# CSU Dominguez Hill Mathematics Colloquium

Location: NSM A 115 C

Time: 2:45pm--3:45pm

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

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# Spring 2017

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.

# Spring 2016

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: 4/6

Time: 2:30 - 3:45 PM

Location: TBD

Title: 12th Annual Archimedes Prize Math Competition

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: 4/27

Time: 2:45 - 3:45 PM

Speaker: Henry Tucker (USC)

Title: TBD

Abstract: TBD

# Fall 2015

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.

# 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

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.

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.

# Spring 2013

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 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

# 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)

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.