Algebraic geometry

Categorical quotient

In algebraic geometry, given a category C, a categorical quotient of an object X with action of a group G is a morphism that (i) is invariant; i.e., where is the given group action and p2 is the projection.(ii) satisfies the universal property: any morphism satisfying (i) uniquely factors through . One of the main motivations for the development of geometric invariant theory was the construction of a categorical quotient for varieties or schemes. Note need not be surjective. Also, if it exists, a categorical quotient is unique up to a canonical isomorphism. In practice, one takes C to be the category of varieties or the category of schemes over a fixed scheme. A categorical quotient is a universal categorical quotient if it is stable under base change: for any , is a categorical quotient. A basic result is that geometric quotients (e.g., ) and GIT quotients (e.g., ) are categorical quotients. (Wikipedia).

Video thumbnail

Quotient group example

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

Video thumbnail

PreCalculus | Finding the difference quotient: Example 3

We present a few examples of calculating the difference quotient. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

From playlist PreCalculus

Video thumbnail

Quotient groups

The idea of a quotient group follows easily from cosets and Lagrange's theorem. In this video, we start with a normal subgroup and develop the idea of a quotient group, by viewing each coset (together with the normal subgroup) as individual mathematical objects in a set. This set, under

From playlist Abstract algebra

Video thumbnail

Visual Group Theory, Lecture 3.5: Quotient groups

Visual Group Theory, Lecture 3.5: Quotient groups Like how a direct product can be thought of as a way to "multiply" two groups, a quotient is a way to "divide" a group by one of its subgroups. We start by defining this in terms of collapsing Cayley diagrams, until we get a conjecture abo

From playlist Visual Group Theory

Video thumbnail

PreCalculus | Find the difference quotient: Example 1

We present a few examples calculating the difference quotient of a function. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

From playlist PreCalculus

Video thumbnail

On finite dimensional omega-categorical structures (...) - P. Simon - Workshop 1 - CEB T1 2018

Pierre Simon (Berkeley) / 31.01.2018 On finite dimensional omega-categorical structures and NIP theories The study of omega-categorical structures lies at the intersection of model theory, combinatorics and group theory. Some classes of omega-categorical structures have been classified,

From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields

Video thumbnail

Simplify a rational expression by using properties of exponents

👉 Learn how to simplify expressions using the quotient rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the exponents of the term in the numerat

From playlist Simplify Using the Rules of Exponents | Quotient Rule

Video thumbnail

Quiver moduli and applications, Markus Reineke (Bochum), Lecture 3

Quiver moduli spaces are algebraic varieties encoding the continuous parameters of linear algebra type classification problems. In recent years their topological and geometric properties have been explored, and applications to, among others, Donaldson-Thomas and Gromov-Witten theory have

From playlist Felix Klein Lectures 2020: Quiver moduli and applications, Markus Reineke (Bochum)

Video thumbnail

Applying the quotient rule to simplify a rational expression

👉 Learn how to simplify expressions using the quotient rule and the negative exponent rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the expon

From playlist Simplify Using the Rules of Exponents

Video thumbnail

Simplify rational expression using the rules of exponents

👉 Learn how to simplify expressions using the quotient rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the exponents of the term in the numerat

From playlist Simplify Using the Rules of Exponents | Quotient Rule

Video thumbnail

Peng Shan: Coherent categorification of quantum loop sl(2)

'Coherent categorification of quantum loop sl(2)' Peng Shan (Tsinghua University) Abstract: We explain an equivalence of categories between a module category of quiver Hecke algebras associated with the Kronecker quiver and a category of equivariant perverse coherent sheaves on the nilpot

From playlist SMRI Algebra and Geometry Online

Video thumbnail

Gromov-Witten Theory and Gauge Theory (Lecture 2) by Constantin Teleman

PROGRAM: VORTEX MODULI ORGANIZERS: Nuno Romão (University of Augsburg, Germany) and Sushmita Venugopalan (IMSc, India) DATE & TIME: 06 February 2023 to 17 February 2023 VENUE: Ramanujan Lecture Hall, ICTS Bengaluru For a long time, the vortex equations and their associated self-dual fie

From playlist Vortex Moduli - 2023

Video thumbnail

Galois theory: Transcendental extensions

This lecture is part of an online graduate course on Galois theory. We describe transcendental extension of fields and transcendence bases. As applications we classify algebraically closed fields and show hw to define the dimension of an algebraic variety.

From playlist Galois theory

Video thumbnail

The affine Hecke category is a monoidal colimit - James Tao

Geometric and Modular Representation Theory Seminar Topic: The affine Hecke category is a monoidal colimit Speaker: James Tao Affiliation: Massachusetts Institute of Technology Date: February 24, 2021 For more video please visit http://video.ias.edu

From playlist Seminar on Geometric and Modular Representation Theory

Video thumbnail

Geordie Williamson: Langlands and Bezrukavnikov II Lecture 14

SMRI Seminar Series: 'Langlands correspondence and Bezrukavnikov’s equivalence' Geordie Williamson (University of Sydney) Abstract: The second part of the course focuses on affine Hecke algebras and their categorifications. Last year I discussed the local Langlands correspondence in bro

From playlist Geordie Williamson: Langlands correspondence and Bezrukavnikov’s equivalence

Video thumbnail

Simplify an expression by applying quotient rule of exponents

👉 Learn how to simplify expressions using the quotient rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the exponents of the term in the numerat

From playlist Simplify Using the Rules of Exponents | Quotient Rule

Video thumbnail

Ralph KAUFMANN - Categorical Interactions in Algebra, Geometry and Physics

Categorical Interactions in Algebra, Geometry and Physics: Cubical Structures and Truncations There are several interactions between algebra and geometry coming from polytopic complexes as for instance demonstrated by several versions of Deligne's conjecture. These are related through bl

From playlist Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday

Video thumbnail

Learn the basics for simplifying an expression using the rules of exponents

👉 Learn how to simplify expressions using the quotient rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the exponents of the term in the numerat

From playlist Simplify Using the Rules of Exponents | Quotient Rule

Related pages

Isomorphism | Quotient by an equivalence relation | Scheme (mathematics) | Geometric invariant theory | GIT quotient | Quotient stack | Algebraic geometry | Geometric quotient | Algebraic variety | Category (mathematics) | Group (mathematics) | Morphism