A dynamical system can be described as a collection of physical quantities that evolves over time. In this talk, we will begin with a quick introduction to dynamical systems to demonstrate the tools used to investigate solutions to different problems qualitatively. We then delineate a specific phenomenon called tipping, which describes a sudden, drastic, irreversible change in the behavior of a solution as a result of a change to the system. Finally, we extend these ideas to mathematical modeling of different climate applications and see how dynamical systems and tipping relate to climate systems.
I will talk about a famous old unsolved problem: is it true that no matter how you draw a continuous, non-intersecting, closed curve in the plane, there necessarily exist four points on the curve at the vertices of a square? Along the way, we will see how complicated curves in the plane can be, and what this problem has to do with Klein bottles in 4-dimensional space. No special background beyond differential calculus will be assumed, but some suspension of disbelief will be helpful.
A "fractal" is often described as a set of points that can be obtained through some recursive procedure. As the recursive procedure repeats on every level of the construction, it tends to create patterns that remain visible at every level of magnification. The resulting set of points could be a line or a surface, or it could be a disconnected dust-like scattering of points. Fractals also tend to have some rather bizarre properties. For example, the von Koch curve has an infinite length, yet the entire curve can be drawn inside of the unit square. This talk introduces some of the core concepts and applications of fractals by discussing several examples ranging from the classic Cantor set to more general random fractals.
Knots in mathematics are little pieces of strings whose ends are joined together. If you went to the Distinguished Lecture Series this semester, you know that there are many very simple questions about knots which we still do not have answers to, and that they are fundamental even to how our cells interpret our DNA. On the other hand, curves are equations like y^2=x^3, which you might learn how to differentiate or graph in calculus. In this talk, I want to persuade you that these two things are related--if we simply try to graph our curves over the complex numbers. The result is the starting point for many remarkable stories about knots and geometry. No prerequisites are needed, though knowledge of differential calculus, parametric equations, and/or complex numbers may be helpful.
In 1637, on the margins of his copy of Diophantus's Arithmetica, a French magistrate and amateur mathematician named Pierre de Fermat scribbled down what he claimed was a theorem stating that there are no non-trivial examples of the obvious analogue of Pythagorean triples for powers higher than 2. It is very unlikely however that he had a proof, since the first complete and correct demonstration was not available till 1994, when the English number theorist Andrew Wiles tied together an immense amount of mathematics for this purpose.
In this talk, I'll try to give a breezy mathematical history of the events leading up to this proof, which was the result of an unexpected confluence of ideas. Some familiarity with the basic theory of groups will be helpful.
Elliptic curves are geometric objects defined by cubic equations which possess very rich structure. They appear in a variety of contexts, from the purely theoretic (such as the proof of Fermat’s Last Theorem) to highly practical (modern cryptography).
In this talk, I’ll give a brief overview of what an elliptic curve is, and describe an especially interesting property possessed by some elliptic curves: complex multiplication (CM). Roughly speaking, elliptic curves with CM have “extra symmetry.” Time permitting, I’ll also talk about how one might find equations of CM elliptic curves, which will require some surprisingly deep concepts from number theory and complex analysis.
Prerequisites: A solid background in algebra, especially field theory, will be helpful. While knowledge of complex analysis is very useful, I’ve tried to structure the talk so that it is not strictly necessary.
If you take apart a Rubik’s Cube and reassemble it at random, does it remain solvable? In this talk, we explore the mathematics behind two famous puzzles: the Rubik's Cube and (briefly) the 15-puzzle. We will see how the legal moves of these puzzles form groups, and how their structure determines which configurations are reachable and which are not. Along the way, we will discuss how to compute the size of the Rubik's Cube group, how to detect illegal positions, and understand the order of individual moves, concepts that play a central role in solving the Rubik’s Cube. If time permits, we will conclude with a brief discussion of how these ideas lead to solution methods.
Prerequisites: I will assume a background in linear algebra. Knowledge of groups is useful, but I plan to provide the necessary definitions and results.
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