University of California, Irvine, Department of Mathematics
Time:
Wednesday, June 3, 2015 - 11:30am to 1:30pm
Location:
RH 340P Ph.D. Defense
In many types of modern communication, a message is transmitted over a noisy medium. When this is done, there is a chance that the message will be corrupted. An error-correcting code adds redundant information to the message which allows the receiver to detect and correct errors accrued during the transmission. We will study the famous Reed-Solomon code (found in QR codes, compact discs, deep space probes, ...) and investigate the limits of its error-correcting capacity. It can be shown that understanding this is related to understanding the "deep hole" problem, which is a question of determining when a received message has, in a sense, incurred the worst possible corruption. We partially resolve this in its traditional context, when the code is based on the finite field F_q or F_q^*, as well as new contexts, when it is based on a subgroup of F_q^* or the image of a Dickson polynomial. This is a new and important problem that could give insight on the true error-correcting potential of the Reed-Solomon code.
University of California, Irvine, Department of Mathematics
Time:
Tuesday, June 2, 2015 - 11:00am
Location:
NS2 1201
Abstract:
Over finite fields, if the image of a polynomial map is not the entire field, then its cardinality can be bounded above by a significantly smaller value. Earlier results bound the cardinality of the value set using the degree of the polynomial, but more recent results make use of the powers of all monomials.
In this talk, we explore the geometric properties of the Newton polytope and show how they allow for tighter upper bounds on the cardinality of the multivariate value set. We then explore a method which allows for even stronger upper bounds, regardless of whether one uses the multivariate degree or the Newton polytope to bound the value set. Effectively, this provides an alternate proof of Kosters' degree bound, an improved Newton polytope-based bound, and an improvement of a degree matrix-based result given by Zan and Cao.
University of California, Irvine, Department of Mathematics
Time:
Tuesday, March 3, 2015 - 10:00am
Location:
340N Rowland Hall
The purpose of this thesis is to use the tools of Inner Model Theory to the study of notions relative to generic embeddings induced by ideals. We seek to apply the Core Model Induction technique to obtain lower bounds in consistency strength for a specific stationary catching principle called StatCatch*(I), related to the saturation of an ideal I of omega_2. This principle involves the central notion of self-genericity in its formulation, introduced by Foreman, Magidor and Shelah. In particular, we show that assuming StatCatch*(I) (plus some additional hypothesis in the universe), we can obtain, for every finite n, an inner model with n Woodin Cardinals.