Graduate Student Advisory Committee (GSAC) Colloquium Schedule:

Welcome back.

  THERE WILL BE FREE PIZZA.

Elliptic curves in cryptography

Many modern cryptographic systems rely on the availability of algebraic structures which come with operations that can be quickly computed but not quickly reduced to trivial cases.  For the past 25 years, elliptic curves have been a source of several such structures.  In this talk we'll look at examples of novel cryptographic techniques which arise naturally out of the structure of elliptic curves.

Sets & Spaces, Schemes & Spaces, Stacks & Spaces

People often think of algebraic geometry as a technical and mysterious branch of mathematics.  This is probably due in no small part to the fact that algebraic geometers tend to study objects with cryptic names, like sheaves and schemes.  My goal is to dispel some of the mystery behind these names, and to show how each can be seen as something natural and worthy of study.  Beginning with sets, we'll see how we might naturally be led to define and study schemes, algebraic spaces, stacks and more.  Despite all the creepy words I'll use, the talk will be non-technical and (hopefully) not at all scary.

Poincare's Dream

In 1904, after developing the fundamental group, Henry Poincare conjectured that the only simply connected closed 3-manifold was in fact the three sphere. After this paper little progress was made until Bill Thurston in the 1970's announced a stunning result and subsequent conjecture.  Thurston's conjecture is known as the Geometrization Conjecture and was recently proved by Grigori Perelman thus completing the proof of the Poincare Conjecture. In this talk we will endeavor to explain the statement of the Geometrization Conjecture through examples.  noted.

The Central Limit Theorem: A Thermodynamic Perspective

Upon completing a class in probability, one might reasonably assert that the strong law of large numbers and the central limit theorem could be called the first and second fundamental theorems of probability. In some limited contexts, the strong law of large numbers can be intuitively understood to say that a large random sample "looks like" the population from which in was chosen and therefore statistics gathered about that sample should closely estimate population parameters which are in some way analogous.  The central limit theorem, however, is a bit harder to come to grips with.  In particular, why is the limiting distribution of a sum of i.i.d. random variables (with finite variance) normal?  Why isn't this limiting distribution something/anything else? Why is the limiting distribution the same regardless of what the distribution of the source variables is? In this talk, I will attempt to provide an intuitive answer to these questions.  I will do so by invoking entropy, the second law of thermodynamics and the mathematical connections that bridge the gap between these (admittedly somewhat slippery) physical concepts and the world of abstract probability.

A (Topological) Introduction to Bass-Serre theory

Bass-Serre theory is a convenient way to find structure in a group G by analyzing how G acts on trees. There is a nice topological interpretation using covering theory. In this talk, I will describe the main ideas of Bass-Serre theory, graphs of groups, graphs of spaces, and give a couple of applications. This is an elementary talk, and I won't assume too much background in algebraic or geomeric topology.

Fall Break.

The Borel-Cantelli Lemmas

The Borel-Cantelli lemmas are quite useful tools in modern probability (meaning measure theory based) involving sequences of events, especially with tail events.  In the direct form, the lemma asserts that for a sequence of events whose probabilities sum to a finite number implies that the infintely many of those events occur with probabilty zero.  The converse is false, but becomes true if the events are independent.  I will present proofs of the lemma and examples including a special form of the strong law of large numbers and the more comical example showing the expected time to see an offer on a house that is lower than the first offer is infinite, given some reasonable assumptions of course.

Schur-Weyl duality

Two groups that appear everywhere are the symmetric group (permutations) and the general linear group (invertible matrices with complex entries). Often one is lead to study their actions on various spaces, their representations. I will explain a beautiful classical relation between the representations of these two groups: the Schur-Weyl duality. The talk will be accessible to everyone.

Voting and Group Choice (click to open notes in a new window)

As today is an election day it seems appropriate for a talk on voting and group choice. In this talk, we'll discuss different voting systems that seem, at first glance, to be fair, and then explore the limitations and problems of each of them. We'll then discuss what kind of properties we'd expect a fair voting system to have, and then see how restrictive these seemingly simple properties are on our possible voting systems. We'll end with a proof of Arrow's Impossibility Theorem, which is a surprising result about the possible voting systems allowed given a few simple criteria. The talk assumes no background in the subject, and should

Game Theory: Step 0

von Neuman's minimax theorem states that for every two-person, zero-sum game with finite strategies, there exists a value v and a mixed strategy for each player, such that Player A's strategy guarantees him a payoff of v regardless of Player B's strategy, and similarly Player B can guarantee himself a payoff of -v. According to von Neumann, the inventor of Game Theory, "there could be no theory of games … without that theorem … there was nothing worth publishing until the Minimax Theorem was proved." The above can be stated as sup_x inf_y f(x,y) = inf_y sup_x f(x,y)   (i.e. f has a saddle point) for some function f(x,y) that represents the gain of player A if he adapts strategy x while player B adapts strategy y. In the above case, the function f happens to be linear in each variable. I will state and prove a more general theorem that will imply the one above. The proof is self-contained. The only prerequisite is knowing what a compact topological space is.

Ramsey Theory on the Integers: History, Developments, and Contributions

If the positive integers are colored by a finite number of colors, we can ask whether a monochromatic predetermined configuration will appear. Questions of this kind were first addressed at the beginning of the 20th century by Schur, Van der Waerden, and Rado, and later evolved into a branch of mathematics known as Ramsey theory on the integers.  In our talk, first we will survey some historical background, then we will address major developments, and conclude by describing some results obtained in a recent research under the Brian and Gayle Hill Fellowship. The talk is of an elementary nature.

Introduction to the Mapping Class Groups of Surfaces

The study of the mapping class groups was initiated in the 1920's by Dehn and Nielsen and was developed by famous mathematicians such as Thurston, Harvey, Hatcher, Birman and many others. In addition to being a central object of the geometric group theory, these groups also play an important role in algebraic geometry, where they are known as modular groups. In this talk I will give an introduction to the mapping class groups based on examples.

The Fundamental Observation of Geometric Group Theory

This talk will showcase an observation due to Efremovich, Schwarz and Milnor that is an essential tool in geometric group theory. The result illustrates the remarkable interplay between geometry and algebra, by showing that if a group acts on a metric space in a nice way then the large scale geometry of the group is same as the space on which it acts. The talk will include all necessary background material, examples and many applications of this result.

Mathematical Models of "One-Shot" Molecular Engines

Cells have perfected over the course of millions of years extraordinary mechanisms for driving directed motion in the face of considerable thermal noise. In order to produce useful work all these tiny machines have developed a way of rectifying Brownian Motion. Feynman (1962) showed that structural anisotropy is not sufficient to drive motion at equilibrium unless coupled with external driving forces.  We discuss mathematical models of force generation for some so called "One-Shot" motors, which rectify Brownian motion by using the energy of polymerization/depolymerization of biopolymers. In contrast with conventional protein motors, these motors do their increment of work and are disassembled and reassembled at another place in the cell. We derive and compare  the load-velocity relationships for two such motors : actin polymerization ratchet motor and power stroke kinetochore motor.