Maximum modulus principle In this video, I talk about the maximum modulus principle, which says that the maximum of the modulus of a complex function is attained on the boundary. I also show that the same thing is true for the real and imaginary parts, and finally I discuss the strong max
From playlist Complex Analysis
Number Theory | Congruence Modulo n -- Definition and Examples
We define the notion of congruence modulo n among the integers. http://www.michael-penn.net
From playlist Modular Arithmetic and Linear Congruences
Alina Ostafe: Dynamical irreducibility of polynomials modulo primes
Abstract: In this talk we look at polynomials having the property that all compositional iterates are irreducible, which we call dynamical irreducible. After surveying some previous results (mostly over finite fields), we will concentrate on the question of the dynamical irreducibility of
From playlist Number Theory Down Under 9
Set Theory (Part 2): ZFC Axioms
Please feel free to leave comments/questions on the video and practice problems below! In this video, I introduce some common axioms in set theory using the Zermelo-Fraenkel w/ choice (ZFC) system. Five out of nine ZFC axioms are covered and the remaining four will be introduced in their
From playlist Set Theory by Mathoma
In this video we continue discussing congruences and, in particular, we discuss solutions of linear congruences. The content of this video corresponds to Section 4.4 of my book "Number Theory and Geometry" which you can find here: https://alozano.clas.uconn.edu/number-theory-and-geometry/
From playlist Number Theory and Geometry
Bas Spitters: Modal Dependent Type Theory and the Cubical Model
The lecture was held within the framework of the Hausdorff Trimester Program: Types, Sets and Constructions. Abstract: In recent years we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and c
From playlist Workshop: "Types, Homotopy, Type theory, and Verification"
This lecture is part of an online graduate course on Galois theory. We define the discriminant of a finite field extension, ans show that it is essentially the same as the discriminant of a minimal polynomial of a generator. We then give some applications to algebraic number fields. Corr
From playlist Galois theory
Abundant, Deficient, and Perfect Numbers ← number theory ← axioms
Integers vary wildly in how "divisible" they are. One way to measure divisibility is to add all the divisors. This leads to 3 categories of whole numbers: abundant, deficient, and perfect numbers. We show there are an infinite number of abundant and deficient numbers, and then talk abou
From playlist Number Theory
Richard Taylor "Reciprocity Laws" [2012]
Slides for this talk: https://drive.google.com/file/d/1cIDu5G8CTaEctU5qAKTYlEOIHztL1uzB/view?usp=sharing Richard Taylor "Reciprocity Laws" Abstract: Reciprocity laws provide a rule to count the number of solutions to a fixed polynomial equation, or system of polynomial equations, modu
From playlist Number Theory
Complex analysis: Maximum modulus principle
This lecture is part of an online undergraduate course on complex analysis. We prove the maximum modulus principle, and use to to prove the fundamental theorem of algebra and to find the symmetries of the unit disk. For the other lectures in the course see https://www.youtube.com/playli
From playlist Complex analysis
Theory of numbers: Congruences: Introduction
This lecture is part of an online undergraduate course on the theory of numbers. This lecture introduces congruences. We give some examples of using congruences to study the problem of which integers can be written as a sum of 2 or 3 squares or 3 cubes. For the other lectures in the
From playlist Theory of numbers
Supercongruences for Apery-like numbers by Brundaban Sahu
Program Workshop on Additive Combinatorics ORGANIZERS: S. D. Adhikari and D. S. Ramana DATE: 24 February 2020 to 06 March 2020 VENUE: Madhava Lecture Hall, ICTS Bangalore Additive combinatorics is an active branch of mathematics that interfaces with combinatorics, number theory, ergod
From playlist Workshop on Additive Combinatorics 2020
Galois Representations 2 by Shaunak Deo
PROGRAM : ELLIPTIC CURVES AND THE SPECIAL VALUES OF L-FUNCTIONS (ONLINE) ORGANIZERS : Ashay Burungale (California Institute of Technology, USA), Haruzo Hida (University of California, Los Angeles, USA), Somnath Jha (IIT - Kanpur, India) and Ye Tian (Chinese Academy of Sciences, China) DA
From playlist Elliptic Curves and the Special Values of L-functions (ONLINE)
Cryptography and Network Security by Prof. D. Mukhopadhyay, Department of Computer Science and Engineering, IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in
From playlist Computer - Cryptography and Network Security
Henri Darmon: Andrew Wiles' marvelous proof
Abstract: Pierre de Fermat famously claimed to have discovered “a truly marvelous proof” of his last theorem, which the margin in his copy of Diophantus' Arithmetica was too narrow to contain. Fermat's proof (if it ever existed!) is probably lost to posterity forever, while Andrew Wiles' p
From playlist Abel Lectures
Number Theory: Primitive Roots - Oxford Mathematics 2nd Year Student Lecture
Like many Universities around the world, Oxford has gone online for lockdown. So how do our student lectures look? In this, the second online lecture we are making widely available, world-renowned mathematician Ben Green introduces and delivers a short lecture on Primitive Roots, part of t
From playlist Oxford Mathematics 2nd Year Student Lectures
A number theory problem from Morocco!
We solve a viewer suggested number theory problem from a math contest in Morocco. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.com/stores/michael-penn-math Personal Website: http://www.michael-penn.net Randolph College Math: http:/
From playlist Math Contest Problems
Introduction To Beilinson--Kato Elements And Their Applications 4 by Chan-Ho Kim
PROGRAM : ELLIPTIC CURVES AND THE SPECIAL VALUES OF L-FUNCTIONS (ONLINE) ORGANIZERS : Ashay Burungale (California Institute of Technology, USA), Haruzo Hida (University of California, Los Angeles, USA), Somnath Jha (IIT - Kanpur, India) and Ye Tian (Chinese Academy of Sciences, China) DA
From playlist Elliptic Curves and the Special Values of L-functions (ONLINE)
Introduction to number theory lecture 29. Rings in number theory
This lecture is part of my Berkeley math 115 course "Introduction to number theory" For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj53L8sMbzIhhXSAOpuZ1Fov8 We show how to write several results in number theory, such as the Chines remainder theorem
From playlist Introduction to number theory (Berkeley Math 115)
Prove Infimums Exist with the Completeness Axiom | Real Analysis
The completeness axiom asserts that if A is a nonempty subset of the reals that is bounded above, then A has a least upper bound - called the supremum. This does not say anything about if greatest lower bounds - infimums exist for sets that are bounded below, but we can use the completenes
From playlist Real Analysis Exercises