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# Linear Algebra MA 55400

Purdue

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This 6 page Class Notes was uploaded by Dorothea Bode on Saturday September 19, 2015. The Class Notes belongs to MA 55400 at Purdue University taught by Jiu-Kang Yu in Fall. Since its upload, it has received 66 views. For similar materials see /class/208116/ma-55400-purdue-university in Mathematics (M) at Purdue University.

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Date Created: 09/19/15

Courses and Seminars of Interest to Graduate Students offered by the Mathematics Department Spring 2009 MA 55400 CRN 22179 LINEAR ALGEBRA Instructor Prof lK Yu of ce Math 604 phone 49767414 email jyu mathpurdueedu Time MWF 830 Prerequisite a basic course in linear algebra familiarity with the notion of a eld the ring of integers and the ring of polynomials over a el Description The course will cover most material from chapters 36 in the text various normal forms modules over a principal ideal domain bilinear and hermitian forms together with additional topics References Roman Advanced linear algebra Text W Adkins and S Weintraub Algebra MA 58400 CRN 34045 ALGEBRAIC NUMBER THEORY Instructor Prof F Shahidi of ce Math 650 phone 49741917 e mail shahidi mathpurdueedu Time MWF 930 Prerequisite MA 55300 adn 55400 Description Dedekind domains norm discriminant different niteness of class numbers Dirichlet unit theorem quadratic and cyclotomic extensions quadratic reciprocity decomposition and inertial groups completions and local elds References S Lang Algebraic Number Theory Text G Janusz Algebraic Number Fields GSM7 MAS 1996 MA 59800 CRN 34039 ACOUSTICS OF POROUS MEDIA THEORY NUMERICS AND APPLICATIONS Instructor Prof J Santos of ce Math 416 emailsantosmathpurdueedu Time TTh 1200115 Description The course will describe the theory of wave propagation in uidsaturated porous media with applications to detection and characterization of hydrocarbon reservoirs monitoring of C02 sequestration after injection and characterization of partially frozen porous media among others The equations describing wave propagation in saturated porous media will be solved using nite element techniques Computer implementation willl be discussed Numerical upscaling techniques used to represent highly heterogeneous fluidi lled porous media will also be presented Course Contents 1 Derivation of the constitutive relations and equations of motion for uidsaturated porous media Biot s media Relation with Darcyls law and thermodynamic considerations 2 Determination of the coefficients in the constitutive relations and the viscodynamic coefficients in Biotls equations of motion in terms of the properties of the individual solid and uid phases Introduction of viscoelasticity employing the Correspondence Principle 3 Analysis of the phase velocities and attenuation coefficients for the different types of body waves propagating in Biotls media 4 Review of the Finite Element Method Description of some nite element spaces in 1D 2D and 3D Analysis of the interpo lation error Mixed nite element spaces methods for solving elliptic and Maxwell equations 5 Solution of elliptic problems using nite element methods Error analysis 6 Numerical solution of Biotls equations of motion using the nite element method in the 1D and 2D cases Global and iterative parallelizable domain decomposition nite element procedures 7 Extension of Biotls theory for the case when the porous matrix is composed of weakly coupled solids Plane wave analysis A 39 39 to wave 1 A 39 in h d bearing sediments 8 De nition of numerical upscaling procedures in Biotls media to determine associated complex frequency dependent plane wave and shear moduli A quot 39 to wave 1 A 39 in t h t t d porous media 9 Numerical solution of the coupled Biotls and Maxwellls equation in Biotls media using the nite element method Seismo electric and Electroseismic applications Description of Homework Assignements 1 Calculation of the coefficients in Biotls equations of motion for some materials using fortran or similar computer language 2 Calculation of the phase velocities and for some 4 39d t t d porous materials 3 Numerical simulation for 1D wave propagation in Biotls media Application to the analysis of attenuation and dispersion effects in partially saturated porous media 4 Numerical simulation of 2D wave propagation in Biotls media Serial and parallel implementations 5 Computer implementation of numerical upscaling procedures in Biotls media Application to seismic monitoring in C02 sequestration sites 6 Numerical modeling of coupled electromagnetic and seismic waves in 1D uidsaturated porous media A l m l Courses and Seminars of Interest to Graduate Students offered by the Mathematics Department Spring 2009 7 page 1 MA 59800 CRN 34040 MATHEMATICS OF DISPERSIVE PROCESSES Instructor Prof J Cushman of ce Math 416 phone 49748040 email jcushman mathpurdueedu Time TTh 130245 Description Dispersive processes take place in almost all natural and engineered environments that involve uctuations in a eld variable such as velocity in turbulence and porous media ows topographic elevation in landscapes conductance in a semiconductor price in an option contract or stock and etc This course focuses on developing a uni ed framework for these apparently disparate elds We will cover many of the standard approaches for developing theories of dispersion continuous time random walks projection operator methods of statistical mechanics renormalization group and generalized central limit theorems as applied to stochastic differential equations and their associated Fokkeriplanck equations and stochastic perturbation and Green s functions approaches to name a few The eld equations that result from the various approaches will all be shown to have a universal form Applications will be stressed MA 59800 CRN 34041 MODULI SPACES AND STACKS Instructor Prof R Kaufmann of ce Math 710 phone 49741205 email rkaufmaanmathpurdueedu Time TTh 9001015 Content We will discuss stacks in the topological algebraic geometric and differentiable categories Applications will include moduli spaces orbifolds gerbes and a dash of GromoviWitten theory Abstract When considering questions like what is the quotient of a space by a group action which is not necesarily nice or what kind of space classi es all objects of a certain type 7 these spaces are usually called moduli spaces 7 one realizes that the quotient or universal space are not of the same type as those one started with For instance taking the quotient of a manifold by a Lie group action might not lead to a manifold Also the moduli space of all Riemann surfaces does not behave as expected if one considers it merely as a space The solution to these problems are orbifolds or stacks There has been an avid interest in stacks from many different aspects in the last years and the literature is growing exponen tially as classical results are being transferred and extended to this new setting Stacks play a central role in algebraic geometry in GromoviWitten theory but they also appear in number theory topology and as orbifolds in differential geometry In particular the derived algebraic geometry approach in topology heavily uses stacks The interest has also been stoked by physics where stacks naturally appear as quotients of systems by symmetries In the course we will start with the motivating examples of group actions and moduli spaces We will then de ne stacks in topology differential and algebraic geometry and look at basic properties of them As further examples we will study orbifolds in more detail and look at a dash of GromoviWitten theory MA 59800 CRN 34042 THE RADON AND XiRAY TRANSFORMS AN INTRODUCTION TO INVERSE PROBLEMS Instructor Prof P Stefanov of ce Math 448 phone 49767330 email stefanovmathpurdueedu Time MWF 230 Prerequisite Familiarity with distributions and Fourier transform MA 54200 is highly desired Description The course will focus on the Radon transform integrals over hyperplanes and on the Xiray transform integrals along lines We will study questions of invertibility stability estimates description of the range and Helgason type of support theorems The attenuated ray trnasform and the Doppler transform integrals of vector elds will be covered as well The last part of the course will be devoted to local tomography and recovery of the wave front set Radon types of transforms play fundamental role in applications like medical imaging and geophysics The course is intended as an introduction to Inverse Problems through Radon type of transforms We will also brie y discuss the generalization of those problems to nonEuclidean geometries MA 59800 CRN 34044 TOPICS IN COMPLEX ANALYSIS Instructor Prof A Eremenko of ce Math 450 phone 49741975 email eremenko mathpurdueedu Time TTh 122001215 Description This is a second graduate course in onedimensional complex analysis The prerequisits are MATH 53000 and MATH 54400 The textbook is Rudin Real and Complex Analysis I plan to cover most of the second part of this book MA 59800 CRN 22196 BRIDGE TO RESEARCH SEMINAR Instructor Prof S Bell of ce Math 628 phone 49741967 email bell mathpurdueedu Time M 430 Description The seminar has two main goals both aimed at helping students early in their graduate career nd their place in the department The rst is to help students discover what area of mathematics they might be interested in researching as well as who they might like to work with The second is to provide students with an opportunity to interact with faculty in a casual setting This is achieved by having professors from the department give brief talks about their research area at a level that is accessible to those in their rst and second year of graduate study Courses and Seminars of Interest to Graduate Students offered by the Mathematics Department Spring 2009 7 page 2 MA 611007 CRN 340467 METHODS OF APPLIED MATHEMATICS I Instructor Prof D Danielli7 of ce Math 6207 phone 497419207 email daniellimathpurdueedu Time MWF 1030 Prerequisite MA 511007 54400 Description The purpose of this course is to present the most fundamental theorems of functional analysis7 keeping applications in mind Topics covered include metric spaces Banach spaces linear transformations the Fredholm RieszSchauder theory and elements of spectral theory for compact operators Hilbert spaces and spectral theory for selfadjoint operators Applications to ordinary and partial differential equations7 as well as to integral equations7 will be discussed Text A Friedman7 Foundations of modern analysis7 Dover MA 615007 CRN 221987 NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS I meets With CS 61500 Instructor Prof Z Cai7 of ce Math 4127 phone 497419217 email zcai mathpurdueedu Time TTh 10301145 Prerequisite MA 514007 MA 52300 Authorized equivalent courses or consent of instructor may be used in satisfying course prerequisites Description Finite element method for elliptic partial differential equations weak formulation nitedimensional approxi mations error bounds algorithmic issues solving sparse linear systems nite element method for parabolic partial differential equations backward difference and C l N39 h l V t f 39 quot to nite difference methods for elliptic7 parabolic7 and hyperbolic equations stability7 consistency7 and convergence discrete maximum principles MA 643007 CRN 222017 METHODS OF PARTIAL DIFFERERENTIAL EQUATIONS II Instructor Prof P Bauman7 of ce Math 7187 phone 497419457 email bauman mathpurdueedu Time MWF 930 Prerequisite MA 64200 Description Continuation of MA 642 Topics to be covered are Lp theory for solutions of elliptic equations7 including Moser s estimates7 Aleksandrov maximum principle7 and the CalderonZygmund theory Introduction to evolution problems for parabolic and hyperbolic equations7 including Galerkin approximation and semigroup methods Applications to nonlinear problems MA 661007 CRN 222027 MODERN DIFFERENTIAL GEOMETRY Instructor Prof H Donnelly7 of ce Math 7167 phone 497419447 email hgd mathpurdueedu Time MWF 130 Prerequisite MA 56200 Description A foundational course in Riemannian geometry Topics include the Leviicivita connection7 geodesics7 normal coordinates7 Jacobi elds Emphasis on curvature and its relation with topology Familiarity with differentialble manifolds7 tensor elds7 and differential forms is assumed Text John M Lee7 Riemannian Manifolds An Introduction to Curvature7 Springer7Verlag7 1997 MA 665007 CRN 34676 ALGEBRAIC GEOMETRY II Instructor Prof D Arapura7 of ce Math 642 phone 497419837 email dvb mathpurdueedu Time TTh 1200115 Description Imagine starting an introductory course on integral calculus with the de nition of Lebesgue measure While logically correct7 this would be a disaster pedagogically The choices for a second course in algebraic geometry7 such as this one7 are analogous I could go through chapters II and III of or I and II of GIl methodically7 but by the end of the semester you may have no idea what this stuff is good for Or7 I could take certain technical constructions as black boxes and apply them in hopefully interesting ways I prefer to do the latter To focus ideas7 I will be concentrate on the so called Liiroth problem The problem asks whether a eld between C and a eld of rational functions over C is also a eld of rational functions In one variable7 this is true by Liirothls theorem Although stated algebraically7 this is a basic result in the theory of algebraic curves or Riemann surfaces In two variables7 the problem has a positive solution by Castelnuovo7s rationality theorem for algebraic surfaces This is a somewhat deeper result which will require 39 A A 39 s eaves7 39 quot theory for divisors7 and it will take a good part of the semester to set up everything The story doesn7t end here By the 19707s7 counterexamples were found in three or more dimensions If time permits7 I will say something about the ClemensGrif ths example7 which will require quick tour of Hodge theory7 Abelian varieties7 and intermediate Jacobians This is a second course in algebraic geometry So I will assume that everyone knows basic algebraic geometry for example from Prof Wlodarczyk s class from the fall References 1 GH Griffiths7 Harris Principles of Algebraic Geometry 2 Hartshorne7 Algebraic Geometry 3 KSC Kollar7 Smith7 Corti77 Rational and Nearly Rational Varieties Courses and Seminars of Interest to Graduate Students offered by the Mathematics Department7 Spring7 2009 7 page 3 MA 69000 CRN 22204 TOPICS IN ALGEBRA AND ALGEBRAIC GEOMETRY Instructor Prof S Abhyankar of ce Math 600 phone 49741933 email ram mathpurdueedu Time TTh 300415 Description I shall discuss several topics in algebra and algebraic geometry There are no prerequisites All interested students are welcome I shall use my new book Lectures on Algebra Volume I published by World Scienti c as a textbook which the students are expected to purchase Although this is an advanced book the course will move much slower and can be taken as a basic course understandable to scientists and engineers During the course a softer version of the book will be produced The students may also nd it desirable to read my userfriendly book Algebraic Geometry For Scientists And Engineers published by American Mathematical Society Here is a list of some of the topics which may be covered 1 Expansions of polynomials of any degree in terms of sequences of other polynomials 2 Resultants Discriminants and solutions of higher degree polynomial equations in several variables 3 Newton7s Theorem on Fractional Expansions 4 Implicit Function Theorem and Inverse Function Theorem 5 Intersection Theory and Bezout s Theorem 6 Classi cation and Resolution of Singularities of Curves Surfaces and Higher Dimensional Varieties 7 Divisors Differentials and Genus Formulas MA 69000 CRN 22205 TOPICS IN COMMUTATIVE ALGEBRA Instructor Prof Heinzer of ce Math 636 phone 49741980 email heinzer mathpurdueedu Time MWF 330 Description The course is planned to be a continuation of MA 690B of this fall I hope to cover various topics in commutative algebra related to material in the text by W Bruns and J Herzog titled CohenMacaulay Rings revised edition Students enrolled in the course will be encouraged to actively participate by presenting material in class MA 69000 CRN 22210 BRUHATTITS THEORY Instructor Prof J K Yu of ce Math 604 phone 49767414 email jyu mathpurdueedu Time MWF 930 Description BruhatiTits theory is the structural theory of radic reductive groups developed by Bruhat and Tits Such groups have extremely rich structures and a large part of them is encoded in a geometric object called the BruhatiTits building The theory has many applications to number theory representation theory and geometry We will cover the theory for split and quasiisplit groups the BruhatiTits models the descent theory and Moyiprasad theory Lecture notes will be handed out MA 69200 CRN 22211 INTRODUCTION TO SPECTRAL METHODS FOR SCIENTIFIC COMPUTING Instructor Prof J Shen of ce Math 406 phone 49741923 email shen mathpurdueedu Time TTh 130245 Prerequisite A good knowledge of calculus linear algebra numerical analysis and some basic programming skills are essential Some knowledge of real analysis and functional analysis will be helpful but not necessary Description This is an introduction course on spectral methods for solving partial differential equations PDEs We shall present some basic theoretical results on spectral approximations as well as practical algorithms for implementing spectral methods We shall specially emphasize on how to design efficient and accurate spectral algorithms for solving PDEs of current interest The course is suitable for advanced undergraduate students in mathematics and graduate students in sciences and engineering Text Jie Shen and Tao Tang Spectraland High Order Methods with Applications MA 69200 CRN 22213 TOPICS IN IMAGING SPECTRAL THEORY OF THE EARTH Instructor Prof M de Hoop of ce Math 422 phone 49766439 email mdehoop mathpurdueedu Time TTh 10301145 MA 69300 CRN 34047 COMMUTATIVE AND NONiCOMMUTATIVE HARMONIC ANALYSIS Instructor Prof L Lempert of ce Math 728 phone 49741952 email 1empert mathpurdueedu Time TTh 122001215 Prerequisite Mathematics as required on the Quali er Examinations basic general topology MA 571 is more than enough notions of a differentiable manifold Hilbert space Description The leitmotiv of this course is that a large part of mathematics can be understood in terms of symmetries or in terms of group representations This will be illustrated historically starting with XVII12h century number theory and probability and concluding with XXth century quantum mechanics symmetries lurk behind all Topics touched upon Early probabilities representing integers by quadratic forms Dirichlet7s work on Fourier series Gauss sums primes in arithmetic progressions the birth of noncommutative representation theory partial differential equations Weyl s work on representation theory of compact Lie groups quantum mechanics symmetries in quantum mechanics according to Neumann and Wigner Recommended Text G M Mackey The Scope and History of Commutative and Noncommutative Harmonic Analysis American Mathematical Society 1992 Courses and Seminars of Interest to Graduate Students offered by the Mathematics Department Spring 2009 7 page 4 MA 69300 CRN 34048 ENTIRE FUNCTIONS Instructor Prof de Branges of ce Math 800 phone 49746057 email brangestathpurdueedu Time MWF 930 Description The theory of entire function as it developed in the nineteenth and twentieth centuries intended applications to zeros of zeta functions A proof of the Riemann hypothesis has been obtained which uses some of these now classical methods and also new methods which can be seen as extensions of the classical ones A basic course in entire functions which requires only information taught for qualifying examinations is now offered for the purpose of presenting the proof of the Riemann hypothesis Special techniques taught include weighted Hardy spaces and derived Hilbert spaces of entire functions as presented in Hilbert Spaces of Entire Functions 1968 on reserve in the library MA 69300 CRN 34049 TOPICS IN Cquot ALGEBRAS AND GROUP REPRESENTATIONS Instructor Prof M Dadarlat of ce Math 708 phone 49741940 email mdd mathpurdueedu Time MWF 230 Prerequisite Math 54400 and some background in Functional Analysis such as Math 54600 Description The rst few lectures will be devoted to the basic theory of Calgebras including the Gelfand transform Then we will cover topics such as group Calgebras the classic representation theory of compact groups induced representations Mackey s machine References Jacques Dixmier Calgebras Barry Simon Representations of Finite and Compact Group I Raeburn and D P Williams Morita Equivalence and Continuous Trace C Algebras Gerald B Folland A Course in Abstract Harmonic Analysis Text We will follow no speci c textbook MA 69400 CRN 34103 TOPICS IN STOCHASTIC DIFFERENTIAL EQUATIONS Instructor Prof F Baudoin of ce Math 438 phone 49741406 email fbaudoin mathpurdueedu Time MWF 1130 Description The purpose of this course is to propose an introduction to some advanced topics in the theory of stochastic differential equations and their applications In a rst part of the course we will focus on small time behaviour properties and study the stochastic Taylor expansion for solutions of stochastic differential equations and show that it provides a powerful tool in o The study of heat kernels for parabolic equations in small times 0 Numerical approximation methods for solutions of parabolic and related functional inequalities In a second part we will focus on long time behaviour properties and related functional inequalities A good knowledge in Stochastic processes is required and the course will partially follow the book References F Baudoin An introduction to the geometry of stochastic flows Imperial College Press 2005 MA 69600 CRN 34038 INTRODUCTION TO KAEHLER GOEMETRY Instructor Prof Yeung of ce Math 712 phone 49741942 email yeung mathpurdueedu Time MWF 1030 Prerequisite MA 56200 53000 Description The purpose of the course is to introduce foundational materials for Kaehler geometry bring in some analytic techniques and study their applications in complex manifolds and algebraic geometry The nal goal is to understand higher dimensional complex manifolds or algebraic varieties from an analytic or geometric point of view The rst one third of the course will be devoted to introductory materials essentially with no prerequisite We can nd this in the rst few chapters of books of Kodaira and Morrow Griffiths and Harris or Mok Then we will go through some standard techniques from analysis or partial differential equations such as L2 estimates and harmonic maps Finally we will get into more advanced topics including multiplier ideal sheaves and its applications such as invariance of plurigenera and existence of canonical metrics such as metrics with constant curvature on a line bundle and extremal metrics References 1 Kodaira K Morrow l Complex manifolds 2 Griffiths P Harris J Principle of algebraic geometry 3 Mok N Metric rigidity theorems on locally hermitian symmetric spaces Seminars Algebra and Algebraic Geometry Seminar Prof Abhyankar Time Thursday 4307600 Applied Math Lunch Seminar Prof Buzzard Time Friday 1130 Courses and Seminars of Interest to Graduate Students offered by the Mathematics Department Spring 2009 7 page 5 Automorphic Forms and Representation Theory Seminar7 Profs Goldberg and Yu Time Thursdays7 130 Bridge to Research Seminar7 Prof Bell Time Mondays 430 Commutative Algebra Seminar7 Profs Heinzer and Ulrich Time Wednesdays 430 Computational and Applied Math Seminar7 Prof Shen Time Fridays 330 Computational Finance Seminar7 Profs FigueroaLopez and Viens Time Mondays 430 Function Theory Seminar7 Prof Eremenko Time exible Geometric Analysis Seminar7 Prof Yeung Time Monday 330 Foundations of Analysis Seminar7 Prof de Branges Time Thursday 930 Number Theory Seminar7 Prof Goins Time Thursday7 330 Operator Algebras Seminar7 Prof Dadarlat Time Tuesdays7 230 PDE Seminar7 Prof Bauman Time Thursdays7 330 Probability Seminar7 Prof Sellke Time Mondays 330 Spectral and Scattering Theory Seminar7 Prof sea Barreto Time Wednesday 430 Topology Seminar7 Prof Kaufmann Time Thursday 330 Working Algebraic Geometry Seminar7 Profs Arapura and Matsuki Time Wednesday 330500 Courses and Seminars of Interest to Graduate Students offered by the Mathematics Department7 Spring7 2009 7 page 6

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