Lectures related to Pentagonal Number Theorem and Jacobi Identity

Harmonic Forms: Main Theorem

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Taylor Series: Convergence and Applications

Explores Taylor series convergence and applications in approximating functions and solving mathematical problems.

Fundamental Groups

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Differentiating under the integral sign

Explores differentiating under the integral sign and continuity of functions in integrals.

Theta functions: Properties and Transformations

Explores the properties and transformations of theta functions, including modular forms and lattice levels.

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Distribution & Interpolation Spaces

Explores distribution and interpolation spaces, showcasing their importance in mathematical analysis and the computations involved.

Weak Derivatives: Definition and Properties

Covers weak derivatives, their properties, and applications in functional analysis.

Implicit Functions Theorem

Covers the Implicit Functions Theorem, providing a general understanding of implicit functions.

Meromorphic Functions & Differentials

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

Sobolev Spaces in Higher Dimensions

Explores Sobolev spaces in higher dimensions, discussing derivatives, properties, and challenges with continuity.

Cartesian Product and Induction

Introduces Cartesian product and induction for proofs using integers and sets.

Probability Theory: Integration and Convergence

Covers topics in probability theory, focusing on uniform integrability and convergence theorems.

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Fourier Series: Extension and Periodicity

Covers the extension and periodicity of Fourier series and the interpretation of coefficients.

Linear Recurrence Relations

Explores linear recurrence relations, including examples like the Fibonacci numbers and the proof of related theorems.

Fourier Series: Understanding Periodicity and Coefficients

Explores t-periodic functions in Fourier series, discussing intervals, propositions, and variable changes for coefficient calculation and series convergence.

Lattices: Properties and Theta Functions

Explores lattices in Rd, Gram matrices, theta functions, and modular properties.

Central Limit Theorem

Covers the Central Limit Theorem and its application to random variables, proving convergence to a normal distribution.