# 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

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

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

### 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

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.