2013 Talks
Polytope Bounds on Value Sets
Luke Smith
April 17th, 2013 - 4:00 - 4:50pm - RH 440R
Abstract
Over finite fields, if the image of a polynomial (a.k.a. the value set) is not the entire field, its cardinality is often quite a bit smaller. Earlier results bound this cardinality using the degree of the polynomial. We improve upon these bounds for multivariable polynomial vectors over a finite field using the polytope of the polynomial.
About the Speaker
Luke Smith is a 4th year math grad working in number theory. When not doing math, Luke awaits his next opportunity to explore the outdoors and to travel. He also really loves experimenting with cooking.
Advisor and Collaborators
Luke's advisor is Daqing Wan.
A Friendly Introduction to Large Cardinals and Consistency Strength
Ryan Holben
February 20th, 2013 - 4:00 - 4:50pm - RH 440R
Abstract
We will talk about the Continuum Hypothesis and how we tackle it in the field of set theory. Learn about large cardinals, independence and consistency proofs, and elementary embeddings, and why they are important!
About the Speaker
Ryan received his B.A.'s in math and physics at Colby College, and is now in his last year of graduate school. When not working on research he enjoys biking and communing with the Irvine rabbits.
Advisor and Collaborators
Ryan's advisor is Martin Zeman.
Bounding the Regularity of a Graded Module
Roger Dellaca
November 4th, 2014 - 4:00 - 4:50pm - RH 440R
Abstract
Given a set X of d points in the complex plane, every complex-valued function on X is the restriction of a polynomial of degree at most d-1; some configurations of points allow the maximal degree of such polynomials to be strictly smaller than d-1, while others do not. This is an example of the notion of (Castelnuovo-Mumford) regularity of the algebraic set. Regularity measures the complexity of the geometric object, and its associated algebraic object as well. A general definition of regularity can be given in terms of a minimal free resolution of a graded module, which is used in several explicit techniques in Commutative Algebra, including the Hilbert function and Hilbert polynomial. A theorem of Gotzmann takes an odd-looking way of writing a Hilbert polynomial, and uses it to bound the regularity of all ideals with that Hilbert polynomial. This was used to give an explicit construction of the set of all such ideals as a geometric space, called the Hilbert scheme. I will show how this technique can be used to bound the regularity of certain classes of modules with a given Hilbert polynomial. This allows an explicit construction of a generalization of the Hilbert scheme.
About the Speaker
Roger is a 6th-year student. He has a M.S. in Math from CSULB and a M.S. in Management Science from CSUF. His research is in Commutative Algebra and Algebraic Geometry. He enjoys cycling.
Advisor and Collaborators
Ali's advisor is Vladimir Baranovsky.
Mathematical Modeling of Language
Jacquelyn Rische
January 23rd, 2013 - 4:00 - 4:50pm - RH 440R
Abstract
In this talk, we will look at mathematical modeling of language using computer simulations. Using these models, we study how individuals with language spread through a population of individuals without language. We consider a population without language on one- and two-dimensional grids. Language will appear in the population through a genetic mutation. To study how the language group will grow, we focus on the effects of talking and movement. If two individuals with language are next to each other on the grid, they can communicate. We consider their ability to talk to be advantageous, giving them a higher reproduction rate. Individuals are also able to move around on the grid and reproduce within a certain radius, called the jump radius. We are looking at how these affect the time it takes for the individuals with language to invade the population. We find that, for a two-dimensional grid, a jump radius that is too small or too large will increase the time it takes to invade. For a one-dimensional grid, we do not see the same effect. The time to invasion decreases as the jump radius increases.
About the Speaker
Jacquelyn Rische received her B.A. in mathematics from Whittier College in 2007 and her M.S. in mathematics from University of California, Irvine in 2009. In the little free time Jacquelyn has, she enjoys cross-stitching and making greeting cards.
Advisor and Collaborators
Jacquelyn's advisor is Natalia Komarova.
