Andrew Critch

Office 852 Evans Hall Department of Mathematics, UC Berkeley Berkeley, California 94720-3840 Office Hours:   MonWed 6:00-7:00 @ 71 Evans E-mail:  critch at math dot berkeley dot edu

Quick links: Berkeley weather Student calendar Printer statuses Math faculty Math grads

about me | teaching | learning | videos | documentaries | textbooks | random math | random AG | work

About me

I'm Canadian, which means I am from Canadia. I was organically grown in Hillview, Newfoundland, where I learned how to be happy! I grew up speaking Newfoundish, and learned American from television. I graduated from Clarenville High School in 2004, then I studied math at Memorial University for almost two years, then at the University of Toronto for two years, and now I'm at Berkeley, and facebook is still evil :)

Why would anyone want to be a mathematician? (for those who don't buy the "it's just so beautiful" defense)
Pi = 0, or, how I learned to stop worrying and let delta depend on x
How to learn what an étale topos is

Teaching ('09 fall)

Math 1A (Single Variable Calculus) with Prof. Michael Christ: MW 5:00pm-6:00pm @ 71 Evans Hall
How to lose marks on math exams: a guide to getting less than you deserve!

Learning ('09 fall)

20 Questions Seminar, coorganized with Pablo Solis.
Student Seminar on Arithmetic Geometry with Martin Olsson.
Commutative Algebra and Algebraic Geometry Seminar with David Eisenbud
Student Algebraic Geometry Seminar, organized by Morgan Brown and Charley Crissman.
Seminars and colloquia this week at UC Berkeley.

Videos (fun math for everyone!)

How to turn a bubble inside out (part 1 / part 2). [10 mins / 10 mins] An excellent narrated animation for non-mathematicians which discusses the problem of turning a bubble inside-out, and a solution. Probably to avoid silly disputes, the video just calls it a sphere whose surface can pass through itself and stretch but cannot be creased or pinched infinitely tightly.

How to turn a bubble inside out efficiently. [22 secs] A super-efficent way to turn a bubble inside-out; slightly more difficult to describe over coffee.

Moebius transformations revealed. [2 mins] A nice video which explains Möbius transformations of the (complex) plane and shows how to visualize them using the Riemann sphere.

How to think about a 4th spatial dimension. [7 mins] I am always slightly disappointed when I hear people say things like "it's impossible to visualize 4 dimensions," when in fact it is possible, using projections, something we are already using when we visualize 3 dimensions! Carl Sagan explains this using some excellent choices of visual aids.

Look around you - Maths. [8 mins] Just watch it :)

New Math. [4 mins] A video illustrating Tom Lehrer's song about new-fangled math curricula for children.

Documentaries

Saturday morning science. [47 mins] A series of fun and simple experiments carried out on board the international space station, with possibly the most endearing narrator of all time. Hey, physics is math!

The boy with the incredible brain. [47 mins] A documentary about a savant with a capacity for vast mental calculation and memorization, not unlike the character "Rain Man" from the eponymous popular film. What's most incredible is that this man is does not suffer from any marked social handicap and is actually able to describe to us mere mortals what he is experiencing.

Fermat's last theorem. [45 mins] A documentary about the proof of Fermat's Last "theorem", a centuries-old mathematical problem that was finally solved in the 1990's; very inspiring.

Textbooks I often recommend

Henri Cartan - Elementary Theory of Analytic Functions of One or Several Complex Variables. "Complex analysis done right", in my opinion. Short and sweet (200 small pages), it's one of two math texts I've ever read perfectly in order from cover to cover. For the level of rigorous understanding of complex analysis it provides in a short time, it's a must read!

John M. Lee - Introduction to Smooth Manifolds. "Manifolds done right", the second of two math texts I've read in oder from cover to cover. This book is 600 pages of nicely planned, rigorous, pedagogical exposition on manifolds that won't let you down :)

Atiyah-Macdonald - Introduction to Commutative Algebra. Short and sweet (only 120 pages!), if you plan to study number theory or algebraic geometry, this is a great book to get you started with the bare essentials of commutative algebra. I especially like that it introduces Spec (the prime spectrum) of a commutative ring in the very first seciton of exercises.

Random math

• Self-assigned student numbers. How can N students assign their own unique student numbers without any student knowing anything but his own number? If you want the answer, email me! I don't think you'll find it anywhere else ;)

• Compact convergence does not imply local uniform convergence. A friend asked me if uniform convergence on compact sets implies local uniform convergence. On a locally compact space, these notions are equivalent. The space I made up for a counterexample to the general claim was interesting enough to me that I decided to write it up.

Random algebraic geometry

• Divisors and invertible sheaves (an arrowful tale). This is a one-page annotated commutative diagram summarizing the mappings that relate Weil and Cartier divisors and invertible sheaves, with references. (The green arrow is the one from my talk at the MAGIC seminar.)

• The long exact sequence of Tor (a worked example). Computing Tor modules can be done using any choice of projective resolution, and this is usually enough for most applications. However, finding the maps involved in the long exact sequence connecting them (which arises from tensoring a short exact sequence of modules with some other module) requires the "Snake Lemma" applied to an exact sequence of complexes constructed using the "Horseshoe Lemma." This is an example we worked out for our homological algebra seminar.

• Stalk-local detection of irreducibility. If a locally Noetherian scheme is connected and stalk-locally irreducible, then it is irreducible. This is sort of geometrically intuitive when you think of how, for example, reducibility of the variety {xy=0} in the plane can be detected in an "infinitely small" neighborhood of the origin; what I find neat is how this can be seen directly by using elementary algebra and Noetherianity instead of higher theorems (see link), and so maybe this would make a good introductory textbook exercise. Thanks to my officemate Adam Boocher for sharing an interest in this problem!

  (As an applicaiton of this, the "integral" hypothesis can be removed from Hartshorne's AG II.6.11.)

Some of my work in electronic format

• Generalized limits and limit extrema in topology. (Andrew Critch, Summer 2006) Here I provide a treatment of limits of relations in topological spaces, and introduce an operator which I call the "limit closure" that can often be used for topological arguments where "limsup" and "liminf" operators would be used in real analysis. I think it provides an interesting perspective for undergraduate analysis students.

• Resolving the Banach-Tarski Paradox: Inseparability of Rigid Bodies. (Andrew Critch, Spring 2006) Here I prove the Banach-Tarski theorem, which is usually interpreted to mean that a solid ball can be "taken apart" into finitely many rigid pieces and "put back together" to form two solid balls identical to the original. I then demonstrate that classical proofs of the theorem (including mine), which use radial symmetry of the ball, produce pieces which are "interlocked" like those of a jigsaw puzzle, and so in fact cannot be taken apart at all.

Old Content

Teaching ('09 summer)

Math 53 (Multivariable Calculus): MTWRF 12:00pm-2:00pm @ 3107 Etcheverry Evans Hall

Learning ('09 summer)

Toric Varieties workshop at MSRI with David Cox and Hal Schenck.
Seminars and colloquia this week at UC Berkeley.

Teaching ('09 spring)

Math 53 (Multivariable Calculus): MWF 5:00pm-6:00pm @ 75 Evans Hall

Learning ('09 spring)

Math 256B - Algebraic Geometry with Arthur Ogus.
Math 220 - Stochastic Methods in Applied Mathematics with Alexandre Chorin.
MAGIC seminar (Many Algebro-Geometrically Important Concepts), coorganized with Adam Boocher, Mike Daub, George Melvin, Damien Mondragon, Pablo Solis, Harold Williams, and Paul Ziegler.
MSRI 2009 Algebraic Geometry program, organized by William Fulton, Joe Harris, Brendan Hassett, János Kollár, Sándor Kovács, Robert Lazarsfeld, and Ravi Vakil.
Seminars and colloquia this week at UC Berkeley.

Teaching ('08 fall)

Math 53 (Multivariable Calculus): MWF 3:00pm-4:00pm @ 81 Evans Hall

Learning ('08 fall)

Math 256A - Algebraic Geometry
Math 254A - Number Theory
Math 300 - Teaching Workshop
HAPPY group (Hartshorne Additional Practice Problem Youth group)
HARD seminar (Homological Algebra Reading and Discussion seminar)
Seminars and colloquia this week at UC Berkeley.