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Lecture# Harmonic Forms and Riemann Surfaces

Description

This lecture covers the concept of harmonic forms on Riemann surfaces, exploring the Main Theorem of compact Riemann surfaces and the uniqueness of solutions to harmonic equations. It delves into the properties of holomorphic and anti-holomorphic forms, as well as the Riemann bilinear identity. The instructor discusses the representation of differential forms and the conjugation operator. The lecture emphasizes the importance of harmonic differential forms and their applications in complex analysis.

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In course

Instructor

MATH-680: Monstrous moonshine

The monstrous moonshine is an unexpected connection between the Monster group and modular functions. In the course we will explain the statement of the conjecture and study the main ideas and concepts

Related concepts (384)

Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".

Proof theory

Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Proof (truth)

A proof is sufficient evidence or a sufficient argument for the truth of a proposition. The concept applies in a variety of disciplines, with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written communication such as conversation, dialog, rhetoric, etc., a proof is a persuasive perlocutionary speech act, which demonstrates the truth of a proposition.

Proof by contradiction

In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved.

Quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K. If , and the quadratic form equals zero only when all variables are simultaneously zero, then it is a definite quadratic form; otherwise it is an isotropic quadratic form.

Related lectures (1,000)

Harmonic Forms: Main Theorem

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Differential Forms Integration

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

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Explores meromorphic functions, poles, residues, orders, divisors, and the Riemann-Roch theorem.

Differential Forms on Manifolds

Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.

Open Mapping Theorem

Explains the Open Mapping Theorem for holomorphic maps between Riemann surfaces.