Analytic geometry | Lattice points | Discrete groups | Lie groups
In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension which spans the vector space . For any basis of , the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a regular tiling of a space by a primitive cell. Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory and group theory. They also arise in applied mathematics in connection with coding theory, in percolation theory to study connectivity arising from small-scale interactions, cryptography because of conjectured computational hardness of several lattice problems, and are used in various ways in the physical sciences. For instance, in materials science and solid-state physics, a lattice is a synonym for the "framework of a crystalline structure, a 3-dimensional array of regularly spaced points coinciding in special cases with the atom or molecule positions in a crystal. More generally, lattice models are studied in physics, often by the techniques of computational physics. (Wikipedia).
What is a Group? | Abstract Algebra
Welcome to group theory! In today's lesson we'll be going over the definition of a group. We'll see the four group axioms in action with some examples, and some non-examples as well which violate the axioms and are thus not groups. In a fundamental way, groups are structures built from s
From playlist Abstract Algebra
Definition of a group Lesson 24
In this video we take our first look at the definition of a group. It is basically a set of elements and the operation defined on them. If this set of elements and the operation defined on them obey the properties of closure and associativity, and if one of the elements is the identity el
From playlist Abstract algebra
Abstract Algebra: We define the notion of a subgroup and provide various examples. We also consider cyclic subgroups and subgroups generated by subsets in a given group G. Example include A4 and D8. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-
From playlist Abstract Algebra
Abstract Algebra | The dihedral group
We present the group of symmetries of a regular n-gon, that is the dihedral group D_n. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Abstract Algebra
Group Definition (expanded) - Abstract Algebra
The group is the most fundamental object you will study in abstract algebra. Groups generalize a wide variety of mathematical sets: the integers, symmetries of shapes, modular arithmetic, NxM matrices, and much more. After learning about groups in detail, you will then be ready to contin
From playlist Abstract Algebra
A group is (in a sense) the simplest structure in which we can do the familiar tasks associated with "algebra." First, in this video, we review the definition of a group.
From playlist Modern Algebra - Chapter 15 (groups)
Group Theory: The Center of a Group G is a Subgroup of G Proof
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Group Theory: The Center of a Group G is a Subgroup of G Proof
From playlist Abstract Algebra
Groups in abstract algebra examples
In this tutorial I discuss two more examples of groups. The first contains four elements and they are the four fourth roots of 1. The second contains only three elements and they are the three cube roots of 1. Under the binary operation of multiplication, these sets are in fact groups.
From playlist Abstract algebra
Dihedral Group (Abstract Algebra)
The Dihedral Group is a classic finite group from abstract algebra. It is a non abelian groups (non commutative), and it is the group of symmetries of a regular polygon. This group is easy to work with computationally, and provides a great example of one connection between groups and geo
From playlist Abstract Algebra
Gabriele NEBE - Lattices, Perfects lattices, Voronoi reduction theory, modular forms, ...
Lattices, Perfects lattices, Voronoi reduction theory, modular forms, computations of isometries and automorphisms The talks of Coulangeon will introduce the notion of perfect, eutactic and extreme lattices and the Voronoi's algorithm to enumerate perfect lattices (both Eulcidean and He
From playlist École d'Été 2022 - Cohomology Geometry and Explicit Number Theory
Phong NGUYEN - Recent progress on lattices's computations 1
This is an introduction to the mysterious world of lattice algorithms, which have found many applications in computer science, notably in cryptography. We will explain how lattices are represented by computers. We will present the main hard computational problems on lattices: SVP, CVP and
From playlist École d'Été 2022 - Cohomology Geometry and Explicit Number Theory
Alessandra Sarti, Old and new on the symmetry groups of K3 surfaces
VaNTAGe Seminar, Feb 9, 2021
From playlist Arithmetic of K3 Surfaces
Counting points on the E8 lattice with modular forms (theta functions) | #SoME2
In this video, I show a use of modular forms to answer a question about the E8 lattice. This video is meant to serve as an introduction to theta functions of lattices and to modular forms for those with some knowledge of vector spaces and series. -------------- References: (Paper on MIT
From playlist Summer of Math Exposition 2 videos
Mod-01 Lec-3 Symmetry in Perfect Solids (Continued)
Condensed Matter Physics by Prof. G. Rangarajan, Department of Physics, IIT Madras. For more details on NPTEL visit http://nptel.iitm.ac.in
From playlist NPTEL: Condensed Matter Physics - CosmoLearning.com Physics Course
History of science 7: Did Witt discover the Leech lattice?
In about 1970 the German mathematician Witt claimed to have discovered the Leech lattice many years before Leech. This video explains what the Leech lattice is and examines the evidence for Witt's claim. Lieven Lebruyn discussed this question on his blog: http://www.neverendingbooks.org/w
From playlist History of science
Infinite Generaton of Non-Cocompact Lattices on Right-Angled Buildings - Anne Thomas
Anne Thomas University of Sydney, NSW April 6, 2011 SPECIAL LECTURE Let Gamma be a non-cocompact lattice on a right-angled building X. Examples of such X include products of trees, or Bourdon's building I_{p,q}, which has apartments hyperbolic planes tesselated by right-angled p-gons and
From playlist Mathematics
The Square Lattice via group D4 and its hypergroups | Diffusion Symmetry 5 | N J Wildberger
Hypergroups are remarkable probabilistic/ algebraic objects that have a close connection to groups, but that allow a transformation of non-commutative problems into the commutative setting. This gives powerful new tools for harmonic analysis in situations ruled by symmetry. Bravais latti
From playlist Diffusion Symmetry: A bridge between mathematics and physics
The One Dimensional Random Walk Hypergroup | Diffusion Symmetry 6 | N J Wildberger
We introduce affine transformations, in the form of the ax+b group, to describe the classical (group theoretic) symmetries of the simplest lattice - the one dimensional integral lattice. We meet the fundamental idea of representation theory: assigning concrete matrices to abstract algebrai
From playlist Diffusion Symmetry: A bridge between mathematics and physics
MIT 3.60 | Lec 6a: Symmetry, Structure, Tensor Properties of Materials
Part 1: 2D Plane Groups, Lattices (cont.) View the complete course at: http://ocw.mit.edu/3-60F05 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 3.60 Symmetry, Structure & Tensor Properties of Material
Abstract Algebra: We introduce the notion of a group and describe basic properties. Examples given include familiar abelian groups and the symmetric groups. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.o
From playlist Abstract Algebra