" This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry."
This textbook is an introductory coverage of algorithms and data structures with application to graphics and geometry.
Table of Contents
Part I: Programming environments for motion, graphics, and geometry
1. Reducing a task to given primitives: programming motion
2. Graphics primitives and environments
3. Algorithm animation
Part II: Programming concepts: beyond notation
4. Algorithms and programs as literature: substance and form
5. Divide-and-conquer and recursion.
7. Syntax analysis
Part III: Objects, algorithms, programs.
8. Truth values, the data type 'set', and bit acrobatics
9. Ordered sets
11. Matrices and graphs: transitive closure
14. Straight lines and circles
Part IV: Complexity of problems and algorithms
15. Computability and complexity
16. The mathematics of algorithm analysis
17. Sorting and its complexity
Part V: Data structures
18. What is a data structure?
19. Abstract data types
20. Implicit data structures
21. List structures
22. Address computation
23. Metric data structures
Part VI: Interaction between algorithms and data structures: case studies in geometric computation
24. Sample problems and algorithms
25. Plane-sweep: a general-purpose algorithm for two-dimensional problems illustrated using line segment intersection
26. The closest pair
This course is an introduction to the calculus of functions of several variables. It begins with studying the basic objects of multidimensional geometry: vectors and vector operations, lines, planes, cylinders, quadric surfaces, and various coordinate systems. It continues with the elementary differential geometry of vector functions and space curves. After this, it extends the basic tools of differential calculus - limits, continuity, derivatives, linearization, and optimization - to multidimensional problems. The course will conclude with a study of integration in higher dimensions, culminating in a multidimensional version of the substitution rule.
This is a variation on 18.02 Multivariable Calculus. It covers the same topics as in 18.02, but with more focus on mathematical concepts.
This course is an introduction to differential geometry. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature.
" Double affine Hecke algebras (DAHA), also called Cherednik algebras, and their representations appear in many contexts: integrable systems (Calogero-Moser and Ruijsenaars models), algebraic geometry (Hilbert schemes), orthogonal polynomials, Lie theory, quantum groups, etc. In this course we will review the basic theory of DAHA and their representations, emphasizing their connections with other subjects and open problems."
Seminar on a selected topic from Renaissance architecture. Requires original research and presentation of a report. The aim of this course is to highlight some technical aspects of the classical tradition in architecture that have so far received only sporadic attention. It is well known that quantification has always been an essential component of classical design: proportional systems in particular have been keenly investigated. But the actual technical tools whereby quantitative precision was conceived, represented, transmitted, and implemented in pre-modern architecture remain mostly unexplored. By showing that a dialectical relationship between architectural theory and data-processing technologies was as crucial in the past as it is today, this course hopes to promote a more historically aware understanding of the current computer-induced transformations in architectural design.
This text is intended for a brief introductory course in plane geometry. It covers the topics from elementary geometry that are most likely to be required for more advanced mathematics courses. The only prerequisite is a semester of algebra.
The emphasis is on applying basic geometric principles to the numerical solution of problems. For this purpose the number of theorems and definitions is kept small. Proofs are short and intuitive, mostly in the style of those found in a typical trigonometry or precalculus text. There is little attempt to teach theorem-proving or formal methods of reasoning. However the topics are ordered so that they may be taught deductively.
The problems are arranged in pairs so that just the odd-numbered or just the even-numbered can be assigned. For assistance, the student may refer to a large number of completely worked-out examples. Most problems are presented in diagram form so that the difficulty of translating words into pictures is avoided. Many problems require the solution of algebraic equations in a geometric context. These are included to reinforce the student's algebraic and numerical skills, A few of the exercises involve the application of geometry to simple practical problems. These serve primarily to convince the student that what he or she is studying is useful. Historical notes are added where appropriate to give the student a greater appreciation of the subject.
This book is suitable for a course of about 45 semester hours. A shorter course may be devised by skipping proofs, avoiding the more complicated problems and omitting less crucial topics.
Access also available here: https://academicworks.cuny.edu/ny_oers/6/
This book is designed for a semester-long course in Foundations of Geometry and meant to be rigorous, conservative, elementary and minimalistic.
Access also available here: http://anton-petrunin.github.io/birkhoff/
Table of Contents
2 The Axioms
4 Congruent triangles
5 Perpendicular lines
6 Similar triangles
7 Parallel lines
8 Triangle geometry
9 Inscribed angles
11 Neutral Geometry
12 Hyperbolic plane
13 Geometry of h-plane
14 Affine geometry
15 Projective geometry
16 Spherical geometry
17 Projective model
18 Complex coordinates
19 Geometric constructions
This course is an intensive introduction to architectural design tools and process, and is taught through a series of short exercises. The conceptual basis of each exercise is in the interrogation of the geometric principles that lie at the core of each skill. Skills covered in this course range from techniques of hand drafting, to generation of 3D computer models, physical model-building, sketching, and diagramming. Weekly lectures and pin-ups address the conventions associated with modes of architectural representation and their capacity to convey ideas. This course is tailored and offered only to first-year M.Arch students.
The Module on Geometry starts by looking at the historical development of knowledge that the humankind gather along centuries and became later, about 300 BC, the mathematical subject called “Euclidian Geometry” because of the great work of Euclid. The inductive-deductive reasoning which characterizes this subject will be developed through investigation of your own conjectures on geometric objects and properties. You will explore geometry by using basic mechanical instruments (compass and straightedge) and computer software.
As you progress you will treat the Euclidian geometry using a referential system to locate points. The orthogonal Cartesian system of coordinates that you already know from secondary school is the most common referential system you will use in both two and three dimensions. You will also learn some other systems of coordinates that will empower you to do research in geometry and in other mathematical modules as well.
Going deeper in analyzing the axiomatic construction of Euclidean geometry you will learn new geometrical structures, generally designated as Non-Euclidian geometry. So, summarily speaking, this Module is about Euclidean geometry treated in both syntactical and analytical ways and encloses an introduction to Non-Euclidean Geometry, handled synthetically only.
This course is oriented toward US high school students. Its structure and materials are aligned to the US Common Core Standards. Foci include: formulas for calculating the volume of prisms, cylinders, pyramids, cones, and spheres; and assist you in using geometric modeling to solve problems involving three-dimensional figures.
In this course, you will study the relationships between lines and angles. You will learn to calculate how much space an object covers, determine how much space is inside of a three-dimensional object, and other relationships between shapes, objects, and the mathematics that govern them.
A rigorous introduction designed for mathematicians into perturbative quantum field theory, using the language of functional integrals. Basics of classical field theory. Free quantum theories. Feynman diagrams. Renormalization theory. Local operators. Operator product expansion. Renormalization group equation. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and string theory.
This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations.
Studies how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Models of randomized computation. Data structures: hash tables, and skip lists. Graph algorithms: minimum spanning trees, shortest paths, and minimum cuts. Geometric algorithms: convex hulls, linear programming in fixed or arbitrary dimension. Approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms.
In this undergraduate level seminar series topics vary from year to year. Students present and discuss the subject matter, and are provided with instruction and practice in written and oral communication. Some experience with proofs required. The topic for fall 2008: Computational algebra and algebraic geometry.
Seminar for mathematics majors. Students present and discuss the subject matter, taken from current journals or books and write up exercises. Topic for spring 2003: Elementary topological properties of differentiable manifolds. Topics covered include Sard's theorem, the Thom transversality theorem, vector fields and the Poincare-Hopf theorem, and cohomolgy via differential forms. Prerequisites subject to negotiation with the instructor. Instruction and practice in oral communication provided. In this course, students take turns in giving lectures. For the most part, the lectures are based on Robert Osserman's classic book A Survey of Minimal Surfaces, Dover Phoenix Editions. New York: Dover Publications, May 1, 2002. ISBN: 0486495140.
The main aims of this seminar will be to go over the classification of surfaces (Enriques-Castelnuovo for characteristic zero, Bombieri-Mumford for characteristic p), while working out plenty of examples, and treating their geometry and arithmetic as far as possible.
Topics vary from year to year. Fall Term: Numerical properties and vanish theorems for ample, nef, and big line bundles and vector bundles; multiplier ideals and their applications