Felipe Voloch explains elliptic curves and how they give us groups and how these groups are used in the security of Bitcoin.

Professor Bill Beckner presents his lecture "Enroute to the Land of Fourier: Fair Coins, Smooth Bumps, Symmetry, Uncertainty" to the Math Club.

I will explain in my talk how ideas from representation theory can be applied to a solution of a basic "real life" problem. The problem that I will discuss arise in structural biology and concerns determination of the three dimensional structure of a large molecule from the set of images, taken by an electron microscope (3D Cryo Electron Microscopy). Complete solution to this problem will constitute an important scientific breakthrough since it promises to be an entirely general technique...

When trying to find rational solutions to an algebraic equation, a favorite trick is to first look for local solutions. That is, we seek solutions over the reals and modulo all powers of all primes. Sometimes local solutions exist if and only if a rational solution exists. This is when we say that the local-to-global principle holds. We will illustrate this principle by discussing the problem of representing integers by quadratic forms over the rationals.

The Brunn-Minkowski inequality: On the Volume of the Sum of Two Sets

In this talk, I will show you a glimpse of one of the most exciting facets of research in modern number theory: arithmetic geometry. We will start with a (gentle) introduction to this area of research through some familiar examples. Then we will move on to a not so familiar example where we count solutions of equations mod p. I will end by answering two of the oldest and most mystifying questions in mathematics: how does this work fit into the bigger picture, and who cares?

In the early 1980's W. Stothers and (independently) R. Mason discovered a new and simple inequality about the zeros of polynomials. The proof uses only basic facts about derivatives. As a corollary Mason obtained a very simple proof of "Fermat's Last Theorem" for polynomials. Inspired by Mason's observations, Masser and Oesterl e proposed an analogous inequality for integers, which has come to be known as the ABC conjecture. In this talk I will discuss Mason's inequality, some of its...

Suppose we are given a set of segments of track from a toy train, and wish to make interesting designs which are simple closed curves. What options are available to us? Surprisingly, even a simple question like this leads us to use tools from linear algebra and number theory, and serves as a model for more general questions about positioning objects in space.

Professor Todd Arbogast presents his topic "Fully Conservative Characteristic Methods for Transport"

Cool insights about circles and basic geometry give magical proofs to some calculus results such as the derivative and integral of sin(x). In fact, some basically calculus insights preceded calculus itself by a couple thousand years. For example, Archimedes used ideas about levers and limits to deduce the equation for the volume of the sphere.

Professor Voloch will speak about how linear algebra is used in the theory of error-correcting codes, which are extensively used in telecommunications. He will also cover the modern ideas that go into the new "low-density parity check" (LDPC) codes, as well as the mathematical challenges associated with them.

Geometric and topological objects frequently come in families parameterized by other such objects. Depending on how intricate and/or symmetrical these objects are, the families can be "twisted.'' We will investigate the source of this behavior in concrete cases, and outline a program to understand such twisting in a large range of cases using topology.

Professor Vick explains why you should drop everything you're doing and start studying topology.