Location: NSM A 115 C

Time: 2:45pm-3:45pm

We will have cookies and coffee starting at 2:30pm

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Date: 2/7

Time: 2:45pm - 3:45 PM

Speaker: Vincent Tran (CSUDH)

Title: Generalized Birthday Problem for homogeneous and inhomogeneous case

Consider the following problem involving r balls and n urns: we seek to compute the probability that no box contains more than one ball if the r balls are randomly tossed into the n boxes. We distinguish between the homogeneous case where all boxes have the same probabilities versus the more general inhomogeneous case. We will find general formulas and compare the two cases in several settings. If there are n = 365 boxes and the balls are considered to be birthdays of r random people, this probability corresponds to the change that at least two people in the group having the same birthday. We will look at further applications such as the protection of privacy in public data sets, and also to so-called birthday attacks of cryptographic hashing functions.

Speaker: Alexander Wittmond (CSUDH)

Title: Partition Problems and a Pattern of Vertical Sums

We give a possible explanation for the mystery of a missing number in the statement of a problem that asks for the non-negative integers to be partitioned into three subsets. Based on a pattern of sums of certain elements in the three sets, we nd a more standard solution to the problem, using only congruence modulo ve. We also show that the original statement plays a special role among all statements that satisfy the same pattern of the sums. This is joint work with Eunice Krinsky and Serban Raianu.

Speaker: Alexander Wittmond (CSUDH)

Title: The Design and Implementation of a Function Reactive Game Engine

Game engines are typically designed using either hierarchical or entity/component models and written in imperative languages. We examine the implementation of a game engine that was built using a data flow model and written in Haskell, a functional language. We also look at how function reactive programming can be applied to game engine design using Netwire, an arrow based functional reactive programming library.

Date: 10/11

Time: 2:45pm - 3:45 PM

Speaker: Sally Moite (CSUDH)

Title: First Open Numbers And Goldbach's Conjecture

Choose remainders r, one for each prime up to a last prime (LP). From the numbers 1,2,3,4,5,... eliminate numbers congruent to +/- any of these remainders mod the respective prime leaving a first open number (FON). A problem is to find the maximum first open number (MFON) for any choice of remainders. Some computed results are presented for last primes up to 43, as well as a conjecture on an upper limit for the MFON as a function of LP. If this conjecture is true, it would prove Goldbach's Theorem. Some elements of the computations are presented, along with some results for partial computations for last primes up to 2753.

Date: 4/12

Time: 2:45pm - 3:45 PM

Speaker: Gene Kim (USC)

Title: Distribution of Descents of Fixed Point Free Involutions

Recall that a permutation \(\pi \in S_n\) has a *descent*
at position \(i\) if \(\pi(i+1) > \pi(i)\), and the *descent number*
of \(\pi\), \(d(\pi)\), is the number of positions \(i\) for which \(\pi\) has
a descent at position \(i\). It is well known that the distribution of
\(d(\pi)\) in \(S_n\) is asymptotically normal. In this talk, we discuss
the distribution of \(d(\pi)\) of a specific conjugacy class of \(S_n\):
the fixed point free involutions (otherwise known as matchings). We also
explore an interesting bijection that we discovered.

Date: 3/22

Time: 2:30pm - 3:45 PM

Speaker: Francis Edward Su (Harvey Mudd College)

Title: Voting in Agreeable Societies

When does a candidate have the approval of a majority? How does the geometry of the political spectrum influence the outcome? What does mathematics have to say about how people behave? When mathematical objects have a social interpretation, the associated results have social applications. We will show how some classical mathematics can be used to understand voting in "agreeable" societies. This talk also features research with undergraduates.

Date: 2/8

Time: 2:45pm - 3:45 PM

Speaker: Natalia da Silva (CSUDH)

Title: Numbers That Can be Written as Differences of Harmonic Numbers

Our research started with a conference by Hendrik Lenstra, in which he discusses harmonic numbers as defined by Philippe de Vitry around 700 years ago. A number is called harmonic if it can be written as a power of 2 times a power of 3. Gersonides proved around the same time that there are only four pairs of consecutive harmonic numbers. His result can be interpreted as follows: the famous abc-conjecture is true if instead of looking at all natural numbers we restrict attention to harmonic numbers only. Mersenne primes are prime numbers which can be written as a prime power of 2 minus 1. We investigate in how many ways Mersenne primes can be written as a difference of harmonic numbers, and we find that all the ones greater than 31 can only be written as the original power of 2 minus 1. One of our main results is that we can extend the set on which the ABC conjecture is true by adding the set of Mersenne primes to the set of harmonic numbers. This is a report on work in progress joint with Serban Raianu and Hector Salgado, with support from PUMP and NSF Grant DMS-1247679.

Speaker: Hector Salgado (CSUDH)

Title: Numbers That Cannot be Written as Differences of Harmonic Numbers

Harmonic numbers are numbers written as products of powers of two and three. Their differences have been looked at before (e.g. by Lenstra, De Vitry, and Gersonides ) and it was noticed that there is a finite number of ways that differences of harmonics numbers are equal to one. Looking at the harmonic numbers below one thousand, and looking at their differences, we noticed that eleven numbers below one hundred could not be written as a difference of harmonic numbers and gave a proof of this fact. We note that 41 is the first number that is not a difference of harmonic numbers, and it has appeared as a notable number in the works of Euler and many other (according to Smend even Bach, the composer). Following this, we show there are infinitely many numbers that cannot be written as a difference of harmonic numbers. We connect this with the famous ABC conjecture from number theory, and show that the conjecture is true on the union of set of harmonic numbers with finitely many of non-differences of harmonic numbers. This is a report on work in progress joint with Natalia da Silva and Serban Raianu, with support from PUMP and NSF Grant DMS-1247679.

Date: 3/8

Time: 1:00 - 2:00 PM

Speaker: John Rock (Cal Poly Pomona)

Title: A Tabular Method for Integration by Parts

Abstract: Integration by Parts (IBP) is a very useful technique that has a undeserved bad reputation. IBP allows us to solve a wide variety of problems in calculus and even provides a way to prove Taylor's Theorem with remainder, but the manner in which this technique is typically taught is woefully and unnecessarily inefficient. In this lecture, a tabular approach to IBP that is designed to reduce such inefficiency will be discussed and several examples will be considered. Note that this tabular method is not a shortcut. Rather, it simply avoids redundancy. Also, the talk will begin with an introduction to the excellent PUMP program.

Date: 3/24

Time: 2:45 - 3:45 PM

Speaker: Guangbin Zhuang (USC)

Title: Hopf Algebras of Finite Gelfand-Kirillov Dimension

Abstract: Hopf algebras occur naturally in group theory, in Lie theory, in group scheme theory, and in numerous other places across the fields of mathematics and physics. Since the popularization of quantum groups (which can be deemed as a special kind of Hopf algebras) around 1980s, a great number of noncommutative Hopf algebras have been introduced and the study of them remain active ever since. In the last few years, a lot of effort has been devoted to the classification of Hopf algebras of finite Gelfand-Kirillov dimension. For example, in a very recent preprint, Wu, Liu and Ding complete the classification of prime regular Hopf algebra of GK-dimension one, which was initiated by Lu-Wu-Zhang and Brown-Zhang. Also, some interesting examples has been discovered in the classification of connected Hopf algebras of low GK-dimension. In the talk, I am going to mention some basics of Gelfand-Kirillov dimension and Hopf algebras. I will also talk about some classification results on Hopf algebras of low GK-dimension.

Date: 4/20

Time: 2:45 - 3:45 PM

Speaker: Cynthia Parks and Keith Ball (CSUDH)

Title: Wronskians and Linear Dependence for Formal Power Series Ring

Abstract: In 2010 Bostan and Dumas proved that the vanishing of generalized Wronskians for a finite family of formal power series over a characteristic zero field implies their linear dependence over the base field. In this project, we generalize Bostan and Dumas' result to formal power series ring in countably many variables. We take advantage of the fact that this ring is isomorphic to the ring of \(K\)-valued arithmetic functions which naturally comes with the log-Wronskian if \(K\) contains \(log(n)\) for each \(n \ge 1\). We then remove this assumption on \(K\) by some basic facts in field theory and linear algebra.

Date: 4/28

Time: 2:45 - 3:45 PM

Speaker: Henry Tucker (USC)

Title: Fusion categories, their invariants, and classifications via operator algebra methods

Abstract: The objects of fusion categories generalize the properties of complex representations of finite groups: they can be decomposed into sums of irreducible objects, tensor products and duals are well-defined, and their morphisms are linear maps on finite dimensional vector spaces. They appear as invariants of knots and quantum field theories, as representations of quantum groups and vertex algebras, and as order parameters for topological states of matter. A great deal of recent progress has been made in classifying these categories and in understanding their invariants. In this talk I will discuss my ongoing study of the so-called near-group fusion categories; that is, those with only one non-invertible object. I have established formulae for their Frobenius-Schur indicators (categorical generalizations of the classical indicators for finite groups) in important cases and I am beginning work toward completing their classification by realizing these abstract fusion categories as systems of endomorphisms on Cuntz C* algebras and their non-unitary generalizations. From this realization we expect to establish an understanding of the classification parameters for near-group fusion categories in terms of Weil representations of quadratic forms.

Date: 9/9

Speaker: Matthias Aschenbrenner (UCLA)

Title: Elimination theory for transseries

Abstract: The concept of a "transseries" is a natural extension of that of a Laurent series, allowing for exponential and logarithmic terms. The germs of many naturally occurring real-valued functions of one variable have asymptotic expansions which are transseries. Since the late 1990s, van den Dries, van der Hoeven, and myself, have pursued a program to understand the algebraic and logical aspects of the differential field of transseries. Recently we were able to make a significant step forward. My goal for this talk is to give a gentle introduction to transseries, to explain our recent work, and to state some open problems.

Date: 9/30

Speaker: Joshua Sack (CSULB)

Title: Duality for Quantum Structures

Abstract: This talk presents dualities between two types of quantum structures. One type is a lattice-ordered algebraic structure, called a Hilbert lattice, that serves as a discrete analog to the Hilbert space for reasoning about testable properties of a quantum system; Hilbert lattices are central to the original quantum logic developed by Birkhoff and von Neumann. The other, called a quantum Kripke frame, is a relational graph-like structure that, like a labelled transition system, is used to model how a computation evolves through time; quantum Kripke frames give meaning to the logic of quantum actions. This duality connects two different perspectives on quantum structures, one a static perspective about testable properties, and the other a dynamic perspective concerning the results of quantum actions. This duality connects two different perspectives on quantum structures, one a static perspective about testable properties, and the other a dynamic perspective concerning the results of quantum actions.

Date: 10/7

Speaker: James Freitag (UCLA)

Title: Isogenies of elliptic curves and differential equations

Abstract: An isogeny of elliptic curves is a surjective morphism of algebraic groups which has finite kernel. Elliptic curves are classified by their j-invariants, and we call the collection of j-invariants of all elliptic curves isogenous to a given elliptic curve an isogeny class. Several number theoretic special points conjectures, which we will describe, concern the intersection of products of isogeny classes with algebraic varieties. Part of the challenge of understanding such intersections is that the products of isogeny classes are countable discrete sets. The idea of this talk concerns the replacement of isogeny classes with an object more like an algebraic variety, the solution set to a system of differential equations. Analyzing the appropriate differential equations leads to results regarding the isogeny class intersections. We will give a general exposition of elliptic curves, isogenies, and special points conjectures.

Date: 10/21

Speaker: Daniele Struppa (Chapman University)

Title: Regularity for functions on quaternions

Abstract: Back in the thirties, Fueter developed a theory of analyticity
for functions of a quaternionic variable; such functions are known as Fueter
regular. His theory, based on a clever extension of the Cauchy-Riemann
operator (to what is now known as Cauchy-Fueter operator) was very
successful, but difficult to extend to the case of several variables. In
the first part of my talk I will discuss some of my work from 1996-2006,
which allowed the construction of a stable and deep theory of analyticity in
several quaternionic variables. Since 2006 I have worked to remedy the major
shortcoming of the Fueter theory, namely the fact that polynomials and power
series are not analytic in the sense of Fueter. In 2006 I introduced a new
notion of what my coauthor and I called slice regularity, that successfully
includes power series as a special case of regularity. The second part of
my talk will describe the first rudiments of such a theory.

References: For the first part of the talk: F. Colombo, I. Sabadini,
F. Sommen, D.C. Struppa, Analysis of Dirac Systems and Computational Algebra,
Birkhauser, 2004.
For the second part of the talk: G. Gentili, C. Stoppato, D.C. Struppa,
Regular Functions of a Quaternionic Variable, Springer, 2013.

Date: 11/18

Speaker: Katarzyna Wyka (CUNY School of Public Health)

Title: The application of latent transition analysis to large scale disaster data: modeling PTSD in a population of disaster workers.

Abstract: Sophisticated statistical methodologies are needed in order to analyze large, population-based datasets, such as screening projects, following disasters. The purpose of this paper is to demonstrate the utility of latent transition analysis (LTA) in disaster research. The persistence of posttraumatic stress symptoms resulting from the World Trade Center (WTC) disaster exposure has been well documented. However, little is known about whether the developmental trajectories of these symptoms are associated with their distinct phenotypic expressions. Based on 5 annual waves of data (2003-2008), four posttraumatic symptom profiles were identified among the WTC disaster workers (n=2960). Symptomatic profile was characterized by high probability of endorsing the majority of 17 posttraumatic stress symptoms and the highest symptom severity (profile prevalence: T1 1%, T2 7%, T3 4%, T4 3%, T5 2%). Intermediate-Avoidance and Intermediate-Numbing profiles had similar symptom severity but distinct probabilities of endorsing the avoidance and numbing symptoms, respectively (prevalence: T1 22%, T2 18%, T3 12%, T4 8%, T5 6% and T1 11%, T2 8%, T3 5%, T4 4%, T5 4%). Non-symptomatic profile prevalence was 58%, 66%, 80%, 85% and 87% over time. The profiles with elevated symptoms showed relatively moderate stability (34%-53%) and distinct prognostic trajectories, particularly with regard to symptom remission. These finding have implications for post-disaster interventions and may help inform etiological models of PTSD.

Date: 4/16

Speaker: Yinhuo Zhang (University of Hasselt, Belgium)

Title: Brauer groups

Abstract: This is a survey talk on the developing of the Brauer groups of structured algebras. The classical Brauer group of a field K classifies the central division algebras over K. The Brauer-Wall group or the super Brauer group for a field K classifies finite-dimensional graded central division algebras over the field. In this talk, we show how these Brauer groups have been generalized to the Brauer groups of (braided) tensor categories.

Date: 04/08

Speaker: Serban Raianu (CUSDH)

Title: External Homogenization: from graded rings to corings via Hopf algebras

Abstract: External homogenization is a construction/method used in the 1980's to prove results about graded rings and modules. It was then extended to Hopf algebras coacting on algebras. We give a coring version of it and also provide a coring version of a Maschke-type theorem.

Date: 03/03 (Tue)

Speaker: John Rock (Cal Poly Pomona)

Title: An Introduction to Fractal Geometry

Abstract: The word `fractal' was first coined by Benoit Mandelbrot in 1975 to describe mathematical monsters that exhibit highly irregular and counter-intuitive structure. However, the use of the word monster to describe such fractal objects turns out be rather ironic. Indeed, to be monstrous is to be unnatural, and yet Mandelbrot's seminal book, The Fractal Geometry of Nature , reveals that nature itself exhibits fractal structure in a seemingly endless variety of ways. In this talk, we will discuss examples of fractals that arise from various natural and mathematical contexts and look into some of the mathematical tools that have been created to analyze these fascinating objects, including some recent work done by graduate and undergraduate students at Cal Poly Pomona. Also, information about grant opportunities for CSU math majors will be discussed at the start of the talk.

Date: 02/18

Speaker: Christopher Lee (UCLA)

Title: Intelligent Agents in Evolutionary Game Theory: the Transition to Tyranny

Abstract: The Prisoner's Dilemma is a classic problem in game theory, and has been intensively studied in many fields, such as the evolution of cooperation. One of its long-standing results has been that simple strategies such as Tit-for-Tat (TFT) out-perform more complex strategies. Recently, an exciting new class of first-order Markov strategies called Zero Determinant (ZD) strategies has been discovered, as an outstanding example of this principle, and has been shown under certain assumptions to be universally robust to invasion by other strategies. In this work, however, we report a very different, non-Markov strategy in the form of "intelligent agents" that are capable of self-recognition, which we find alters these conclusions in several ways: 1. this makes the selection of an optimal strategy vector to use against a population of Markov opponents depend strongly on the intelligent agents' population fraction; e.g. when in the minority they might fare best by cooperating with Markov opponents, whereas in the majority by defecting. 2. ZD strategies are not universally robust against such agents, and empirically our real-world agent implementation greatly out-performed the best Markov-1 strategies such as TFT, Win-Stay-Lose-Shift, and ZD. 3. Indeed, such agents make the criterion of "universal robustness" look weak; we prove that such agents can attain a far greater fitness advantage limit, that we define as "maximal resident advantage" (MRA), which obligates them to attack (defect against) their opponents. 4. Agents can attain MRA whenever their population fraction rises above a threshold that depends on the opponents' strategy, but is never higher than 50%. Below this threshold, some level of cooperation with opponents is favored; above it, never. This "transition to tyranny" arises for any group of agents capable of self-recognition. 5. We show using geometric considerations that for some score matrices, agents' optimal level of cooperation with any Markov-1 opponent decreases monotonically as the agents' population fraction increases. We illustrate our work using the Prisoners Dilemma, but our approach is applicable to a wide variety of games.

Here is the abstract in MS Words.

Date: 11/20

Speaker: Corey Dunn (CSU San Bernardino)

Title: Relating linear dependence of algebraic curvature tensors
to simultaneous diagonalization of operators

Abstract: The "curvature" of a surface is a tricky object to
define, and it wasn't formally done so until 1827 by
Gauss. Generally, on surfaces of dimension greater than two
this object is quite complicated. As a result, it is sometimes advantageous
to study an algebraic portrait of this curvature, known as an "algebraic
curvature tensor". In this talk, we introduce these algebraic
curvature tensors and describe an open problem of current interest: how
efficiently can one express curvature? We describe some of what is
known about this efficient expression of curvature, and illustrate how it
is related to simultaneous diagonalization of linear operators. Any
student with a knowledge of basic linear algebra should understand almost
everything, and there will also be more advanced perspectives that should
be of interest to the faculty as well.

Date: 11/05

Speaker: Daniel Katz (CSU Northridge)

Title: Sequences, Correlations, and Number Theory

Abstract:
Many problems in engineering require sequences of +1s and -1s having
low

autocorrelation, that is, they do not resemble translated
versions of

themselves. Interestingly, this is equivalent to
a much-studied problem

in complex analysis. Random sequences
are not particularly good, and

the best known sequences come from
constructions in number theory.

Proving that these constructions
work is challenging, and both algebra

and analysis play crucial
roles.

Date 10/22

Speaker: Ranjan Bhaduri (Sigma Analysis & Management)

Title: Some Musings of a Mathematician about the Hedge Fund Space

Room: NSM C 213 (notice the room change)

Abstract: The hedge fund space has grown into a multi-trillion dollar business, and there are several quantitative and systematic hedge funds in existence. In addition, certain mathematical techniques are invoked in the hedge fund industry. This talk gives some insights about the mathematics utilized in the hedge fund world. In addition, it gives some nuggets of wisdom to students (both undergraduate and graduate) looking to have success in the business and finance world.

Here is the pointpower presentation of Ranjan's talk.

Date: 10/9

Speaker: George Jennings (CSUDH)

Title: The Poincare' disk for mortals

Abstract: The Poincare' disk is an discuss some background of lattice
problems and computational complexity, and then concentrate on a special
very useful class of cyclic lattices, on which SVP and SIVP turn out to
be equivalent with positive probability.
# Spring 2013

Date: 11/18

Speaker: Peter Petersen (UCLA)

Title: Two Curious Results for Planar Curves

Abstract: This is not a research talk. Rather I will explain two more or less well-known theorems about planar curves. One is the “Four Vertex Theorem”. About 20 years ago Osserman discovered a purely descriptive proof that works for all simple closed curves. In texts one generally only sees an older more analytic proof that only works when the curve is strictly convex. The other result is a theorem about general planar curves discovered by Frabicius-Bjerre. It turns out that such curves have an algebro-geometric relationship between double points, double tangents, and inflection points. The proof is also quite simple and descriptive. Henceforth you’ll be allowed to create doodles and know that these have mathematical significance.

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.

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.

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.

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.

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.

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.

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.

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.

Date: 9/14

Speaker: Alexander Tyler (CSU Dominguez Hills)

Title: MathFest Advanture and Los Toros Math Competition.

Abstract:

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

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.

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.

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.