# MathCS Seminar 2014

## Fall 2014

### Thursday, December 18th, 2014 at 4pm (tea and cookies at 3:30pm)

#### *Speaker:* **Lander Cnudde, University of Ghent, Belgium**

*Title:* **Fourier transforms in commutative and non-commutative multicomplex settings**

*Abstract:* This seminar addresses the generalization of the
classical Fourier transform to multicomplex settings. Inspired by a
successful case study on the slices of the non-commutative Clifford
algebra $Cl_{m+1}$, a more conceptual approach to the matter is
established. Using operator relations, we construct a general
background that allows to create Fourier analogues in more general
non-commutative as well as commutative settings. Finally we illustrate
this claim and the underlying line of thoughts by setting up a Fourier
transform for the bicomplex numbers which turns out to be in
accordance to our expectations. The framework uses concepts of both
analysis and algebra, with key roles for the Mehler formula and the
Hille-Hardy formula.

### Wednesday December 10th 2014 at 4PM (tea and cookies at 3.30PM)

#### *Speaker:* **Luke Smith, Graduate Student, Department of Mathematics, University of California, Irvine**

*Title:* **Polytope Bounds on Multivariate Value Sets**

*Abstract:* Over finite fields, if the image of a polynomial map is
not the entire field, then its cardinality can be bounded above by a
significantly smaller value. Earlier results bound the cardinality of
the value set using the degree of the polynomial. However, these
bounds can be improved significantly if our bounds depend on the
powers of all monomials in a polynomial map, rather than just the one
with the highest degree. The Newton polytope of a polynomial map is
one such object constructed by each of these monomials, and its
geometry provides sharp upper bounds on the cardinality of the value
set. In this talk, we will explore the geometric properties of the
Newton polytope and show how allows for an improvement on the upper
bounds of the multivariate value set.

*Bio:* Luke Smith is a 6th year PhD student at UCI. His research
interests involves number theory, finite fields, value sets, and Witt
vectors. He also enjoys teaching and has recently been involved in
mathematics educational outreach with the UCI Math circle and MIND
Research Institute.

### Friday October 24th 2014 at 12.30 (tea and cookies at noon)

#### *Speaker:* **Dr. Brendan Fahy, Postdoctoral Fellow, KEK High Energy Research Organization, Tsukuba, JapanTBA**

*Title:* **Linear combination interpolation, Cuntz relations and infinite products (joint work with I. Lewkowicz, P. Jorgensen and D. Volok)**

*Abstract:* Calculating observable quantities in QCD at low energies
requires a non-perturbative approach. Lattice QCD is a
non-perturbative solution which quantities can be estimates using
Monte Carlo methods. However many quantities such as multi-hadron
operators require large amounts of computational power to
compute. Using the stochastic LapH method the costly matrix inverses
required are estimated rather than computed exactly drastically
reducing the computational costs. These modern computation techniques
allow for the computation of a large number of operators including
multi-hadron operators. Results of the spectrum of energies for the
lowest 50 bound states in a finite box are presented for the rho-meson
channel.

### Tuesday October 21st 2014 at 4pm (tea and cookies at 3:30pm)

#### *Speaker:* **Prof. Daniel Alpay, Ben-Gurion University of the Negev, Israel**

*Title:* **Linear combination interpolation, Cuntz relations and infinite products (joint work with I. Lewkowicz, P. Jorgensen and D. Volok)**

*Abstract:* We introduce the following linear combination
interpolation problem: Given $N$ distinct numbers $w_1,..., w_N$ and $N+1$
complex numbers $a_1,..., a_N $and $c$, find all functions f(z) analytic
in a simply connected set (depending on f) containing the points
$w_1,...,w_N$ such that $\sum_{u=1}^N a_u f(w_u)=c$. To this end we prove
a representation theorem for such functions f in terms of an
associated polynomial p(z). We first introduce the following two
operations, substitution of p, and multiplication by monomials $z^j$ ,
$0<= j < N$. Then let M be the module generated by these two operations,
acting on functions analytic near 0. We prove that every function f,
analytic in a neighborhood of the roots of p , is in M. In fact, this
representation of f is unique. To solve the above interpolation
problem, we employ an adapted systems theoretic realization, as well
as an associated representation of the Cuntz relations (from
multi-variable operator theory.) We study these operations in
reproducing kernel Hilbert space): We give necessary and sufficient
condition for existence of realizations of these representation of the
Cuntz relations by operators in certain reproducing kernel Hilbert
spaces, and offer infinite product factorizations of the corresponding
kernels.

### CECHA Workshop on Integral transforms, boundary values and generalized functions, Fall 2014

Schedule: October 17th - October 21st 2014

#### Friday October 17th 2014

Chairperson: Irene Sabadini

** 10:50am-11:05am Registration/ Welcome **

** 11:05am-11:55am Michael Shapiro, Instituto Politecnico Nacional, Mexico **

Title: “On the Hilbert and Schwarz Formulas and Operators”

** 11:55am-12:10pm Discussion Session **

** 12:20pm-1:30pm Lunch, Athenaeum **

** 2:00pm-2:50pm Mircea Martin, Baker University **

Title: “Spin Operator Theory”

** 2:50pm-3:10pm Discussion Session **

** 3:10pm-4:00pm M. Elena Luna Elizarraras, Instituto Politecnico Nacional, Mexico **

Title: “A Bicomplex Model of Lobachevsky Geometry”

** 4:00pm-4:20pm Discussion Session **

#### Saturday October 18th 2014

Chairperson: Paula Cerejeiras

** 10:00am-10:50am Matvei Libine, Indiana University Bloomington **

Title: "Geometric Properties of Conformal Transformations on $R^{p,q}$"

** 10:50am-11:05am Discussion Session **

** 11:05am-11:55am Ahmed Sebbar, Institut de Mathématiques de Bordeaux **

Title: “Motions of Critical points of Green's functions”

** 11:55pm-12:00pm Discussion Session **

** 12:00pm-1:15pm Lunch, Sandhu **

** 1:30pm-2:20pm Fabrizio Colombo, Politecnico di Milano, Italy **

Title: “The Fueter-Sce Mapping and its Inverse”

** 2:20pm-2:30pm Discussion Session **

** 2:30pm-3:20pm Adrian Vajiac, Chapman University **

Title: “Multicomplex Hyperfunctions”

** 3:20pm-3:30pm Discussion Session **

#### Sunday October 19th 2014

Chairperson: Mihaela Vajiac

** 10:00am-10:50am Irene Sabadini, Politecnico di Milano, Italy **

Title: “Monogenic Hyperfunctions in One and Several Variables”

** 10:50am-11:05am Discussion Session **

** 11:05am-11:55am Uwe Kӓhler, University of Aveiro **

Title: “Crystallographic structures: how to make an effective reconstruction by the spherical X-ray transform?”

** 11:55pm-12:00pm Discussion Session **

** 12:00pm-1:15pm Lunch, Sandhu **

** 1:30pm-2:20pm Paula Cerejeiras, University of Aveiro **

Title: “Diffusive Wavelets for Nilpotent Groups”

** 2:20pm-2:30pm Discussion Session **

** 2:30pm-3:20pm Daniel Alpay, Ben-Gurion University of the Negev, Israel **

Title: “Spaces of stochastic (commutative and non commutative) distributions and applications”

** 3:20pm-3:30pm Discussion Session **

#### Monday October 20th 2014

Chairperson: M. Elena Luna Elizarraras

** 10:00am-10:50am Craig Nolder, Florida State University **

Title: “Conjugate Harmonic Components of Monogenic Functions and Symmetry”

** 10:50am-11:05am Discussion Session **

** 11:05am-11:55am Graziano Gentili, Università di Firenze **

Title: “Spherical power expansion and a Mittag-Leffler theorem for semi-regular functions”

** 11:55am-12:10am Discussion Session **

** 12:20pm-1:30pm Lunch, Athenaeum **

** 2:00pm-2:50pm Dana Clahane, Fullerton College **

Title: “Complex, Bicomplex, and Quaternionic Gaussian Moat Problems”

** 2:50pm-3:10pm Discussion Session **

** 3:10pm-4:00pm Lander Cnudde, Universiteit Gent, Belgium **

Title: “Slice Fourier transform: definition, properties and corresponding convolutions”

** 4:00pm-4:20pm Discussion Session **

** 6:30-8:30pm Social Dinner **

#### Tuesday October 21st 2014

** 10:00am-12:10pm Discussion Session **

** 12:20pm-1:30pm Lunch, Athenaeum **

** 2:00pm-4:00pm Discussion Session **

### Thursday October 9th 2014 at 4pm (tea and cookies at 3:30pm)

#### *Speaker:* **Prof. Ahmed Sebbar, Institut de Mathematiques de Bordeaux**

*Title:* **On a Remarkable Power Series**

*Abstract:* We consider the sequence, defined by

$s_{2n} = s_{n}, n \geq 1; s_{2n+1} = (-1)^{n}, n \geq 0 $

or equivalently

$s_{n} = (-1)^{b} $ if $n=2^a(1+2b); a,b \in \mathbf{N}$

We explain how it is related to paperfolding and we give a precise analysis at $x = 1$ of the power series

$f(x) = \sum s_n x^n$

### Thursday, September 25th 2014, at 4pm (tea and cookies at 3:30pm)

#### *Speaker:* **Prof. Christopher Lyon, CalState Fullerton**

*Title:* **Two notions of mirror symmetry for certain K3 surfaces**

*Abstract:* In the mid-1990s, the physicists Berglund and Hubsch
proposed a way to construct a ``mirror partner* for certain kinds of*
Calabi-Yau manifolds. When the manifold has (complex) dimension 2,
these are examples of K3 surfaces. Around the same time, Dolgachev
and others conceived of a version of mirror symmetry that applies to
more general families of K3 surfaces. In this talk, we will introduce
these special kinds of K3 surfaces, which are defined as hypersurfaces
in weighted projective space. Then we will discuss the issue of
compatibility between the aforementioned versions of mirror symmetry.
While the question is open in general, we will highlight a particular
collection of surfaces where the compatibility can be proved. This is
joint work with Paola Comparin, Nathan Priddis, and Rachel Webb.

### Thursday, September 11th 2014, 4pm (tea and cookies at 3:30pm)

#### *Speaker:* **Prof. Ahmed Sebbar, Institut de Mathematiques de Bordeaux**

*Title:* **Equivariant functions**

*Abstract:* An equivariant function is a special meromorphic
function on the Poincare upper half-plane. A concrete non trivial
example was given by Don Zagier answering a question of the physicist
Werner Nahm. We show how to construct all the equivariant functions
by using ideas from complex analysis, modular forms and projective
differential geometry. The talk is based on a joint work with
Abdellah Sebbar from The university of Ottawa.

## Spring 2014

### Wednesday, May 14, 2014 from noon to 12:50 pm in VN 116

#### *Speaker:* **Dr. Roman Buniy, Chapman University**

*Title:* **Generalized helicity and Beltrami fields**

*Abstract:* We propose covariant and non-abelian generalizations of the magnetic helicity and Beltrami equation. The gauge invariance, variational principle, conserved current, energy-momentum tensor and choice of boundary conditions elucidate the subject. In particular, we prove that any extremal of the Yang-Mills action functional $\frac{1}{4}\int_\Omega{F_{\mu\nu}F^{\mu\nu}}\,d^4x$ subject to the
local constraint $\epsilon^{\mu\nu\alpha\beta}{F_{\mu\nu}F_{\alpha\beta}}=0$ satisfies
the covariant non-abelian Beltrami equation.

### Thursday, May 8, 2014 from 4pm to 5pm in Von Neumann Hall

#### *Speaker:* Dr. Hendrik De Bie, Clifford Research Group, Ghent University, Belgium

*Title:* **Convolutions for the quaternion Fourier transform with applications in image processing**

*Abstract:* In this lecture I will explain how a quaternion Fourier
transform may be defined, an how a suitable convolution product can be
associated with it.I will subsequently discuss an application in color image processing,
namely a color edge detector. If time allows, I will also show how some of these results may be
generalized to higher dimensions using Clifford algebras.

### Thursday, April 24, 2014 from 4pm to 5pm in Von Neumann Hall

#### *Speakers:* Ryan Burns, Isaac Lien, Scott Lien, David Tyler

*Title:* **GrandPAD: Finally technology so easy that it can be used by a people ages 75 to 105.**

*Abstract:* GrandPAD is a technology startup that was born at Chapman that is focused on providing an easy way for seniors citizens over the age of 75 to stay connected with their friends and family.
GrandPAD was founded by Isaac Lien (CIS Major Class of 2017) and Scott Lien (Father of Isaac and former executive with Intuit.) Other key members of the team include Ryan Burns (Computer Science Class of 2014) and David Tyler (Computer Science Class of 2014).
Come and learn about the technology we have used to build GrandPAD and see the product in action.
GrandPAD -- Simply Connected

See more at: http://www.GrandPAD.net

### Thursday, April 17, 2014 from 4pm to 5pm in Von Neumann Hall

#### *Speaker:* Professor Andrew Moshier, Chapman University

*Title:* **String diagrams for categories of posets**

*Abstract:* String diagrams, e.g., for braiding and symmetric monoidal categories, have
proven useful in Physics and Theoretical Computer Science, thanks to coherence
theorems that tell us that diagrams equivalent under suitable topological invariants
denote equal morphisms. In this talk, we extend a similar courtesy to categories of
partially ordered sets by adding an order on diagrams and allowing for topologically
"lossy" diagram rewrites. As an application, we characterize the objects in such
categories that behave as semi-lattices, lattices and distributive lattices.

PDF copy of the abstract containing a diagram.

### Thursday, April 10, 2014 from 4pm to 5pm in Von Neumann Hall

#### *Speaker:* Dr. Ali Nayeri, Chapman University

*Title:* **String Gas Cosmology and the Blue Tilt of the Primordial Gravitational Waves**

*Abstract:* The BICEP-2 team has reported the detection of primordial cosmic microwave background B-mode polarization, with hints of a suppression of power at large angular scales relative to smaller scales. Provided that the B-mode polarization is due to primordial gravitational waves, this might imply a blue tilt of the primordial gravitational wave spectrum. Such a tilt would be incompatible with standard inflationary models, although it was predicted some years ago in the context of a mechanism that thermally generates the primordial perturbations through a Hagedorn phase of string cosmology. It has recently been shown that a Hagedorn phase of string gas cosmology can provide a causal mechanism for generating a nearly scale-invariant spectrum of scalar metric fluctuations, without the need for an intervening period of de Sitter expansion.The purpose of this talk is to encourage greater scrutiny of the data with priors informed by a model that is immediately falsifiable, but which predicts features that might be favored by the data-- namely a blue tensor tilt with an induced and complimentary red tilt to the scalar spectrum, with a naturally large tensor to scalar ratio that relates to both.

### Monday, April 7, 2014 at noon in Von Neumann Hall

#### *Speaker:* Dr. Cary Deck, University of Arkansas

*Title:* **Behavior in Interconnected Strategic Contests**

*Abstract:* Many strategic situations can be described as a contest where agents make unrecoverable investments in the hope of claiming a prize. This presentation discusses a series of projects where outcomes depend on a set of interconnected contests. One looks at the benefit of alliance formation when there is an opportunistic challenger who can decide which contest to enter. The other considers the case of payoff complementarities across sub-contests. In both cases the comparative statics of the respective model hold in aggregate, however individual behavior systematically differs from what is predicted.

### Wednesday, March 5, 2014 at noon in Von Neumann Hall

#### *Speaker:* Professor Asen Dontchev, University of Michigan, Ann Arbor

*Title:* **Inverse Function Theorems: Old and New**

*Abstract:* The classical inverse/implicit function theorems revolves around solving an equation in terms of a parameter and tell us
when the solution mapping associated with this equation
is a differentiable function with respect to the parameter. In this talk we move into a much wider territory in replacing
equation-solving by problems involving nonsmooth equations, as well as
models of optimization, equilibrium, and control theory.

It turns out that if we put aside differentiability and focus on Lipschitz continuity only, or even more general metric regularity properties of mappings, we can cover a wider range of models and get estimates of the solution changes resulting from approximations of the model. As illustrations we will present implicit function theorems for inequalities and variational inequalities.

### Thursday, February 27, 2014 from 4pm to 5pm in Von Neumann Hall

#### *Speaker:* Professor Michael Campbell from CSUF

*Title:* **Phase Transitions in Economic Models: Hot and Cold Market Equilibria in Bounded-Rational Potential Games**

*Abstract:* In economic “games” for which there exists a potential (Shapley & Monderer), a dynamical model for which each agent’s strategy adjustment follows the gradient of the potential (perfectly rational part that wants to maximize payoff) along with a normally distributed random perturbation (part that considers errors in judgment, miscalculations, emotional bias, etc), is shown to equilibrate to a Gibbs measure for a finite number of agents. There is also a non-dynamical way (large deviation theory) to impose this “bounded rationality” using Shannon entropy of agents arbitraging information within the system, and the equilibrium measure is again the Gibbs measure just as in statistical mechanics. For an infinite number of agents, more than one equilibrium measure may occur, which is the analogy of a phase transition in statistical mechanics (similar to water changing to ice or steam). Here, the fluctuation variable that allows us to adjust the influence of the “irrational” element of decisions is related to “temperature” in statistical mechanics. A simple discrete Cournot oligopoly with increased local competition among agents has a phase transition. This model has a rich phase diagram with an "antiferromagnetic" checkerboard state, striped states and maze-like states with varying widths, and finally a "paramagnetic" unordered state. Such phases have economic implications as to how agents compete given various restrictions on how goods are distributed. The standard Cournot model corresponds to a uniform distribution of goods, whereas the power-law variations correspond to goods for which the distribution is more localized.

### Thursday, January 23 -- Tuesday, January 28, 2014 in Von Neumann Hall

#### 6th Annual CECAT Workshop in Pointfree Mathematics

**Hosted by the Center of Excellence in Computation, Algebra and Topology (CECAT)**

Held at Chapman University, Von Neumann Hall (545 W. Palm Ave, Orange, CA 92866)

**Program**

**Thursday 23rd**

10.00am-11.30am **Bernhard Banaschewski**

"Further thoughts on Pointfree Function Rings"

12.30pm-1.30pm **Fred Dashiell**

"A Geometric description of the second dual of C(K)"

**Friday 24th**

11.30am-12.30pm **Bernhard Banaschewski**

"Further thoughts on Pointfree Function Rings" cont.

1.00pm-2.00pm **Peter Jipsen**

"Planar (semi) modular lattices"

**Saturday 25th**

11.30am-1pm **Mark Sioen**

"Some remarks (and more questions) about the topology of uniform convergence on preferred sublocales "

**Monday 27th**

noon-1.30pm **Bernhard Banaschewski**

"Archimeadean Kernels and Function Rings"

2pm-3.30pm **Drew Moshier**

"Gelfand Naimark Duality for Ordered Spaces"

**Tuesday 28th**

10am-11.30pm **Bernhard Banaschewski**

"Realcompact Alexandroff Frames"

12.30pm-1.30pm **Open session**

### Monday, January 13, 2014 from 1pm to 2pm in Von Neumann Hall

#### Speaker: Dr. Adrian Nistor, UIUC (now Chapman University)

*Title:* **Detecting and Repairing Performance Bugs**

*Abstract:* Software bugs and ineffective testing cost the US economy
tens of billions of dollars each year. My research develops novel
software and hardware techniques for combating software bugs. My
research so far focuses on performance and concurrency bugs.

In this talk, I will present Toddler and Lullaby, two novel techniques for automatically detecting and repairing performance bugs---programming mistakes that slow down program execution. Unlike profilers, which identify code regions that take a long time to execute, Toddler and Lullaby focus on execution and code patterns that are highly indicative of programming mistakes. Toddler and Lullaby found over 150 new performance bugs in widely used Java and C/C++ applications, including Groovy, Lucene, Google Core Libraries, GCC, MySQL, and Chromium. Over 100 of these bugs have already been fixed by developers.

And 4pm to 5pm in Von Neumann Hall

*Title:* **Detecting Concurrency Bugs**

*Abstract:* Multi-core processors are ubiquitous. To utilize the
processing power of these processors, developers need to write
concurrent code. Unfortunately, writing concurrent code is
notoriously difficult and prone to subtle concurrency bugs. In this
talk, we will review the challenges faced by developers writing
concurrent code and techniques to help developers cope with
concurrency bugs.