Diophantine geometry | Lattice points | Euclidean geometry
In mathematics, the n-dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are n-tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. is the simplest example of a root lattice. The integer lattice is an odd unimodular lattice. (Wikipedia).
The integers modulo n under addition is a group. What are the integers mod n, though? In this video I take you step-by-step through the development of the integers mod 4 as an example. It is really easy to do and to understand.
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
The elements of a set can be ordered by a relation. Some relation cause proper ordering and some, partial ordering. Have a look at some examples.
From playlist Abstract algebra
Determine if the Binary Operation Defined by the Table is Commutative and Associative
In this video we determine whether or not a binary operation is commutative and associative. The binary operation is actually defined by a table in this example. I hope this video helps someone.
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
Now that we know what a quotient group is, let's take a look at an example to cement our understanding of the concepts involved.
From playlist Abstract algebra
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
Algebraic Structures: Groups, Rings, and Fields
This video covers the definitions for some basic algebraic structures, including groups and rings. I give examples of each and discuss how to verify the properties for each type of structure.
From playlist Abstract Algebra
A set might contain many inverse elements under some binary operation. To have such an element, this set must also contain an identity element under the binary operation in question. An element is an inverse element of another element in a set if performing the binary operation between t
From playlist Abstract algebra
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
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
Lecture 29 | The Fourier Transforms and its Applications
Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood continues to lecture on the general stretch theorem and begins covering medical imaging. The Fourier transform is a tool for solving physical p
From playlist Lecture Collection | The Fourier Transforms and Its Applications
[ANT13] Dedekind domains, integral closure, discriminants... and some other loose ends
In this video, we see an example of how badly this theory can fail in a non-Dedekind domain, and so - regrettably - we finally break our vow of not learning what a Dedekind domain is.
From playlist [ANT] An unorthodox introduction to algebraic number theory
Daniel Dadush: Integer Programming and the Kannan-Lovasz Conjecture
In this talk, I will give a (very) high-level overview of the lattice theoretic and convex geometric tools needed to solve n-variable integer programs in O(n)n time. In the process, I will introduce the Kannan-Lovasz conjecture, which posits the existence of very sparse lattice projections
From playlist Workshop: Parametrized complexity and discrete optimization
Mathematical Ideas in Lattice Based Cryptography - Jill Pipher
2018 Program for Women and Mathematics Topic: Mathematical Ideas in Lattice Based Cryptography Speaker: Jill Pipher Affiliation: Brown University Date: May 21, 2018 For more videos, please visit http://video.ias.edu
From playlist Mathematics
Pi hiding in prime regularities
A story of pi, primes, complex numbers, and how number theory braids them together. Mathologer on why 4k + 1primes break down as sums of squares: https://youtu.be/DjI1NICfjOk Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply sh
From playlist Neat proofs/perspectives
The (Coarse) Moduli Space of (Complex) Elliptic Curves | The Geometry of SL(2,Z), Section 1.3
We discuss complex elliptic curves, and describe their moduli space. Richard Borcherd's videos: Riemann-Roch Introduction: https://www.youtube.com/watch?v=uRfbnJ2a-Bs&ab_channel=RichardE.BORCHERDS Genus 1 Curves: https://www.youtube.com/watch?v=NDy4J_noKi8&ab_channel=RichardE.BORCHERDS
From playlist The Geometry of SL(2,Z)
Short vector problems and simultaneous approximation
Short vector problems and simultaneous approximation, by Daniel E. Martin, presented at ANTS XIV.
From playlist My Students
Modular forms: Theta functions in higher dimensions
This lecture is part of an online graduate course on modular forms. We study theta functions of even unimodular lattices, such as the root lattice of the E8 exceptional Lie algebra. As examples we show that one cannot "her the shape of a drum", and calculate the number of minimal vectors
From playlist Modular forms
Orbit of a set in abstract algebra
In this video we start to take a look at the orbit-stabilizer theorem. Our first stop is the orbit of a set. The orbit is created by taking an arbitrary element of a set and acting on that element by all the elements in the set of an an arbitrary group. In this video, we look at a few p
From playlist Abstract algebra