A Few Interesting Answers to Some Hard Questions

Transcription

1 A Few Interesting Answers to Some Hard Questions Continuum Hypothesis Ordinal Numbers Odd Perfect Numbers N vs. NP - resolved!

2 I think developing an elevator speech is important, and I encourage all students to do this before they graduate. -Judy Walker In general, it's good to give people an idea that we're still working. Math isn't done. I don't play the role of a monk sitting in my office, keeping the store of all mathematical knowledge safe to pass on to the next generation. New technology like GPS or Google requires new mathematical thought and progress. You can't just look up how to do it in a book. -Tom Marley

3 Operator Theory What I usually do is discuss the idea of transformation via the idea of electronic communication: think of changing a signal such as your voice into a digital format. The sound waves are sampled and then converted into sequences of 0s and 1s. This is an example of a linear transformation. I then tell people that I don't study these transformations themselves, but I study collections of things like them, which have certain algebraic properties. -David Pitts One of the big ideas of 20th century science was quantum mechanics, and the key thing quantum mechanics found was that there's a fundamental limit to how well you can measure some things (speed vs position). Physicists found it comes down to the fact that in ordinary arithmetic when you multiply two numbers, it doesn't matter what order you do it in, but the quantum view builds a model of algebra where a times b and b times a need not be the same. Now the operator theory I study doesn't work with quantum theory directly, but looks at the algebra you get with families of these non-commuting variables together. So operator theory really just works to develop other parts of mathematics with the quantum mechanical perspective. -John Orr

4 Beatles Unknown "A Hard Day's Night" Chord Mystery Solved Using Fourier Transform instantly recognizable twang For 40 years, no one knew what chord George Harrison was playing on his 12-string Rickenbacker. Jason Brown of Dalhousie University applied Fourier Transforms to decompose the sound into its original frequencies, and parse the notes. Lennon played a six-string, Paul had his bass, Ringo was drums... none of them quite fit. In the end, Brown discovered that there had to be a piano playing an F note, which was impossible to play with the other notes on the guitar. Science 2.0, October 2008,

5 Diff Eq and PDEs Differential equations arise in many different areas of the physical sciences and engineering and also in diverse subjects such as medicine, psychology, economics, operations research to mention just a few. As simple examples, differential equations arise in the theory of electric circuits, gravitational equilibrium of a star, learning of tasks, vibrating strings, heat flow, wave motion, and so on. -Lynn Erbe In the sciences and engineering, mathematical models are developed to aid in the understanding of physical phenomena. These models often yield an equation that contains some derivatives of an unknown function. Such an equation is called a differential equation. Two examples of models developed in calculus are the free fall of a body and the decay of a radioactive substance. -"Fundamentals of Differential Equations"

6 Visual Effects in the Movies Fluid Animation: water, fire, air, smoke, snow, even fur or cloth You choose how, and at what rate, particles are emitted in your space, and you control their speed. Then you determine what kind of forces or acceleration they should be subject to. (gravity field) Fountain, Smoke, Sand The 30 second water sphere scene in Harry Potter took six months to simulate. Trade-off between accuracy (implementing the complex mathematical equations describing each physical process) and efficiency. +Plus Magazine, June 2009, plus.maths.org

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8 Math model may decrease phantom traffic jams Some traffic jams have no apparent cause no accident, no stalled vehicle, no construction. High Traffic Density Traffic jams can be modeled as a selfsustaining wave, similar to detonation waves produced by explosions. Jamitons MSNBC, June 2009

9 Coding Theory There are many examples of instances in which information needs to be transmitted reliably across a channel --- satellite pictures from outer space, cellular phone conversations, , etc.; even data storage (on a hard disk or a flash drive or cd, for example) can be thought of in this way. No matter how careful we are, errors are bound to occur. It is the goal of coding theory to find ways of adding redundancy to the information so that these errors can be detected and efficiently corrected. -Judy Walker

10 Detecting Altered Photos Consider the shadows... Tampering with an image leaves statistical traces in the file. If something is removed, part of the background will be copied, and the two parts of the file will be identical. AMS Math Moments, 2006

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14 Mathematician Receives Grammy Dr. Kevin Short of the University of New Hampshire won a Grammy award (historical category) in 2008 for his restoration of a 1949 wire recording of a live Woody Guthrie concert. It is the only known recording of Guthrie performing in concert. It was recorded by a Rutgers college student onto spools of wire. Once it was carefully transferred to a digital formal, Short used Chaotic Compression Technology to fill in the broken signal. Signal Processing: cell phone downloads, better hearing aids, more sensitive explosive materials detectors ScienceNews, 2008

15 Math Education Most simply, mathematics education research is inquiry by carefully developed research methods aimed at providing evidence about the nature and relationships of many mathematics learning and teaching phenomena. It seeks to clarify the phenomena, illuminate them, explain how they are related to other phenomena, and how this may be related to undergraduate mathematics course organization and teaching. - Mathematics Education Research: A Guide for the Research Mathematician by McNight2, Magid, & Murphy

16 Gut Instinct's Surprising Role in Math Two distinct number systems Approximate number system vs genuine computation Emphasize the power of the ballpark figure Fermi Problems Test The New York Times, September 2008

17 Truro Zoning Decision There was a vote on how soon hotels can be converted to condominiums, and the measure needed 2/3 vote to pass. The vote was 136 to 70. The town accountant computed 206 x.66 = But using.6666 a more accurate version of twothirds the affirmative vote needed to be 137 instead of 136, according to an anonymous caller to town hall and to the Times. Cape Cod Times, April 2009

18 Functional Integration You're doing Calculus but instead of using points on the real line (or complex plane), your points are functions. So you're integrating functions of functions (called functionals). How to explain Calculus to someone who's never had it? Calculus makes hard problems easy. It is useful for finding max/mins in real applications. -Dave Skoug

19 Do dogs know Calculus? Professor Tim Pennings of Hope College had an idea while playing fetch with his Welsh Corgi, Elvis, near Lake Michigan. Elvis could run faster than he could swim. Was Elvis's path optimal for retrieving the ball as quickly as possible? MAA - The College Mathematics Journal, May 2003

20 Let r denote the running speed, and s the swimming speed. Also, let's call z the total distance from A to B (through D). Finally, we will denote the time it takes to get to the ball as T(y). Then... is minimized by

21 On average, Elvis ran at r = 6.40 m/s and swam at s = m/s, so from before, we get a relationship of y = 0.144x. Pennings ran 35 throw and fetch trials with Elvis, measuring the values of x and y.

22 Commutative Algebra and Algebraic Geometry If I want to explain very briefly what I do, I say that I work with different number systems and finding solutions to systems of equations in these number systems. If I have more time or a chalkboard, I like to describe the geometric connections: you can think of a line in 3-space as the intersection of two planes. What about a surface in 4-space? Is it the intersection of two 3dimensional objects? -Tom Marley The motivating goal in Algebraic Geometry is to understand solution sets of systems of polynomial equations in several variables. Algebraic Geometry thus subsumes linear algebra, where the goal is to understand solutions sets of systems of linear equations in several variables, but non-linear equations turn out to be much harder to deal with. -Brian Harbourne

23 Mathematicians to thank for great graphics 100 powerful supercomputers perform geometrical, algebraic and calculus-based calculations to animate Pixar's characters. Tony DeRose: Without mathematics, we wouldn't have these visually rich environments, and visually rich characters. You didn't see any water in Toy Story, whereas by the time we got to Finding Nemo, we had the computer techniques that were needed to create all the splash effects. I remember as a mathematics student thinking, 'Well, where am I ever going to use simultaneous equations?' And I find myself using them every day, all the time now. The supercomputers run 24 hours a day, seven days a week, and it still takes them five to six hours to render a single frame lasting 1/24th of a second. So, for every second of film, it takes the computer six days. Science Daily, May 2007

24 Math Biology Whenever I say Math Bio, the response is usually a surprise that biology uses mathematics! So, I try to talk about the large amounts of data collection, using Ecology as an example, and get them to realize these are things they already knew Biologists did, and so using mathematics should be obvious. Then I explain that as a math biologist, I try to put together all of this vast amount of data and use it to create equations that will be able to make predictions about the future of the ecosystem. They usually think then that I'm a biology statistician, but it's hard to get across the idea of a model (as opposed to analyzing data from things that have already happened). There are a few examples I use to help with the concept of modeling: pest outbreaks, global climate change, and managing a fishery. -David Logan

25 YouTube Usage Decoded Why are certain videos on YouTube watched millions of times while 90 percent of the contributions find only the odd viewer? Top Videos of All Time Herd-like behavior Tracked viewer statistics for 5 million videos over two years Junk, Viral, and Quality videos Recognizing potential blockbusters early Science Daily, December 2008

26 Math Neuroscience A central challenge in neuroscience is to understand how neurons represent and process information about the world. Our brain is constantly constructing and updating a representation of the world around us, largely from sensory inputs (visual, auditory, tactile, etc.). How is this information represented in the brain? This requires understanding how neurons - and, more generally, populations of neurons working together in neural networks - encode and transform information that comes in through the senses. -Carina Curto

27 Improving Computer Language Recognition With most current programs, if you speak just a little too quickly or slowly, if your pronunciation isn t clear, or if there is background noise, the system often fails to work properly The brain classifies the various signals from the smallest, fast-changing components (single sound units like 'e' or 'u') up to big, slow-changing elements (the topic). Predicting the next speech sound Science Daily, August 2009

28 For emaxlpe... Arocdnicg to rsceearch at Cmabrigde Uinervtisy, it deosn t mttaer in waht oredr the ltteers in a wrod are, the olny iprmoatnt tihng is taht the frist and lsat ltteer are in the rghit pcale. The rset can be a toatl mses and you can sitll raed it wouthit pobelrm. Tihs is buseace the huamn mnid deos not raed ervey lteter by istlef, but the wrod as a wlohe.

29 Discrete Mathematics Usually I say that I study the structure of networks, using the specific example of the Internet and computer networks. -Stephen Hartke Combinatorics is a fancy word for counting. Combinatorics is concerned with determining the number of logical possibilities of some event without necessarily listing all the particular outcomes. One can often perform calculations involving probabilities simply by counting the possible outcomes. Combinatorics often requires counting the number of rearrangements or groupings of a set of objects. -MathCounts

30 Betting on March Madness 64 games determine the winner So a randomly chosen bracket has a 1 in 264 chance of being completely correct. If every person on Earth could fill out a bracket every second, then it would take them roughly one century to fill out all possibilities. Science Daily, March 2006

31 Groups, Semigroups, & Topology When you have an object and you bend it a little bit (or a lot!), the object hasn't really inherently changed; topology is about understanding what it is about the object that hasn't changed. As an example, what makes a ball a ball, and not a doughnut? Knot theory is really the topology of knotted circles; what is different about your shoes when they are tied as opposed to when they aren't? How many different ways can you tie your shoes? Since lots of physical objects can be modeled as knotted objects (molecules, elementary particles="strings"), the concepts and techniques of knot theory can be applied to many physical systems. -Mark Brittenham

32 Geometric Map of the Internet The researchers at San Diego Supercomputer Center and Cooperative Association for Internet Data Analysis at UC - San Diego have created the first geometric "atlas" of the Internet as part of a project to prevent our most ubiquitous form of communication from collapsing within the next decade or so. They discovered a latent hyperbolic, or negatively curved, space hidden beneath the Internet's topology, leading them to devise a method to create an Internet map using hyperbolic geometry. Such a map would lead to a more robust Internet routing architecture because it simplifies path-finding throughout the network. Internet Black Holes Science Daily, September 2010

33 Other Cool Applications of Math For people with very little background in math (say, someone who took high school math 25 years ago) I don't really try to say "what's operator theory." Instead I try to explain what pure math is and how it's different from grade school arithmetic. So I might talk about something from graph theory and say thats what pure mathematicians do, and thats how what they do connects to the modern world. -John Orr People always like real world applications (they never ask about applications of fine arts, but somehow math requires applications, rather than simply appreciating the beauty of the ideas). -David Pitts

34 Gallons per mile Which improvement will save more gas over the course of a year: changing from 34 mpg to 50mpg, or from 18 to 28mpg? How about 12mpg to 15mpg vs 30mpg to 60mpg? Manufacturers should list efficiency in terms of gallons per 10,000 miles driven. Duke University News, June 2008

35 Congressional Districts Two person fair division: I cut. You choose. Theoretically possible to double actual percentage Each party divides their part as they would like. A party with 40% of the vote could get 80% of the seats on their half of the state. If the other party received all the votes on the other half, than the original party will have 40% of the seats. Science News, February 2009

36 Basketball Free Throws Rick Barry NBA Career Free Throw Percentage of 89.3% In , it hit a high of 94.7%. The higher the arc, the larger the window is, so there's more room for error. The optimum angle for the shot is 45 degrees plus half the angle from the top of the player s hand to the rim. Discover Magazine, August 2008

37 The Donovan Index What is the most successful US pro-sports city? Start by dividing, for each franchise, the sum of the number of teams in the league during each year that franchise won a championship (championship points) by the number of seasons the team has existed. For example, the Phillies get 30 championship points for their World Series title in 2009 and 26 for their 1980 title. They've been in existence for 102 years, so that's an index of Doesn't adjust for population Wall Street Journal, The Numbers Guy, February 2009

38 SPF Dermatologists say this is just a numbers game that confuses customers.

39 SPF SPF measures how much the product shields the sun s shorter-wave UVB rays, which can cause sunburn. If adequately applied, sunscreens with sky-high SPFs offer slightly better protection against lobster-red burns than an SPF 30. In 2007, the FDA proposed capping SPF at 50+. Calculated by comparing the time needed for a person to burn unprotected with how long it takes for that person to burn wearing sunscreen Half the sunscreen = square root of the protection The New York Times, May 2009

40 Lewis Carroll A cup contains 50 spoonfuls of brandy, and another contains 50 spoonfuls of water. A spoonful of brandy is taken from the first cup and mixed into the second cup. Then a spoonful of the mixture is taken from the second cup and mixed into the first. Is there more or less brandy in the second cup than there is water in the first cup?

41 Final Random Facts th The 46 Mersenne Prime 42,643,801 (2-1) was one of TIME's best inventions of In 2009, the Wall Street Journal ranked Mathematician as the best job, and Lumberjack as the worst. 2 42,643,801