Kickoff Party 2016!

Today marked the beginning of a new year for the BCMS!

In case you missed it, we talked about the upcoming events for the year, Dawei gave an awesome presentation on flat surfaces, and we had a lot of pizza (nobody seemed to know what type one of the pizzas was… maybe eggplant, but we’re not entirely sure).

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Some Upcoming Events for the Year:

Department Tea

  • Time: Wednesdays 4-5
  • Place: Maloney faculty lounge
  • Who: Anybody and everybody.

Introduction to \LaTeX

  • Time: September 21st at 5:30 pm
  • Place: Maloney 560
  • Who: Anybody new to LaTeX (i.e. are you in Intro to Abstract?) or anyone who wants to brush up on their typesetting skills

Career Panel

  • Time: October 13th at 5:30pm
  • Place: Higgins 310
  • Who: Anybody who wants to learn what math majors can do after college

Piersquared Tutoring

  • Details TBD
  • Talk to Chris Ratigan

Putnam Problem Solving Group

  • Time: Wednesdays (to be confirmed)
  • Place: Maloney 560
  • Who: People who like fun, math, and fun math

Movie Night

  • Time: later in the semester
  • Movie suggestions welcome

 

Problems That Were On The Board Today:

1.  You pick up a meter stick with 100 ants on it. Each ant walks 1 cm/s toward an end of the stick, and it reverses direction any time it encounters another ant. What’s the longest amount of time you’d have to wait for all ants to fall off of the end of the stick?

2.  Start from a place on the earth’s surface, travel south for 100 miles, east for 100 miles, and finally north for 100 miles and you get back to your starting point. Where did you start?

3.  How do you apply +, -, \cdot, /, (, ) to {3, 3, 8, 8} to get 24? Each number can only be used once (for example with {2, 2, 8, 8} we have (2 + 2) * 8 – 8 = 24).

4.  For what n, is n! equal to the number of seconds in 6 weeks?

5.  Let f be a real-valued function on the plane such that for every square ABCD in the plane, f(A)+ f(B)+ f(C)+ f(D) = 0 .  Does it follow that f(P) = 0 for all points P in the plane?

6.  Show that there exists a convex hexagon in the plane such that: (1) all its interior angles are equal and (2) its sides are 1,2,3,4,5,6 in some order

7.  Show that every even integer greater than 2 can be expressed as the sum of two primes (again, if you solve this question, make sure to let me know so we can write a paper together)

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Math is Everywhere

Almost everyone learns growing up that a line is 1 dimensional, a plane is 2 dimensional, and the space we live in is 3 dimensional. One day, a crazy theoretical physicist came along and argued that the universe consists of 4-dimensional space/time. That physicist’s name was Albert Einstein. A slightly less well-known name is Benoit Mandelbroit. Mandelbroit was a mathematician who argued something much more profound: dimensions are not discrete. The standard example of this phenomenon involves measuring the coastline of Britain. A Euclidean scientist would think that measuring Britain’s coastline with a smaller, more refined, ruler would give a more accurate measurement than a larger, less refined ruler would, and that with a small enough ruler, the length of Britain’s coastline could be determined to any degree of precision. This is actually not the case!

In fact, if the coastline of Great Britain is measured in units of 100 km, then its coast is roughly 2800 miles long. However, if our Euclidean scientist tries to refine this measurement by using units of 50 km, then the coast is 3,400 km long. That’s an increase of 600 km!

The reason for this apparent paradox is that unlike a boring line which has no new structure as you zoom in, Britain’s coastline has detailed structure on all levels of magnification. This new structure makes Britain’s coastline a naturally occurring fractal.

A true fractal is an object which is self-similar, meaning that as you zoom in on the fractal, you see smaller copies of it.

Examples of fractals abound. Three particularly interesting ones are: Pre-modern cities, the cells that make up the human body, and the stock market.

In pre-modern cities, particular cities which grew “naturally” out of collections of settlements, each group of people living in the city would want to have everything they need close to themselves. This

The fact that the cells that make up the human body are fractals is being used in cancer research. Any biology student would understand that the surfaces of cancer cells display different proteins than the surfaces of regular cells. Recent research has shown that cancer cells have a different fractal dimension than regular cells. This can be used to help better identify cancer cells and better design drugs targeted to the specific structure of the cancer cells.

The stock market is a very interesting example of a fractal because unlike the other two examples where fractals were used to find structure in pre-modern cities and normal cells, the fractal nature of the stock market means uncertainty. Fractals are not smooth, they jump around a lot on small distances. According to Nassim Taleb, this can be used for a different investment strategy. Instead of trying to predict which stocks will do well based on their history on the market, Taleb proposes investing small amounts in many different high-risk stocks. You will loose some money here and there, but all you need is one fractal spike on the market and you could make millions!

Believe it or not, your resting heartbeat is also a fractal and over time will naturally get to around 120 bpm without your noticing.