Abstract: A partition of a positive integer n is a sum of positive integers that add up to n. For example, 7 = 4 + 2 + 1. How many ways can we do this for a given number? What happens if we only allow the numbers in the sum to be odd? What if we are not allowed to repeat numbers in a sum? These simple kinds of questions can actually be quite hard. In this introductory talk, we will discuss some basic properties of different functions that count certain types of partitions. We will also show that partitions are closely connected to the geometric series
The number 5 has 7 partitions (can you find them?), but the number 30 has 5604 partitions!
September 24, 2015: Dr. Satyanand Singh: “Playing with the Möbius band”
Abstract: Come participate in the creation, exploration and derivation of some properties of the amazing Möbius band. The mysterious properties of the Möbius band are widely exploited in society today on conveyor belts, medical applications and as a source of pathological examples in mathematics. Join us as we play with the Möbius band and heed its warnings to conserve.
October 8, 2015: Dr. Jonas Reitz: “Throw a Party, Fight the Aliens”
Abstract: The talk will be devoted to the Party Problem, a problem from combinatorics and graph theory that is simple, intuitive, and fiendishly difficult to solve (even for numbers as small as five or six). The numbers that arise in this problem are called Ramsey numbers. I will define Ramsey numbers and we will explore them together.
“Suppose aliens invade the earth and threaten to obliterate it in a year’s time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world’s best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack.” – Paul Erdos, as quoted in “Ramsey Theory” by Ronald L. Graham and Joel H. Spencer, in Scientific American (July 1990), p. 112-117
October 15, 2015: Dr. Caner Koca: “How to do Calculus with Complex Numbers?”
Abstract: About a century after the invention of Calculus by Newton and Leibniz, mathematicians such as Euler, Gauss, Riemann and Cauchy discovered and developed a complex-number version of all the key ideas in Calculus, such as differentiation and integration. In this new theory, one looks at complex-differentiable functions from complex numbers to complex numbers, and study their properties. This analogy, though very formal, can sometimes lead to really unexpected, surprising and slightly disturbing facts! For example, the complex sine and cosine functions turn out to be unbounded, or the complex exponential function is periodic.
In this talk, we will see the basics of the theory of complex functions, and highlight some of the similarities and differences between real and complex calculus. I will give special emphasis on how one can visualize and graph some of these complex functions, especially of the so-called multi-valued complex functions, which in turn give rise to some of the fascinating examples of Riemann surfaces.
October 29, 2015: Mr. Charlie Meyers: “Math of Wireless Networks”
Abstract: Math Club will host a lecture by Applied Math student, Charlie Meyers, this week. He will discuss the mathematics of wireless communications networks and discuss his ongoing research about free public wifi in NYC. Come learn how frequency is related to energy, how waves have strange properties, and how graph theory is used every day. We will briefly discuss complexity theory as well as world-wide efforts to make access to the internet a public utility as opposed to a for-profit institution for those who can afford it. Using math and physics, we will determine the least cost approach to wireless modelling in the city and around the world.
November 12, 2015: Dr. Suman Ganguli: “Counting Past Infinity: The Ordinal Numbers and Transfinite Induction”
Abstract: How do we prove a property holds for all integers? Usually we do by induction. We will start by reviewing induction on the integers, and illustrate with a simple proof by induction.
In the 1880s, the mathematician Georg Cantor introduced the infinite ordinals—infinite numbers which allow for transfinite induction. We will discuss how Cantor was led to formulate the infinite ordinals, give an idea of what some (relatively small!) infinite ordinals “look like,” and show how arithmetic on the ordinals is defined.
November 19, 2015: Ms. Yoonhee Kang: “Pascal’s Wager: Betting on God, why not?”
“God is, or He is not.” But to which side shall we incline? Reason can decide nothing here […] A game is being played at the extremity of this infinite distance where heads or tails will turn up […] Which will you choose then? […] Let us weigh the gain and the loss in wagering that God is. […] If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is. (Pascal, Blaise, 1670, Pensées, §233)
Abstract: Is theism justified? How? Traditional philosophical arguments for theism (Anselm’s ontological argument, Aquinas’ cosmological argument, Descartes’s ontological and cosmological arguments, and so on) attempt to provide “proofs” for the existence of God. But Pascal – French philosopher, mathematician and physicist (1623-1662) –is unconvinced by those attempted proofs for God. Pascal develops a rather pragmatic argument providing prudential reasons for believing in God. Pascal claims, roughly to put, we should wager that God exists because it is the best bet!
In this talk, I will introduce Pascal’s Wager argument – his view on probability, voluntarism, and the use of the concept of infinity etc. I will also discuss various possible objections to his argumentation.
December 10, 2015: Dr. Jonathan Ginsberg: “The Mathematics of Wallpaper”
Abstract: The repeating patterns of wallpaper have been a part of art and architecture for centuries. While at a glance, the possibilities for creating original wallpaper patterns seem endless, there actually exist only 17 truly distinct patterns that can be created. This talk will discuss the isometries of the Euclidean plane used to build the 17 wallpaper groups, as well as constructing several of them, and discussing their use in art and architecture (both classical and modern).