Dividing by a number is the same thing as multiplying by a reciprocal. This concept is particularly applicable when dividing by a fraction. Several examples are worked out and explained. From chapter 2 of the Algebra 1 course by Derek Owens
From playlist Algebra 1 Chapter 2 (Selected Videos)
Now that we have defined and understand quotient groups, we need to look at product groups. In this video I define the product of two groups as well as the group operation, proving that it is indeed a group.
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
Units in a Ring (Abstract Algebra)
The units in a ring are those elements which have an inverse under multiplication. They form a group, and this “group of units” is very important in algebraic number theory. Using units you can also define the idea of an “associate” which lets you generalize the fundamental theorem of ar
From playlist Abstract Algebra
We have seen an example of partitioning in the previous video. These partitioned sets are called equivalence sets or equivalence classes. In this video we look at some notation.
From playlist Abstract algebra
From playlist Complex Multiplication
Nihar Gargava - Random lattices as sphere packings
In 1945, Siegel showed that the expected value of the lattice-sums of a function over all the lattices of unit covolume in an n-dimensional real vector space is equal to the integral of the function. In 2012, Venkatesh restricted the lattice- sum function to a collection of lattices that h
From playlist Combinatorics and Arithmetic for Physics: Special Days 2022
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
Olivier Debarre: Periods of polarized hyperkähler manifolds
Abstract: Hyperkähler manifolds are higher-dimensional analogs of K3 surfaces. Verbitsky and Markmann recently proved that their period map is an open embedding. In a joint work with E. Macri, we explicitly determine the image of this map in some cases. I will explain this result together
From playlist Algebraic and Complex Geometry
[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
Donald Cartwright: Construction of lattices defining fake projective planes - lecture 6
Recording during the meeting "Ball Quotient Surfaces and Lattices " the February 28, 2019 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Ma
From playlist Algebraic and Complex Geometry
Short vector problems and simultaneous approximation
Short vector problems and simultaneous approximation, by Daniel E. Martin, presented at ANTS XIV.
From playlist My Students
Ben Howard: Supersingular points on som orthogonal and unitary Shimura varieties
To an orthogonal group of signature (n,2), or to a unitary group of any signature, one can attach a Shimura variety. The general problem is to describe the integral models of these Shimura varieties, and their reductions modulo various primes. I will give a conjectural description of the s
From playlist HIM Lectures: Junior Trimester Program "Algebraic Geometry"
[ANT05] Minkowski's geometry of numbers
Unsurprisingly, many of the pictures we've drawn are honest geometric objects, leaving them open to geometric attacks.
From playlist [ANT] An unorthodox introduction to algebraic number theory
Tim Steger: Construction of lattices defining fake projective planes - lecture 3
Recording during the meeting "Ball Quotient Surfaces and Lattices " the February 26, 2019 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Ma
From playlist Algebraic and Complex Geometry
Lattice multiplication | Multiplication and division | Arithmetic | Khan Academy
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/arithmetic-home/multiply-divide/place-value-area-models/v/lattice-multiplication Introduction to lattice multiplication Watch the next lesson: https://www.khanaca
From playlist Multiplication and division | Arithmetic | Khan Academy
There is no better way of understanding product groups than working through and example. In this video we look at the product group of the cyclic group with two elements and itself. The final result is isomorphic to what we call the Klein 4 group.
From playlist Abstract algebra