## The Quantum Gravity Topological Quantum Field Theory Blues by Scott Carter

Scott Carter has some poetry up on his website. He works on knotted surfaces, and I think this version is just great. Back in the day, I wanted to understand knotted surfaces and I remember looking at his work a bit.

I’ve been calculating

I said I’ve been calculating

calculating all night long

Got a quasi- triangular Hopf algebra

and I wrote down the coproduct wrong.I’ve been integrating

integrating the whole day through

I said I’ve been integrating

integrating the whole day through

Got a Chern-Simons functional integral

and its convergent, too.I’ve been writing down knot diagrams

converting them to braids

Using the Alexander isotopy

you know I’m not afraid

I’ve been

assigning modules

to each of these six strings

been doin’ it for weeks now

and I still don’t understand a thing.I’ve got them old Quantum Gravity

Topological Quantum Field Theory Blues

I’ve got them old Quantum Gravity

Topological Quantum Field Theory Blues

And without NSF funding I think that you would, too.

## Science Rendezvous!

This year, at Science Rendezvous, we shared symmetry and geometry. These areas of math are very beautiful and full of lovely patterns. In particular, we focused on how to connect geometry and symmetry using group theory. This approach was pioneered by Donald Coxeter, one of the most famous mathematicians of the twentieth century, and former professor at the University of Toronto. The big theme of our display was the notion of symmetry groups. This talk Symmetry and Groups by Professor Raymond Flood of Gresham College gives a great introduction to this connection.

Lukas brought his kaleidoscope, and I got it on video!

## Three-Dimensional Kaleidoscope

My highschool student, Lukas Boelling, made this three-dimensional icosahedral/dodecahedral kaleidoscope with his dad, @eric_boelling. Lukas based his models off this excellent paper: Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes by Roe Goodman. Stay tuned for more models!

## Time+Practice = Achievement

If I’ve learned nothing else, it’s that time and practice equal achievement.

## Math Club Geometry Training Session

These are some questions that I prepared for Math Club. The problems follow Paul Zeitz’s excellent book The Art and Craft of Problem Solving. You can find this hand-out is here:

https://pgadey.ca/teaching/2019-math-club/geometry-training-talk/geometry-training-talk.pdf (tex)

**1. Advice and Suggestions **

- Draw big diagrams. Lots of them.
- Use multiple colours to keep track on information.
- Draw several examples with different lengths/angles.
- Avoid coordinates.

**2. Facts and Questions **

Fact 1Congurence conditions for triangles: (SAS, ASA, SSS, AAS)

Side-angle-side, angle-side-angle, side-side-side, angle-angle-side.

Question 2Why is angle-side-sidenota congruence condition?

Question 4Show that using the Fact~3.

Fact 5 (Transversals)and are parallel iff .

Question 6Show that the sum of the interior angles of a triangle is . (Idea: Add a new parallel line.)

Question 7For any triangle , the exterior angle at is the sum of the interior angles at and .

Question 8 (Parallelograms)Use congruence and parallels to show:

- The opposite sides of a parallelogram have the same length.
- The opposite angles of a parallelogram are equal, and adjacent angles are “supplementary” (sum to ).
- The diagonals of a parallelogram bisect each other.

(Idea: Introduce triangles.)

Question 9 (Isoceles triangles)A triangle is isoceles if . Show that: If is isoceles then .

Question 10Let be a triangle. Let , , and be the midpoints of , , and respectively. Show that the lines: , , and , dissect in to congruent triangles.

(Big Idea: Introduce parallels to and at . These will create new “phantom points” on and . These new points will be very helpful because of our theory of parallelograms.)

Fact 11 (Circles and Chords)If any two are true, then all three are true.

- The line passes through the center of the circle.
- The line passes through the midpoint of the chord.
- The line line is perpendicular to the chord.

Fact 12 (Circles and Tangents)A tangent to a circle is perpendicular to the radius at the point of tangency. Also, a perpendicular to a tangent line placed at the point of tangency, will pass through the center of the center of the circle.

Question 13Suppose that is outside a circle. If and are tangent to the circle, then .

Question 14 (Inscribed Angle Theorem)Consider a circle centered at with points on the perimeter of the circle. Show that . (Idea: Chase angles.)

Question 15Consider a semi-circle with base .

If is on the perimeter of the semi-circle, show that .

Question 16 (Circumcircles)The circumcircle of a triangle is a circle passing through , , and .

- Suppose that a circumcircle exists. Show that its center is the intersection point of the perpendicular bisectors of the sides of the triangle. (If a circumcircle exists, then it has a unique center.)
- Given a triangle, any two perpendicular bisectors will intersect in a point equidistant from all three vertices. (Any triangle has a circumcenter.)
- All three perpendicular bisectors intersect in a unique point, the “circumcenter”.

Question 17 (Incircles)The incircle of a triangle is a circle tangent to , , and .

- Suppose that an incircle exists. Show that its center is the intersection of any two angle bisectors.
- Show that the intersection of any two angle bisectors is the center of an incircle and this point is unique.
- All three angle bisectors intersect in a unique point, the “incenter”.

**3. Contest Problems **

Question 18 (Canada 1991)Let be a circle and a point in the plain. Each line through that interesects determines a chord of . Show that the midpoints of these chords form a circle.

Question 19 (Rochester 2012)Let be an isoceles triangle with and . Consider and interior point of this triangle such that and . Find with proof, the measure .

Question 20 (Rochester 2013)Let be a triangle with . Suppose the bisectors of , , and meet , , and at points , , , respectively. Prove that is a right angled triangle.

## Math Club Number Theory Training Session

These are some questions that I prepared for Math Club. The problems follow Paul Zeitz’s excellent book The Art and Craft of Problem Solving. You can find this hand-out is here:

https://pgadey.ca/teaching/2019-math-club/number-theory-training-talk/number-theory-training-talk.pdf (tex)

**1. Advice and Suggestions **

- Try out lots of examples.
- The small numbers are your friends.

**2. Facts and Questions **

Fact 1If we write for the statement “ divides .”

Formally, means for .

Question 2What is the largest such that is divisible by ? Idea: Find a factorization where is a small constant.

Fact 3The “divisors” of are all such that . We say is “prime” if its divisors are . We say that is “composite” if it is not prime.

Fact 4 (Fundamental Theorem of Arithmetic)Any natural number is a product of a unique list of primes.

Question 5Show that is irrational. Generalize!

Question 6Show that there are infinitely many primes. Euclid’s idea: Suppose there are finitely many and consider .

Question 7Show that there are arbitrarily large gaps between primes. That is, show that for any there are consecutive numbers which are all composite.

Question 8 (Germany 1995)Consider the sequence and . Show that this sequence contains infinitely many composite numbers.

**3. Congruence **

Fact 9 (The Division Algorithm)For any there is a unique pair such that and .

Fact 10We write if . For any there is \mbox{} such that . We say that “ is congruent to modulo ”. Congruence preserves the usual rules of arithmetic regarding addition and multiplication.

Question 11Suppose that has digits in decimal notation.

- Show that .
- Show that .
- Show that .
- Show that .

Question 12What are the last two digits of ?

Question 13Show that any perfect square is congruent to or . Conclude that no element of is a perfect square.

Question 14Show that 3 never divides .

**4. The Euclidean Algorithm **

Fact 15The “greatest common divisor” of and is:

Question 16Show that where and is the unique pair of numbers given by the division algorithm.

Question 17The Fibonacci numbers are defined so that , and for . Show that .

The Fibonacci numbers have the following curious property: Consecutive Fibonacci numbers are the worst-case scenario for the Euclidean Algorithm. In 1844, Gabriel Lamé showed: If then the Euclidean algorithm takes at most steps to calculate . Check out this great write-up at Cut the Knot.

** 4.1. Parity **

Question 18Suppose that is odd and is a permutation. Show that the number

must be even.

Question 19A room starts empty. Every minute, either one person enters or two people leave. Can the room contain people after minutes?

Idea: Consider the “mod-3 parity” of room population.

**5. Contest Problems **

Question 20Show that is not an integer for any .

Idea: Consider the largest power . Divide out by this largest power. This will make all of the denominators odd. (In fancy number theory terms, you’re using a 2-adic valuation.)

Question 21 (Rochester 2012)Consider the positive integers less than or equal to one trillion, i.e. . Prove that less than a tenth of them can be expressed in the form where , , and are positive integers.

Idea: None of , , or can be very big. For example, .

Question 22 (Rochester 2003)An -digit number is “-transposable” if and . For example, is -transposable. Show that there are two 6-digit numbers which are 3-transposable and find them.

\noindent Big Idea: Consider repeating decimal expansions.

Observe that .

Find a number with a repeating decimal of length six.

Question 23Suppose that you write the numbers on the blackboard. You now proceed as follows: pick two numbers and , erase them from the board, and replace them with . Continue until there is a single number left. Does this number depend on the choices you made?

## MAT B41 — Exam Multiple Choice Statistics

MAT B41 wrote their exam today!

The multiple choice have been graded and we got the following information.

It looks like the class did alright, the average is quite good.

*Average:* 70

*Median:* 73

*Standard deviation:* 16.7

These are just the statistics for the multiple choice questions.

The full grades should be available early next week.

## Science Unlimited — Knot Theory and Cat’s Cradle: A Brief Introduction to Storer Calculus

The handout for the talk is available here:

https://pgadey.ca/teaching/talks/science-unlimited-2018-storer-calculus.pdf

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