Location: NSM A 115 C
Time: 3:00pm--4:00pm
We will have cookies and coffee starting at 2:30pm
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Date: 04/26 (Friday)
Speaker:Nathaniel Emerson (USC)
Title: From Polynomial Dynamics to Meta-Fibonacci Numbers
Abstract: We will discuss the dynamics of a complex polynomial. The Julia set of a complex polynomial is the
set where the dynamics are chaotic. Polynomial Julia sets are generally complicated and beautiful
fractals. The structure of a polynomial Julia set is determined by the dynamical behavior of the critical
points of the polynomial. So to understand the Julia set of a polynomial we need only study the
dynamics of a finite number of critical points. A useful way to do this is to consider closest return times
of the critical points. Most simply the closest return times of a point under iteration by a polynomial are
the iterates of the point which are closer to the point than any previous iterate. We consider generalized
closest return times of a complex polynomial of degree at least two. Most previous studies on this
subject have focused on the properties of polynomials which have particular return times, especially the
Fibonacci numbers. We study the general form of these closest return times, and show that they are
meta-Fibonacci numbers. Finally we give conditions on the return times which control the structure of
the Julia set.
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Date: 04/19 (Friday)
Speaker: John Rock (Cal Poly Pomona)
Title: Real and complex dimensions of fractal strings and a
reformulation of the Riemann Hypothesis
Abstract: "Can one hear the shape of a fractal string?" An
affirmative answer, in a context provided by an inverse spectral
problem for fractal strings, is equivalent to the popular and
provocative hypothesis originally posed by Bernhard Riemann—the
nontrivial zeros of the Riemann zeta function lie on the line with
real part one-half. In this talk, we discuss the geometry and spectra
of fractal strings in the context of real and complex dimensions and
their natural relationship with the structure of the zeros of the
Riemann zeta function.
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Date: 03/19
Speaker: Aaron Hoffman (CSUDH)
Title: City of Numbers: The Units over Fields of Prime Order
Abstract: This exploration of the units of the integers mod p will take the
viewer into a City of Numbers, where roads only go one way, and the
central government controls the shape of the districts. The findings
of this journey have results in Number Theory, and relating towards
teaching and learning this subject. Come support the undergraduate
speaker before he goes to represent CSUDH at Cal Poly in May and see
all of the additional material that was left out for the sake of time.
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Date: 02/13 (Room SBS B110)
Speaker: Katherine Stevenson
Title: Symmetries, Coverings, and Galois Theory: A case study in mathematical cross fertilization
Abstract: Group theory arises naturally in many areas of mathematics
as symmetries of objects. These symmetries allow us to understand more
complicated objects as being copies of simple ones "glued" together via
the action of a group of symmetries. We will look at how symmetries help
us understand covering spaces in topology and field extensions in algebra.
Then we will see how these two areas have inspired one another leading to
progress in long outstanding problems and opening new directions of research.
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Date: 9/19
Speaker: Rod Freed (CSU Dominguez Hills)
Title: An isomorphism between the ranges of two representations
Abstract: Let $f$ be a bounded linear isomorphism of a $C^*$ algebra, $X$, onto another $C^*$ algebra, $Y$, and let $U$ and $V$ denote the universal representations of $X$ and $Y$ respectively.
I show that $VfU^{-1}$ extends to a linear isomorphism of $U(X)$ onto $V(Y)$ that is also an ultraweak homeomorphism.
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Date: 10/2
Speaker: Chung-Min Lee (CSU Long Beach)
Title: Influence of straining on particles in turbulence
Abstract: Strain occurs in ocean and atmospheric flows and in many engineering applications, and it produces a large scale geometric change of the flow. We are interested in
seeing its influence in small flow scales. In particular we focus on parametric dependencies of particle movements in the turbulent flows. In this talk we will
introduce numerical methods used for simulating strained turbulence and particle movements, and present distribution and motion statistics of particles with different
Stokes numbers. The implications of the results will also be discussed.
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Date: 10/19
Speaker: Mitsuo Kobayashi (Cal Poly Pomona)
Title: Abundant interest, deficient progress: The study of perfect numbers and beyond
Abstract: The nature of perfect numbers have interested mathematicians from antiquity. These are the natural numbers, like 6, whose proper divisors add to the number itself. However, not much is known about such numbers, and questions such as how many of them exist are unresolved. In modern times, researchers have turned their attention to the nature of abundant and deficient numbers, which together make up the complement of the set of perfects. In this talk we will discuss what is now known about these numbers and in particular how the perfect, abundant, and deficient numbers are distributed in the naturals.
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Date: 11/16
Speaker: Glenn Henshaw (CSU Channel Islands)
Title: Integral Well-Rounded Lattices
Abstract: A well-rounded lattice is a lattice such that the set of vectors that achieve
the minimal norm contains a basis for the lattice. In this talk we will discuss
the distribution of integral well-rounded lattices in the plane and produce a
parameterization of similarity classes of such lattices by the solutions of certain
Pell-type equations. We will discuss applications of our results to the maximiza-
tion of signal-to-noise ratio with respect to well-rounded lattices with a fixed
determinant. Finally we will talk about integral lattices that come from ideals
in algebraic number fields. Under what conditions does the ring of integers of a
quadratic number field contain an ideal that corresponds to a well-rounded lat-
tice in the plane? We will address this and other related questions. Our work on
ideal lattices extend results by Fukshansky and Petersen on well-rounded ideal
lattices. This is joint work with L. Fukshansky, P. Liao, M. Prince, X. Sun, and
S. Whitehead.
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Date: 9/14
Speaker: Alexander Tyler (CSU Dominguez Hills)
Title: MathFest Advanture and Los Toros Math Competition.
Abstract:
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Date: 9/28
Speaker: Lenny Fukshansky (Claremont McKenna College)
Title: On the Frobenius problem and its generalization
Abstract: Let $N > 1$ be an integer, and let $1 < a_1 < \cdots < a_N$ be relatively prime integers. Frobenius number of this $N$-tuple is defined to be the largest positive integer that cannot be represented as a linear combination of $a_1,\ldots ,a_N$ with non-negative integer coefficients. More generally, the $s$-Frobenius number is defined to be the largest positive integer that has precisely $s$ distinct representations like this, so that the classical Frobenius number can be thought of as the 0-Frobenius number.
The condition that $a_1,\ldots ,a_N$ are relatively prime implies that $s$-Frobenius numbers exist for every non-negative integer $s$. The general problem of determining the Frobenius number, given $N$ and $a_1,\ldots ,a_N$, dates back to the 19-th century lectures of G. Frobenius and work of J. Sylvester, and has been studied extensively by many prominent mathematicians of the 20-th century, including P. Erdos. While this problem is now known to be NP-hard, there has been a number of successful efforts by various authors producing bounds and asymptotic estimates on the Frobenius number and its generalization. I will discuss some of these results, which are obtained by an application of techniques from Discrete Geometry.
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Date: 10/12
Speaker: Michael Krebs and Anthony Shaheen (CSU Los Angeles)
Title: How to Build Fast, Reliable Communications Networks: A Brief Introduction to Expanders and Ramanujan Graphs
Abstract: Think of a graph as a communications network. Putting in edges (e.g., fiber optic cables, telephone lines) is expensive, so we wish to limit the number of edges in the graph. At the same time, we would like the communications network to be as fast and reliable as possible. We will see that the quality of the network is closely related to the eigenvalues of the graph's adjacency matrix. Essentially, the smaller the eigenvalues are, the better the communications network is. It turns out that there is a bound, due to Alon, Serre, and others, on how small the eigenvalues can be. This gives us a rough sense of what it means for graphs to represent "optimal" communications networks; we call these Ramanujan graphs. Families of k-regular Ramanujan graphs have been constructed in this manner by Lubotzky, Sarnak, and others whenever k-1 equals a power of a prime number. No one knows whether families of k-regular Ramanujan graphs exist for all k.
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Date: 10/26
Speaker: Kiran S. Kedlaya (UC San Diego)
Title: The Sato-Tate conjecture for elliptic and hyperelliptic curves
Abstract: Consider a system of polynomial equations with integer coefficients. For
each prime number p, we may reduce modulo p to obtain a system of
polynomials over the field of p elements, and then count the number of
solutions. It is generally difficult to describe this count as an exact
function of p, so instead we take a statistical point of view, treating
the count as a random variable and asking for its limiting distribution
as we consider increasing large ranges of primes. Conjecturally, this
distribution can be described in terms of the conjugacy classes of a
certain compact Lie group. We illustrate this in three examples:
polynomials in one variable, where everything is explained in terms of
Galois theory by the Chebotarev density theorem; elliptic curves, where
the dichotomy of outcomes is predicted by the recently proved Sato-Tate
conjecture; and hyperelliptic curves of genus 2, where even the
conjectural list of outcomes was only found still more recently.
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Date: 11/9
Speaker: Wai Yan Pong (CSUDH)
Title: Geogebra and SAGE in the Classroom
Abstract: Geogebra is a free-software for Dynamic Geometry and more. SAGE (Software for Algebra and Geometry Experiment) is a free and open-source software for mathematics. I will talk about about I use these software in the classroom.
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