Graduate Student Advisory Committee (GSAC) Colloquium Schedule:

Welcome back!

There will be a short Organizational Meeting (5–5:30 PM). Pizza will be served!

Crystals, Chemistry, and Space Groups

The (sometimes vague) notion of symmetry appears in all fields of mathematics, both pure and applied. While group theorists and representation theorists enjoy studying symmetry for its own sake, the applied sciences of crystallography and chemistry provide interesting examples of how purely abstract constructions can have useful applications in the real world. In this talk, we will discuss a couple of these applications; in particular, we will see how pure mathematical objects called "space groups" and "point groups" can be used to classify every type of crystal that can be found in nature. We will also discuss how representation theory (the bridge between abstract algebra and linear algebra) can be used to describe the periodic behavior of water molecules.

References:

• Sean McAfee's slides for this presentation. (link)
• Souvignier, Bernd. Representations of crystallographic groups. Lecture notes for the 2010 Summer School on Irreducible Representations of Space Groups. (link)
• Hiller, Howard. Crystallography and cohomology of groups. Amer. Math. Monthly 93 (1986), no. 10, 765–779. (link)

Hope for Losers: Parrondo's Paradox

In a game of chess, pieces can sometimes be sacrificed in order to win the overall game. But can two losing gambling games be set up such that, when they are played one after the other, they becoming winning? The answer is yes! This counterintuitive result from game theory is called Parrondo’s paradox. In this talk, we'll try to understand this apparent paradox and discuss its applications to Brownian ratchets. These examples illustrate how random motion and noise can have a constructive effect on a system.

Busy Beavers and Big Numbers

In this talk we will learn a little bit of computational theory. Along the way, we will have a contest, meet some furry creatures, encounter super-astronomical numbers, and see how the ability to express large numbers reflects the progress of civilization. There will be no abstract algebra or PDEs.

Computing in the Natural World: an in vitro and in vivo approach

In the realm of biology, living organisms can perform immensely complicated tasks both quickly and often times without much error! Such processes include gene regulation, active/passive transport, and self assembly. In the field of natural computing, these tasks are viewed as information processing, which the cell/organism has a programmed computational method to "solve" these "problems" in nature. Natural computing investigates and analyzes the computational processes observed in nature, as well as human-designed computing inspired by natural processes. In this talk, we will examine two examples of computation in nature: one in vitro example of DNA computing and one in vivo example of cellular computing.

Cutting and Gluing Polyhedra

Polyhedra are the most simple and naive objects in geometry. However, they have been used to describe several phenomena in most fields of mathematics. In this way, simple properties of polyhedra can be translated in strong theorems in other fields. In this talk, we will discuss Hilbert's third problem:

"Given two polyhedra of equal volume, decide if it is possible to cut out the first in finitely many polyhedral pieces and reassemble them to obtain the second one."

The solutions and ideas around this question generated a revolution in geometry. We will discuss its instant impact as well as the impact in modern theories.

PS: spectators are encouraged to bring scissors, the first part of the talk will be interactive.

A Golden Consequence of the Chinese Reminder Theorem

During the 17th and 18th centuries the Quadratic Reciprocity Law (QRL) was one of the most important open problems in number theory, impulsing the development of the field considerably. Great mathematicians of the size of Euler and Legendre tried unsuccessfully to give a proof of this result. It was not until 1796 that Gauss gave the first proof for QRL; the "Golden Theorem" as the same Gauss used to call it, giving five other different proofs during his life. Certainly, the beauty and importance of this problem inspired much of the work of Gauss and the subsequent studies on number theory. The goal of this talk is to present a very elementary but amazing proof of QRL that lies in something as simple as the Chinese Remainder Theorem (dispensing with Gauss's lemma); due to G. Rousseau (1991). We also will present a little bit of the history behind QRL and give a modern formulation of it. The prerequisites are essentially to know what integers and prime numbers are, but basic knowledge in group theory would be useful.

Fall break!

Bitcoins: Application and Theory of Elliptic Curve cryptography

One well known application of algebraic geometry is to elliptic curve cryptography. In this talk I will give a brief outline of elliptic curve signatures, and how these apply to digital currency. If there is time, I will outline some recent developments such as ring signatures, confidential transactions, aggregate signatures, tree signatures, etc.

The Banach-Tarski Paradox

The Banach-Tarski paradox is the surprising statement that one can partition a ball in 3-space into finitely many pieces which, using only isometries, can be put back together to make two balls of the same volume as the starting ball. This fact has implications concerning the existence of measures on subsets of 3-space with certain nice properties, and is closely related to the non-existence of a countably additive isometry invariant measure defined on all subsets of the real line that assigns the unit interval measure one. This naturally leads into consideration of amenable groups; groups which themselves admit a measure with nice properties. In this talk I will explain how the duplication of a solid ball is possible, explore the possibility of similar "paradoxes" in other dimensions and discuss what these results tell us about the existence of certain types of measures. In particular, we will see that when a group acts on a set in a nice way, the existence of a nice measure that is invariant under the group action depends on whether the group itself admits a nice measure.

Primality Testing and Primality Proving

One of the most important questions in computational number theory is how to find prime numbers, and how to prove that a given number, we believe is prime, is in fact a prime number. While the question is easily answered for small primes, it is much more difficult for large numbers (a large number may have several thousand decimal digits.). We want to have algorithm which have polynomial time complexities. In this talk, we will look at a several algorithms to determine whether a given number is prime or composite. One is very fast but can only say whether a number is composite or 'probably prime'. The second, is deterministic and polynomial time but is conditional on the generalized Riemann hypothesis. The third, is deterministic, unconditional and polynomial time but is hard to implement. Finally, we will present a practical algorithm which is uses the theory of elliptic curves to find primes. It is probabilistic, unconditional, and capable of proving primality and issues a certificate which could be verified in polynomial time. I will also include a very brief introduction to the theory of elliptic curves.

Inverse Galois problems

Galois theory begins as the study of the "algebraic symmetries" of the roots of a polynomial, the fundamental example being the case of a polynomial with rational coefficients. The subject packages these symmetries, for a given polynomial, into a finite group known as the Galois group. A very old, and still wide-open, question then asks what finite groups can arise in this way. First I will introduce this circle of ideas and give some classical examples that indicate what this question has to do with the "internal" arithmetic of Q, eg the infinitude of the primes. My ulterior motive, however, is to introduce some ideas from arithmetic geometry--in this case the arithmetic of elliptic curves--that shed light on further examples of this "inverse Galois problem," and at the same time point toward still more interesting questions in modern number theory.

A tale of p-adic numbers: interesting facts and arithmetic

The field $Q_p$ of p-adic numbers and its subring $Z_p$ of p-adic integers have many intriguing properties that $Q$ does not possess. Visually, $Z_p$ is a Cantor set. Calculus analysis on Q_p has a very different flavor, for example, there exists non constant smooth functions that have zero derivative. Algebra on Q_p is also pretty cool: every polynomial can be solved by radicals! The Galois group over Q_p which reflects the arithmetic of p-adic numbers is much simpler than the Galois group over Q. In this talk, I'll describe all these fun facts with a taste of number theory.

Bifurcations, Groups and Visual Hallucinations

Symmetric bifurcation theory is the study of bifurcations in dynamical systems that are equivariant with respect to the action of a compact Lie group. The symmetric structure of the system reduces the bifurcation problem to a simpler algebraic problem. In particular, one only needs to analyze the absolutely irreducible representations of the Lie group on the vector space where the dynamical system is defined. In this talk I will first discuss the group theoretic preliminaries for the main theorem in symmetric bifurcation theory, known as the Equivariant Branching Lemma. In the second part of the talk I will show how to apply this theory to analyze pattern formation in models of the visual cortex and discuss the implications towards possible mechanisms for visual hallucinations.

Geometry of moduli spaces of curves and their subvarieties

It is known that the structure of subvarieties of codimension one governs the birational geometry of a given variety. The geometric theory of higher codimensional subvarieties is not yet understood. Moduli spaces of curves are a natural test space. Indeed, one of the great features of moduli spaces is the possibility of exhibiting geometrically a subvariety as the locus of elements with an exceptional property. In this talk, I will briefly sketch the history of the subject, and give a flavor of intersection computations with families of curves. Finally, I will present some new results on higher codimensional effective cycles, and end with some open questions.

The complement system: Friend or foe

The complement system is a key component of the innate immune system that provides protection against pathogens as well as transformed and infected cells. This highly regulated system is always “on”, operating at basal level and is amplified when needed. Deregulation of complement leads to chronic inflammation and disease. Given this potential for catastrophe, we are left asking an open question. Is complement our friend or foe? In this talk, I will briefly discuss the biology of the complement system and detail the mathematics of biochemical reactions. Lastly, I will present a class of submodels that I am currently working on.