A linear feedback shift register (LFSR) is a type of electronic circuit which plays a key role in cryptography and error control. In this talk, intended for a general mathematical audience, I will describe the physical circuit, provide a mathematical abstraction, and discuss mathematical methods for designing LFSRs with certain desirable properties.
A problem which arises again and again in almost any discipline is: "find a formula for (some function)". We attack this problem in a very broad generality by considering the function to be some arbitrary computable function, ie, one which a computer or human agent can evaluate in principle using some finite algorithm. Thus we are essentially duplicating work of Alonzo Church, who provided a solution using lambda calculus. However, Church's formulas involve lambda terms which are rather sophisticated. Our achievement is to show the existence of formulas using only machinery that is already familiar, or can be defined very briefly, to students with nothing more than an introductory calculus background. We make this formal and outline the ideas behind the existence proof.
I will attempt to illustrate Combinatorial Design Theory and its relationship to coding theory, primarily through a series of examples. Though the area of coding theory has many applications, the focus of this talk will be on codes as interesting mathematical objects unto themselves. Some examples we will discuss are the Fano Plane, Quadratic and Singer Difference Sets, and the Hamming Code. Depending on time constraints I will also address some current research being done on Cyclic Difference Sets with Singer Parameters. This talk is intended primarily for a general mathematical audience.
I will describe two constructions in algebraic geometry, the state polytope and the secondary polytope, and a nontrivial relationship between them. The talk will include some technical definitions, but the emphasis will be on geometric intuition. The only real prerequisite is a basic understanding of ideals, though some prior exposure to algebraic geometry would also be helpful.
In this talk, I will describe an algorithm (due to R. E. Gomory) for solving a certain type of integer programming problem. Such a problem is one where we want to minimize a linear function with integer coefficients of some integer variables, subject to a system of linear inequalities.
This talk will be about a random family of conformal maps that remove an evolving path from the upper half-plane. These maps are attained by solving the so-called Loewner differential equation. Typically, we put a random function in the Loewner differential equation, so there is randomness involved. In my talk, I will estimate the continuous random process typically included in the Loewner differential equation with a discrete process and attain a discrete family of conformal maps that, in the scaling limit, will converge to the continuous family of conformal maps which solves the equation.
Starting from some prior belief about a population on which we wish to infer, we draw a random sample from the population and update the belief from available data. Many times statisticians face difficulty in computing the subsequent distribution to the exact integrating factor. This is where Gibbs sampling comes into play, by providing an algorithm to draw a sample from the posterior distribution without knowing its integrating factor. This drawn sample is then used to infer about the distribution and hence the population parameters. In my talk, I will illustrate Gibbs sampling for a particular example from Bayesian inference. Not much statistics will be involved, but a basic knowledge of probability concepts and vocabulary will be helpful.
Using the Sylow theorems, it is easy to compute the order of a Sylow p-subgroup of a finite group. But the theorems do not tell us anything about the structure of such a subgroup. In fact, finding a Sylow p-subgroup is in general a very challenging problem. In this talk, I will present an algorithm for constructing a Sylow p-subgroup of a symmetric group.
In 1694 Jacob Bernoulli published an article in Acta Eruditorum on an algebraic plane curve which he called the "lemniscus." Bernoulli's orginal title is the latin word for a ribbon, but today the curve is known as Bernoulli's lemniscate. It has many fascinating properties which can be explored using only calculus and elementary algebra and is thus a gem of an example to have in your teaching bag. This will be a survey talk exhibiting many of these properties and connections with plane differential geometry. We will see use of circles and hyperbolas, trig and hyperbolic functions, completing the square, polar coordinates, slope fields, compass and ruler constructions, the torus of revolution, stereographic projection for the sphere, and envelopes of families enter in our study.
In this work I will discuss some of the results by C.A. Rogers and Luis Montejano which will give sufficient conditions for the existence of centers of symmetry in convex sets in the plane. We will see that it is enough to have a quiral property in different families of chords to assure that the convex set is centrally symmetric.
I bring tales from my voyages to the farthest reaches of campus, where people (apparently voluntarily) study disciplines other than mathematics. We'll discuss the intersection of linguistics and mathematics, focusing primarily on their applications to each other, but addressing along the way methods of detecting cheaters, eliminating spam, and naming new mathematical objects. Finally, we'll address a new methodology for unification called Rooter which could help you get your PhD and runs in an amazing O(log log n) time.