275A, Notes 0: Foundations of probability theory | What's new

21 days ago by joecamel

"Starting this week, I will be teaching an introductory graduate course (Math 275A) on probability theory here at UCLA. While I find myself using probabilistic methods routinely nowadays in my research (for instance, the probabilistic concept of Shannon entropy played a crucial role in my recent paper on the Chowla and Elliott conjectures, and random multiplicative functions similarly played a central role in the paper on the Erdos discrepancy problem), this will actually be the first time I will be teaching a course on probability itself (although I did give a course on random matrix theory some years ago that presumed familiarity with graduate-level probability theory)."

math
probability
statistics
intro
terrytao
blog
post
21 days ago by joecamel

Geometric Algebra / Observable

22 days ago by joecamel

Nice big-picture intro (exterior algebra, differential forms, exterior derivative, cohomology, dirac matrices)

math
intro
geometric.algebra
observable
interactive
22 days ago by joecamel

The Mathematical Surprises of Triangles, Squares and Pentagons - Scientific American Blog Network

10 weeks ago by joecamel

On his 101st birthday, Martin Gardner's legacy continues to inspire new breakthroughs

2015
martin.gardner
tiles
math
10 weeks ago by joecamel

Decades-Old Graph Problem Yields to Amateur Mathematician | Quanta Magazine

april 2018 by joecamel

"By making the first progress on the “chromatic number of the plane” problem in over 60 years, an anti-aging pundit has achieved mathematical immortality." hn: https://news.ycombinator.com/item?id=16860074

math
graph.theory
biology
aubrey.de.grey
quanta
intro
inspiration
april 2018 by joecamel

>> Compare this to number theory, where every interesting extant problem appears... | Hacker News

april 2018 by joecamel

I don't often come here to comment but as someone in progress on an original research masters in number theory I can say this is utter bullshit. I assume your 'interesting' qualification (somehow) excludes obvious candidates like Landau's problems [0]. Some examples. I was taught about the ABC conjecture as an undergrad. You can easily teach the Brun sieve [1] method of working out that the sum of the reciprocal of the twin primes converges...

numbertheory
math
hn
discussion
intro
overview
interesting
inspiration
april 2018 by joecamel

Companion webpage to the book "Mathematics for Machine Learning"

april 2018 by joecamel

We are in the process of writing a book on Mathematics for Machine Learning that motivates people to learn mathematical concepts. The book is not intended to cover advanced machine learning techniques because there are already plenty of books doing this. Instead, we aim to provide the necessary mathematical skills to read those other books.

ml
book
overview
math
april 2018 by joecamel

Kalman exercise

march 2018 by joecamel

"Quick explanation of how a Kalman filter works just to make sure I understood it and I don't forget it, and also for others to learn. "

HN: "Nice summary. I think of the concept at a slightly higher level of abstraction:

It's just Bayesian inference to update the posterior. A theoretical model gives an estimate of what the state of the system should be (call that the "prior"). A sensor measurement gives an independent estimate of the state (call that the "likelihood"). Composing (multiplying) the two gives the posterior. Further, in this formulation it doesn't really matter what you call the prior or the posterior. You can easily combine estimates from many different sources/sensors.

Kalman filters are just the special case where the likelihood and prior are both Gaussian -- the distributions can be specified with a couple of numbers, and there is a simple closed-form expression for composing them. More generally, one can use all the tools available for Bayesian inference." https://news.ycombinator.com/item?id=16575679

kalman.filter
signal.processing
math
robotics
intro
example
HN: "Nice summary. I think of the concept at a slightly higher level of abstraction:

It's just Bayesian inference to update the posterior. A theoretical model gives an estimate of what the state of the system should be (call that the "prior"). A sensor measurement gives an independent estimate of the state (call that the "likelihood"). Composing (multiplying) the two gives the posterior. Further, in this formulation it doesn't really matter what you call the prior or the posterior. You can easily combine estimates from many different sources/sensors.

Kalman filters are just the special case where the likelihood and prior are both Gaussian -- the distributions can be specified with a couple of numbers, and there is a simple closed-form expression for composing them. More generally, one can use all the tools available for Bayesian inference." https://news.ycombinator.com/item?id=16575679

march 2018 by joecamel

Group theory originated in a very visual and geometric ways studying the symmetr... | Hacker News

february 2018 by joecamel

Some discussion on HN about group theory and education

hn
discussion
math
group.theory
algebra
education
february 2018 by joecamel

Clayton Shonkwiler: 15 Views of the Hypersphere - YouTube

january 2018 by joecamel

"This video and talk are aimed at undergraduates and advanced high-schoolers, but anyone interested in mathematics should love the material being presented."

math
video
education
youtube
high.school
visualization
january 2018 by joecamel

Stanford Lecture: Don Knuth's Christmas Tree Lecture 2017 - YouTube

january 2018 by joecamel

From HN: "I attended this on Thursday. Knuth started with a problem about rectangles inside rectangles (https://imgur.com/faWRt2q — it's going to be an exercise in 7.2.2.1 of TAOCP when that's published, currently in the draft version of Pre-Fascicle 5C). He worked through some small cases, made a conjecture, showed a problem submitted to the Monthly, and lots of cool stuff with generating functions. The lecture was also peppered with jokes and cool stories, including a fascinating conjecture by Bill Gosper, who has a long history of coming up with these Ramanujan-like identities. He also showed a wonderful conjecture (involving queens on an infinite chessboard) that he thinks may never be proved, and showed a snippet of his CWEB program for the problem." https://news.ycombinator.com/item?id=15898919

knuth
video
youtube
stanford
talk
lecture
math
cs
january 2018 by joecamel

My unusual hobby | Stephan Boyer

november 2017 by joecamel

"I think of Coq as an extension of my own ability to reason logically. When I’m in doubt about something, I open the Coq IDE and try to prove it. I think the reason this is so valuable to me is that I often mull over functional programming, types, logic, algorithms, etc. These kinds of things are well-suited to formalization."

coq
type.theory
plt
programming
proof
math
november 2017 by joecamel

Chicago undergraduate mathematics bibliography

november 2017 by joecamel

Somehow I became the canonical undergraduate source for bibliographical references, so I thought I would leave a list behind before I graduated. I list the books I have found useful in my wanderings through mathematics (in a few cases, those I found especially unuseful), and give short descriptions and comparisons within each category. I hope that this list may serve as a useful “road map” to other undergraduates picking their way through Eckhart Library. In the end, of course, you must explore on your own; but the list may save you a few days wasted reading books at the wrong level or with the wrong emphasis.

bibliography
books
list
curriculum
math
education
november 2017 by joecamel

career - What's a mathematician to do? - MathOverflow

october 2017 by joecamel

"The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding." Bill Thurston

math
philosophy
mathematics
mathoverflow
discussion
education
october 2017 by joecamel

Mathematics of Juggling - CornellCast

september 2017 by joecamel

Allen Knutson, Cornell professor of mathematics and former world record-holding juggler, gives a public demonstration and lecture on how juggling patterns can be represented the mathematically.

math
juggling
interesting
inspiration
september 2017 by joecamel

Sunset Geometry

september 2017 by joecamel

"Vanderbei’s analysis is an elegant and subtle exercise in classical trigonometry. In this post, I would like to present an alternative analysis in a different language: Geometric Algebra."

geometric.algebra
math
geometry
trigonometry
september 2017 by joecamel

On Norbert Blum’s claimed proof that P does not equal NP | in theory

august 2017 by joecamel

"so let me apologize by summarizing the claims in the paper."

math
tcs
pvsnp
luca.trevisan
2017
overview
august 2017 by joecamel

Statistician Proves Gaussian Correlation Inequality | Quanta Magazine

august 2017 by joecamel

"When a German retiree proved a famous long-standing mathematical conjecture, the response was underwhelming."

math
statistics
research
probability
inequality
quanta
story
academia
august 2017 by joecamel

I am also an ACM ICPC finalist... | Hacker News

july 2017 by joecamel

A comment on skills acquired from prog-competition experience.

hn
programming
competitive.programming
experience
discussion
career
math
education
july 2017 by joecamel

What Tensors Are For! - YouTube

june 2017 by joecamel

An intro video into tensor calculus lectures. Good motivation and high level overview (geometry - algebra).

video
youtube
tensors
tensor.calculus
math
analysis
intro
high.level
overview
june 2017 by joecamel

Random: Probability, Mathematical Statistics, Stochastic Processes

april 2017 by joecamel

"Random (formerly Virtual Laboratories in Probability and Statistics) is a website devoted to probability, mathematical statistics, and stochastic processes, and is intended for teachers and students of these subjects."

math
probability
statistics
tutorial
visualization
interactive
intro
april 2017 by joecamel

The Abel Prize 2017 - Yves Meyer - YouTube

march 2017 by joecamel

"for his pivotal role in the development of the mathematical theory of wavelets."

math
youtube
talk
overview
wavelets
signal.processing
terrytao
intro
abel.prize
2017
march 2017 by joecamel

Is “the theory of everything” merely the ultimate ensemble theory?

february 2017 by joecamel

The paper contains an interesting figure that illustrates relationships between various basic mathematical structures.

math
paper
visualization
overview
algebra
february 2017 by joecamel

Modes, Medians and Means: A Unifying Perspective

february 2017 by joecamel

"We’ve just seen that the mode, median and mean all arise from a simple parametric process in which we try to minimize the average discrepancy between a single number ss and a list of numbers, x1,x2,…,xnx1,x2,…,xn that we try to summarize using ss. In a future blog post, I’ll describe how the ideas we’ve just introduced relate to the concept of LpLp norms."

statistics
intro
blog
post
math
february 2017 by joecamel

Can a Chess Piece Explain Markov Chains? | Infinite Series - YouTube

january 2017 by joecamel

A cool example of stationary distribution theorem (expected time of return to starting state).

youtube
video
math
markov
probability
intro
example
education
january 2017 by joecamel

[1612.09375v1] Basic Category Theory

january 2017 by joecamel

This short introduction to category theory is for readers with relatively little mathematical background. At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties the three together.

math
pdf
arxiv
category.theory
intro
introduction
january 2017 by joecamel

The moving sofa problem — Dan Romik's home page

january 2017 by joecamel

"The mathematician Leo Moser posed in 1966 the following curious mathematical problem: what is the shape of largest area in the plane that can be moved around a right-angled corner in a two-dimensional hallway of width 1? This question became known as the moving sofa problem, and is still unsolved fifty years after it was first asked."

math
example
introduction
visualization
beginner
education
january 2017 by joecamel

Mathematical Components

december 2016 by joecamel

Mathematical Components is the name of a library of formalized mathematics for the Coq system. It covers a variety of topics, from the theory of basic data structures (e.g., numbers, lists, finite sets) to advanced results in various flavors of algebra. This library constitutes the infrastructure for the machine checked proofs of the Four Color Theorem and of the Odd Order Theorem.

proof
coq
book
math
cs
december 2016 by joecamel

Course Catalogue | The Theoretical Minimum

december 2016 by joecamel

"The Theoretical Minimum courses include a core sequence of six courses, plus a set of supplemental courses that teach additional related material. The core sequence is currently being repeated with Statistical Mechanics being taught during Spring quarter, 2013."

course
physics
math
video
list
learning
education
leonard.susskind
december 2016 by joecamel

Clifford Algebra: A visual introduction | slehar

november 2016 by joecamel

"Clifford Algebra, a.k.a. Geometric Algebra, is a most extraordinary synergistic confluence of a diverse range of specialized mathematical fields, each with its own methods and formalisms, all of which find a single unified formalism under Clifford Algebra. It is a unifying language for mathematics, and a revealing language for physics."

algebra
math
clifford
introduction
motivation
november 2016 by joecamel

How Quaternions encode rotations: derivation and sample code | Hacker News

september 2016 by joecamel

Discussion and resources on HN

hn
math
quaternions
3d
discussion
list
september 2016 by joecamel

Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics

september 2016 by joecamel

"We can do better! Research on the design and use of

mathematical systems provides a guide for designing a unified mathematical language for the whole of physics that facilitates learning and enhances physical insight. This has produced a comprehensive language called Geometric Algebra, which I introduce with emphasis on how it simplifies and integrates classical and quantum physics."

algebra
math
mathematics
physics
geometric.algebra
2002
paper
talk
lecture
oersted
pdf
read
mathematical systems provides a guide for designing a unified mathematical language for the whole of physics that facilitates learning and enhances physical insight. This has produced a comprehensive language called Geometric Algebra, which I introduce with emphasis on how it simplifies and integrates classical and quantum physics."

september 2016 by joecamel

[1609.01421] Struggles with the Continuum

september 2016 by joecamel

"Our assumption that spacetime is a continuum leads to many challenges in mathematical physics. Singularities, divergent integrals and the like threaten many of our favorite theories, from Newtonian gravity to classical electrodynamics, quantum electrodynamics and the Standard Model. In general relativity, singularities are intimately connected to some of the theory's most dramatic successful predictions. We survey these problems and the large amount of work that has gone into dealing with them."

math
mathematics
physics
arxiv
pdf
paper
john.baez
overview
history
september 2016 by joecamel

Great Mathematicians on Math Competitions and "Genius" - Less Wrong

july 2016 by joecamel

"As I mentioned in Fields Medalists on School Mathematics, school mathematics usually gives a heavily distorted picture of mathematical practice. It's common for bright young people to participate in math competitions, an activity which is closer to that of mathematical practice. Unfortunately, while math competitions may be more representative of mathematical practice than school mathematics, math competitions are themselves greatly misleading. Furthermore, they've become tied to a misleading mythological conception of "genius." I've collected relevant quotations below."

education
math
mathematics
interviews
blog
post
lesswrong
july 2016 by joecamel

Book of Proof by Richard Hammack

june 2016 by joecamel

This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics' Open Textbook Initiative.

book
math
mathematics
proofs
introduction
june 2016 by joecamel

Singular Value Decomposition Part 2: Theorem, Proof, Algorithm | Math ∩ Programming

june 2016 by joecamel

’m just going to jump right into the definitions and rigor, so if you haven’t read the previous post motivating the singular value decomposition, go back and do that first. This post will be theorem, proof, algorithm, data. The data set we test on is a thousand-story CNN news data set. All of the data, code, and examples used in this post is in a github repository, as usual.

HN: https://news.ycombinator.com/item?id=11710154

math
linearalgebra
algorithm
blog
post
svd
HN: https://news.ycombinator.com/item?id=11710154

june 2016 by joecamel

The HoTT book | Mathematics and Computation

november 2014 by joecamel

We are a group of two dozen mathematicians who wrote a 600 page book in less than half a year.

book
github
math
opensource
writing
hott
category.theory
collaboration
type.theory
2013
november 2014 by joecamel

Modern Algebra | Mathematics | MIT OpenCourseWare

november 2014 by joecamel

This undergraduate course focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics.

math
algebra
lecturenotes
mit
ocw
course
november 2014 by joecamel

How to talk Mathematics - P. R. Halmos

november 2014 by joecamel

The purpose of what follows is to suggest to a young mathematician what he might do (and what he had better not do) the first few times that he gives a public lecture on his subject.

math
halmos
advice
november 2014 by joecamel

Subhash Khot, Unique Games Conjecturer, Is Awarded Rolf Nevanlinna Prize | Simons Foundation

august 2014 by joecamel

Subhash Khot’s bold conjecture is helping mathematicians explore the precise limits of computation.

tcs
video
simons
article
math
prize
august 2014 by joecamel

Maryam Mirzakhani Is First Woman Fields Medalist | Simons Foundation

august 2014 by joecamel

Maryam Mirzakhani’s monumental work draws deep connections between topology, geometry and dynamical systems.

math
science
fields.medal
article
2014
august 2014 by joecamel

Martin Gardner's classic napkin ring puzzle | Hacker News

july 2014 by joecamel

One of those cases where using integrals rather than geometry is much simpler...

hn
math
martin.gardner
discussion
analysis
puzzle
july 2014 by joecamel

Primers | Math ∩ Programming

june 2014 by joecamel

"As a fair warning to the reader, these primers are a bit more terse than what you’d find in your average textbook."

algorithms
programming
math
reference
tutorials
june 2014 by joecamel

How to Conquer Tensorphobia | Math ∩ Programming

june 2014 by joecamel

"This post is an attempt to bridge the gap between the elementary and advanced understandings of tensors. We’ll start with the elementary (axiomatic) approach, just to get a good feel for the objects we’re working with and their essential properties. Then we’ll transition to the “universal” mode of thought, with the express purpose of enlightening us as to why the properties are both necessary and natural."

algebra
math
mathematics
programming
tensors
introduction
explanation
blog
cs
june 2014 by joecamel

Essential Math for Games Programmers

may 2014 by joecamel

"This tutorial deepens the approach of the previous years' Essential Math for Games Programmers, by spending one day on general math topics, and one day focusing in on the topic of physical simulation. It, like the previous tutorials, provides a toolbox of techniques for programmers, with references and links for those looking for more information."

math
game
graphics
programming
list
slides
resources
may 2014 by joecamel

Demystifying the Fourier magic | XRDS

may 2014 by joecamel

"Over the years I have gotten used to seeing many theorems in theoretical computer science being proved using discrete Fourier analysis. The Walsh-Fourier (Hadamard) transform of Boolean functions proved to be extremely useful in virtually every subfield of theoretical computer science, including PCPs, property testing, pseudorandomness, and communication complexity. As it turns out, many seemingly hard problems can be solved by writing the Walsh-Fourier expansion and using basic theorems of harmonic analysis."

cs
math
tcs
article
fourier.transform
pcp
may 2014 by joecamel

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