this year. References 1. R. C. Dalang, D. Khoshnevisan, C. Mueller, D. Nualart and Y. Xiao (2008), A Mini-course on Stochastic Partial Differential Equation, Lcture Note in Mathematics, 1962 2. C. Prevot and M. Rochner (2007), A Concise Course on Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1905 3. Martin Hairer (2009), An Introduction to Stochastic PDEs 4. J.B. Walsh (1986), An Introduction to Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1108

Abstract: Abstract: The Lambert problem concerns the Kepler problem as a boundary value problem. A celebrated theorem by Lambert and Lagrange states that the transfer time is functionally dependent on the distance between the two ends, the sum of orbital distances from the attractive focus, and the semimajor axis. In this talk I will briefly present an intriguing linkage between this beautiful theorem and minimizing solutions of the n-body problem, through certain convex optimization techniques.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： In this seminar series, we plan to study the basic theory of stochastic differential equations. We will mainly follow the lecture note by Prevot-Rochner (2007). The approach is to consider SPDE as a partial differential equation with the noise which is a Wiener process taking values in a Hilbert space or Banach space. This approach leads to the study of martingale theory in infinite dimensional spaces (Hilbert spaces or Banach spaces), and also the stochastic integrals with respect to the Hilbert space valued martingale. We will also have chance to compare different approaches by Walsh (white noise and martingale measure) and also Da Prato-Zabczyk (in Hairer lecture note). Last year, Shang-Yuan Shiu discussed Walsh’s approach. We hope to find chance to come back to this again this year. References: 1. R. C. Dalang, D. Khoshnevisan, C. Mueller, D. Nualart and Y. Xiao (2008), A Mini-course on Stochastic Partial Differential Equation, Lcture Note in Mathematics, 1962 2. C. Prevot and M. Rochner (2007), A Concise Course on Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1905 3. Martin Hairer (2009), An Introduction to Stochastic PDEs 4. J.B. Walsh (1986), An Introduction to Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1108

Abstract: In this talk, we will present some interesting evaluations of hypergeometric functions. To obtain these evaluations, we realize modular forms on Shimura curves in two ways, firstly in terms of solutions of Schwarzian differential equations and secondly as Borcherds forms, and then use Schofer's formula for values of Borcherds forms at CM-points.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: In this talk, we will discuss a generalized Radon transform which is related to the Boltzmann collision operator. This transform has complex structure due to that it contains singularities from space variables and angular variable. Use the tool from microlocal analysis, we can show it has fractional order diffusion effect.

Abstract: This paper proposes a blind audio watermarking scheme based on singular-spectrum analysis (SSA) which relates to several techniques based on singular value decomposition (SVD). SSA is used to decompose a signal into several additive oscillatory components where each component represents a simple oscillatory mode. The proposed scheme embedded a watermark into a host signal by modifying scaling factors of certain components of the signal. Test results show that the proposed scheme satisfies imperceptibility criterion suggested by IHC with the average ODG of 0.18. It is robust against many attacks, such as MP3 and MP4 compression, band-pass filtering, and re-sampling. This paper does not only propose a new watermarking scheme, it also discusses the singular value and reveals its meaning, which has been deployed and played an important role in all SVD-based schemes.

摘要： We consider the stochastic fractional heat equations [SFHE]. We like to show that when the spatial variables are far apart, the solutions at those points are “independence.” This helps to find the right scale of the chaotic behaviors of the SFHE.

Abstract: In this talk, we will consider the multiple zeta values (MZVs) in characteristic p. We call a MZV "Eulerian" if it is a rational multiple of the Carlitz period raised to its weight power. We will present an algorithm to determine when a given MZV is Eulerian or non-Eulerian. This is a joint work with Matt Papanikolas and Jing Yu.

Abstract: The restricted isometry property (RIP) is proposed by Candes and Tao to measure how closely a matrix behaves like an isometry. Such property, as Donoho and Huo's "incoherence", is invented to give criterions in the theory of compressed sensing. More precisely, one can show that if a matrix A has good RIP, then a sparse signal x can be recovered from the measurement Ax by solving a convex optimization problem. In this talk, we will briefly review the idea of compressed sensing, and then give some criterions based on RIP.

Abstract： The celebrated result of Roth (fields medalist 1958) is to assert the existence of arithmetic progressions of length 3 in an integer set with positive density. This result has been improved and generalized by Szemeredi, Gowers, Bourgain, Green and Tao e.t.c. The purpose of this talk is to introduce the proof by Green and Tao on their refinement of Roth's theorem. This talk is almost self-contained as their proofs basically only use finite Fourier analysis. In this talk, we will first recall some basic finite Fourier methods which are needed in the proof, and outline the main ideas of their proofs. The talk should be accessible to a wide audience, in particular the graduate students.