Partition Theory and Euler's Pentagonal Number Theorem

Speaker Name: Derek Habermas (habermds at potsdam dot edu)
Speaker Afiliation: SUNY-Potsdam

Abstract: Take a natural number, like 7. How many ways can you write that number as the sum of positive numbers? For example, 7 = 6+1 = 5+2 = 5+1+1 = ...  What if you require that the numbers be distinct? Or odd? Or congruent to 2 or 3 mod 4? These sums are called partitions, and partition theory has been around for almost 300 years attracting the attention of great minds such as Euler and Ramanujan. We will explore common questions and research tools used in this area, including Ferrers graphs and generating functions, culminating in a proof of Euler's beautiful Pentagonal Number Theorem. We will also survey a brief history of the subject, and state some recent (announced in January 2011) significant progress of someone who was almost my Mathematical grandfather.

Target Audience: We use the formula for the sum of an infinite geometric series, and we introduce generating functions. All other material is elementary.

Two-faced: The Cantor set and notions of size

Speaker Name: Emilie Wiesner (ewiesner at ithaca dot edu)
Speaker Afiliation: Ithaca College

Abstract: The Cantor set is a specially constructed infinite set; it has earned its name from an appearance in an 1883 paper of the mathematician Georg Cantor.  The Cantor set has many remarkable properties, and I will be talking about a few of them.  In particular, I'll discuss how two ways of defining size ("measure" and "cardinality") lead to two very different ideas of how big this set really is!

Target Audience: The talk doesn't require any specific mathematical knowledge.

Using Video to Understand How Leaves Breathe

Speaker Name: Aaron Luttman (aluttman at clarkson dot edu )

Speaker Afiliation: Clarkson University

Abstract: In order to engage in photosynthesis, leaves need to absorb both carbon dioxide and water.


Water is absorbed from the ground through the root system, but carbon dioxide must be absorbed directly into each leaf through pores in the surface called stomata. When these pores open, more CO_2 is absorbed, which is good for photosynthesis, but water is also lost, which is bad for photosynthesis. Thus each leaf must regulate its stomata so they take in enough CO_2 without losing too much water. This is a problem, because leaves have no brain to direct their stomata about what to do. It turns out that the opening and closing of the stomata can be observed and recorded on video, and mathematical methods for analyzing video can be used to describe the patterns that occur. In this presentation, we will show leaves in the act of "breathing" and demonstrate some of the mathematics used to analyze the breathing patterns.

Target Audience: This talk is aimed at anyone with an interest in applications of mathematical image processing to botany. The background required is really only a basic understanding of photosynthesis. Calculus III is a plus, but it is definitely not required.

Using Linear Algebra to Fix Your Pictures of Outer Space

Speaker Name: Aaron Luttman (aluttman at clarkson dot edu )

Speaker Afiliation: Clarkson University

Abstract: Pictures taken by telescopes on the ground - like those at Kitt Peak in Arizona or Mauna Kea in Hawaii - suffer from two primary problems.

The first is that the images are noisy. This means that random errors cause the picture to not look like the real scene of outer space. The second problem is that the pictures are blurry, which is a systematic (i.e. not random) phenomenon caused by the atmosphere. It turns out that we can mathematically "undo" the problems of noise and blur by using a mathematical model that looks a lot like a problem in linear algebra. In fact, some of the basic ideas of solving linear systems in linear algebra can be applied directly to find out what the picture would look like if there were no problems with noise and blur, given only a few basic assumptions about what the objects we're looking at really look like.

Target Audience: This presentation is aimed at students (or faculty) who have some background in linear algebra or at least solving linear systems of equations.

The CSI Effect: Television and Technology in Law and Society

Speaker Name: Aaron Luttman (aluttman at clarkson dot edu )

Speaker Afiliation: Clarkson University

Abstract: Over the last 15 years television has been inundated with programs that focus on the science of solving crimes.

The actual science in these shows is often misrepresented, and the forensic scientists often solve unsolvable problems in a matter of a few hours. This has actually led to a measured change in the way prosecutors and defense attorneys select juries, as it is believed that seeing the "fake" science on television has changed jurors attitudes towards forensic evidence in actual cases. Jurors now want to see better scientific results than is actually possible, because they have seen these good results on TV. We'll discuss the current state of research on the "CSI Effect," and show a particular example to demonstrate the state of the art in forensic imaging and how it is represented in the courtroom.

Target Audience: This is essentially a sociology talk, requiring only high school social studies. Watching TV a plus!

The Mathematics of Your Lifetime: Mathematical Advances of the Last and Next 20 Years

Speaker Name: Aaron Luttman (aluttman at clarkson dot edu )

Speaker Afiliation: Clarkson University

Abstract: Throughout our mathematical training from elementary school through the first few years of college, we're taught mathematics as facts and ideas that were invented or discovered by people hundreds or even thousands of years ago.

This makes it easy for us to think of mathematics as being complete, as if there's nothing left to uncover or create. The reality is that we live in a wonderfully exciting time of mathematical innovation and development, and new breakthroughs are occurring almost daily. In this presentation we will look back at some of the most exciting mathematical developments of the last 20 years - from the solutions of centuries-old problems in pure mathematics like Fermat's Last Theorem and the Poincare Conjecture to mathematical transformations of applications such as medical imaging and quantum computing - and we'll look forward to the possible advances that today's students will have the opportunity to make in the next 20 years.

Target Audience: This talk is aimed at any college students with an interest in mathematics. A first course in calculus will be helpful, but no prior knowledge of the problems discussed will be assumed.

There is value in mathematical games

Speaker Name: Ryan Gantner (rgantner at sjfc dot edu )

Speaker Afiliation: St. John Fisher College

Abstract: There are many simple games that can be played which have interesting mathematical interpretations and analyses.


In this talk, we'll analyze (and play) a few of these games, most of them cousins of the game of nim. We'll develop strategies for playing these games which depend on mathematical properties. We may also talk about the game of Hackenbush.

Target Audience: While there is plenty of math in this talk, there aren't any specific prerequisites aside from being able to add integers (and even that will be thrown out the window!).

The stochastic voter model

Speaker Name: Ryan Gantner (rgantner at sjfc dot edu )

Speaker Afiliation: St. John Fisher College

Abstract: Envision an environment in which there are two candidates for an election and each voter decides on a candidate simply by asking a neighbor at random times for his/her opinion and then conforming.

Under what situations will this process lead to consensus among the people, and under what situations will the population be spatially divided between candidates? We'll look at the answers to these questions by graphically constructing the stochastic voter model and its dual process.

Target Audience: There are many properties from a first course in probability which show up in this presentation (most notably the memoryless property of exponential random variables), and prior knowledge will help in the understanding of this talk, though one should be able to follow the arguments at some level without such knowledge.

If Markov played baseball

Speaker Name: Ryan Gantner (rgantner at sjfc dot edu )

Speaker Afiliation: St. John Fisher College

Abstract: There is a popular way of modeling the game of baseball using a probabilistic concept known as the Markov Chain. This model will be introduced and described in detail. Using actual recent Major League Baseball data, we'll build transition (and related) matrices and be able to do various analyses, such as the value of bunting, the value of base stealing, and the value of hitting home runs. We will talk about recent work which has been done regarding pitch calling and batting orders.

Target Audience: We'll use several concepts from linear algebra, including matrix multiplication, eigenvalues, and eigenvectors. Prior knowledge of these concepts will significantly help understanding. Some basic results of probability and Markov Chain theory will be discussed and used, though no prior knowledge will be assumed. The talk should be accessible to those with only a basic knowledge of how the game of baseball works.

The geometry of voting and Arrow's impossibility theorem

Speaker Name: Ryan Gantner (rgantner at sjfc dot edu )

Speaker Afiliation: St. John Fisher College

Abstract: In this talk, we'll look at some of the peculiarities which can arise when more than two candidates seek election to a position. As an amusing example to illustrate the basic ideas, we'll specifically examine the 1998 Minnesota gubernatorial election in which ex-professional wrestler Jesse Ventura was elected as a third-party candidate. In this example and other examples, we'll build a geometric representation of one particular family of voting schemes and see how the geometry can help us understand the paradoxes which arise. Finally, we'll talk about Arrow's impossibility theorem, which gives a grim conclusion about the possibilities of running a "fair" election.

Target Audience: In this talk, we'll use a few concepts related to 3-dimensional plotting and projection, along with parameterizations of line segments. However, these concepts are fairly easy to pick up on the fly, so really the only math required is a basic knowledge of algebra.

Benford's Law and its applications

Speaker Name: Ryan Gantner (rgantner at sjfc dot edu )

Speaker Afiliation: St. John Fisher College

Abstract: We'll explore what Benford's Law is and the historical contexts surrounding its proposal.
Many, many examples of data sets will be presented that roughly follow Benford's Law (and a few that don't). Some basic statistics (primarily the chi-squared goodness of fit test) will be introduced to quantify how well certain data sets follow the distribution. We'll examine some basic arguments for why this law seems to arise in nature, and talk about some interesting currently-used applications. 

Target Audience: The ideas of logarithm, probability distribution, and a chi-squared goodness of fit test will be presented, but only knowledge of the logarithm will be assumed. This does not require any specific calculus knowledge.

Many Talks!

Speaker Name: Jeff Johannes (johannes at geneseo dot edu)
Speaker Afiliation: SUNY-Geneseo

Abstract: Kaleidoscopic Mathematics
The World That We See: Perspective and Projective Geometry
Mathematics and Music
The Fourth Dimension and Science Fiction
A Concise History of Calculus
Mathematics of the Calendar
An Introduction to Knot Theory
Golden Rectangles Everywhere
Evariste Galois and Hector Berlioz
Evariste Galois and the Solution of Algebraic Equations
 

Clever Counting in Sudoku

Speaker Name: Emilie Wiesner (ewiesner at ithaca dot edu)
Speaker Afiliation: Ithaca College

Abstract: Have you ever played Sudoku? Have you wondered what makes one puzzle harder than another?, what the minimum number of clues could be?, how many puzzles there are? So have other mathematicians!

I'll talk about these questions and, in particular, how mathematicians have tried to count the number of puzzles. This turns out to be a tough count to make, and mathematicians have used clever counting strategies from Combinatorics and Abstract Algebra to do it. 

Target Audience: A large portion of the talk doesn't require any particular mathematical background, but students can follow more easily if they have seen some counting arguments. There is also an optional part of the talk that uses group theory.