Fall 2006 GSS Schedule
Organizers: Robert Beeler and Jeremy Lyle
| Date | Speaker | Topic | Abstract |
|---|---|---|---|
| September 4 | Timur Milgrom Bonnie McAdoo Christine Kraft Jason Howell |
Life in the Department of Mathematical Sciences | In this talk, four graduate students with experience in the M.S. and Ph.D. programs
in the department of Mathematical Sciences at Clemson will discuss some of their
experiences at different levels and how they differed from their expectations. Timur Milgrom
is a second-year master's student here in the department. Bonnie McAdoo
recently completed the M.S. degree and has now entered the
Ph.D. program. Christine Kraft is a Ph.D. student who recently
completed the qualifying examinations. Jason Howell is also a
doctoral student who is currently working on his dissertation.
The main purpose of this talk is to promote candid discussion among new and experienced graduate students about the realities of graduate school here. |
| September 11 | Robert Beeler | Mathematical Resources on the Web | In recent years, the internet has grown to contain a wealth of
information. In this talk, we will survey some of the resources that the web has to
offer. Some of the topics to be discussed include:
1. What resources are available for teaching, research, etc. 2. Where to find free math books. 3. What fun stuff is available. The links for this talk are here |
| September 18 | Shannon Purvis | Random Acts of Vectors | The Probabilistic Method was recently introduced by Erdos and
Renyi in a paper on random graphs. It has become a powerful tool in
Combinatorics, often used in problems involving randomness. In this talk, we will make use of common methods in Probabilistic Combinatorics by exploring a question about random vectors. Linearity of expectation will be used to find the expected size of the null space of a random l x k matrix generated under a given probability model. Time permitting, we will also use asymptotic methods to estimate the size of the null space and determine how large l must be so that the matrix is dependent with high probability. |
| September 25 | Bryan Faulkner | An Open Question | In this presentation the number of points on elliptic curves with coordinates in a finite field will be examined by character sums. Open questions will be stated relating character sums involving "K3 surfaces" and elliptic curves. |
| October 2 | Ethan Smith | Modular Forms and Hurwitz Class Numbers | For an integer N ³ 0, the Hurwitz class number,
H(N), is defined as follows. H(0)=-1/12.If N
º 1 or 2 (mod 4), then H(N)=0.
Otherwise, H(N) is the class number of not necessarily primitive
positive definite quadratic forms of discriminant -N, except that those
forms which are equivalent to a multiple of the form x 2 +y 2 should be
counted with multiplicity 1/2 and those which are equivalent to a multiple of the form
x 2 +xy+y 2 should be counted with multiplicity 1/3.
It is known that if p is a prime, then
In this talk, we will discuss the behavior of the same sum where we add the extra condition that r º c (mod m) for some integers c ³ 0 and m>0. A group of REU students was able to completely characterize these sums by writing down formulae for the cases m = 2 and 3. Their work also led to many conjectures in the cases m = 4, 5, and 7. The proof for the cases m= 2, 3 involved counting isomorphism classes of curves with m-torsion over finite fields. In this talk, we will see how modular forms can be used to provide a nicer proof for the case m = 3. |
| October 9 | Jeremy Lyle | Domatic and Idomatic Numbers of Graphs | In this talk, I will discuss some of the results about the domatic and idomatic number of graphs. This will include complexity results, methods for particular graph classes, probabilistic methods, and constructions for lower bounds. In particular, I will consider graphs which are somewhat regular, with a specific interest in the n-dimensional hypercube. |
| October 16 | Alexander Engau | Let's go fly a kite, or, you're the (c)one that I want! Uh-uh-uh. | It is a common characteristic of decision problems involving multiple
criteria that the definition of a best alternative depends on the
individual preferences of the decision maker. However, all that is
usually known are some general properties of the underlying preference
structure as induced by a set of chosen preference principles. To use
an existing preference model thus requires deciding which principles
to accept and which to reject. In my talk, I will first review several preference principles and illustrate the corresponding preference models. Motivated by some remaining shortcomings of these models, I will then introduce a new "flying-kite" model which can be described by a variable "ice cream" cone. To conclude, I will invite the audience to join in when we break into song for the famous "You're the (c)one that I want!" Uh-uh-uh. |
| October 23 | John Chrispell | Fractional step q-scheme for time-dependent PDE's | Modeling physical applications often leads to systems that are of mixed type, if more than one variable is required in the numerical solution the approximating systems also become very large when fine mesh solutions are desired. This is the case when the approximation of viscoelastic fluid is considered. Here there is a large number of unknowns in the approximating algebraic system (corresponding to velocity, pressure, and stress), and the different mathematical types of the modeling equations. An operator splitting method which decouples the conservation of momentum equation from the constitutive equation is an appealing solution technique. This split reduces the size of the linear systems that need to be solved and separates the parabolic and hyperbolic equations into different substeps. Motivated by the viscoelastic fluid flow problem, we analyze an operator splitting fractional step q-scheme for the numerical approximation of the convection-diffusion problem, showing both theoretical, and numerical results. |
| October 30 | Ben Phillips | Two Topological Proofs | Clemson students tend to miss out on one very fundamental area of pure
mathematics, topology. In this talk, we shall discuss two results which can be
proven using topological arguments. Specifically, we shall show using topology that
there are infinitely many prime numbers. We shall also concern ourselves with
answering the following question:
Let i: S 1 ® D 2 be
the inclusion of the circle into the disk.
Does there exist a continuous function
r: D 2 ® S 1
such that the composite
r o i: S 1
® S 1 is the
identity? We shall give a proof that in fact no such function can exist, using
algebraic topology. Topics to be discussed include topological spaces, categories, functors, homotopy classes of loops, the fundamental group, and many examples. |
| November 6 | Fall Break | ||
| November 13 | Carol Lee | Careers in the Navy Warfare Center | Carol Lee will be talking to graduate students about careers in the Navy Warfare Center. |
| November 20 | Thanksgiving Break | ||
| November 27 | Jang-Woo Park | ||
| December 4 | Nathan Drake | Analyzing the Guruswami-Sudan Algorithm | One of the main goals in coding theory is to attempt to correct as many errors
as possible. It has been known that we can correct up to the floor of (d-1)/2
errors where d is the minimum distance between any two codewords.
However, list decoding allows us to correct more errors than anticipated.
We will discuss the Guruswami-Sudan Algorithm and how it allows us to use
list decoding to decode beyond the anticipated bounds. We will also discuss some variants on this algorithm, specifically that of Koetter and Vardy. |