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.

 

Spring 2015

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.

 

Fall 2014

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

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.

Fall 2012

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 maximization 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 lattice 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.

Fall 2011

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)

Abstract: Let \(N > 1\) be an integer, and let \(1 < a_1 < \cdots < a_N \) be relatively prime integers. The 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.

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.