MathCS Seminar

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This is the homepage of the Chapman University Mathematics and Computational Science seminar

Seminar Organizers: Mihaela Vajiac and Adrian Nistor

Contents

Spring 2015

The seminar talks are in Von Neumann Hall VN 116 (545 W Palm Ave corner of W Palm Ave and railroad, Orange, CA 92866).

See [http://www.chapman.edu/discover/maps-directions/index.aspx Maps and directions], Von Neumann Hall is Building 38 on the [http://www.chapman.edu/discover/_files/CU_CampusMap2012-13-2.pdf Campus map]



Friday, February 13th, 10:00 a.m. to noon

Speaker: Professor Daniel Alpay, Earl Katz Chair in Algebraic System Theory, Department of Mathematics, at Ben-Gurion University of the Negev

Title: Fock spaces and non commutative stochastic distributions. The free setting. Free (non commutative) stochastic processes.

Abstract: We present the non commutative counterpart of the previous talk. We will review the main definitions of free analysis required and then present, and build stationary increments non commutative processes. The values of their derivatives are now continuous operators from the space of non commutative stochastic test functions into the space of non commutative stochastic distributions.

More details at:

http://blogs.chapman.edu/scst/2015/02/02/daniel-alpay/


Thursday, February 12th, 11:00 a.m. to 1:00 p.m.

Speaker: Professor Daniel Alpay, Earl Katz Chair in Algebraic System Theory, Department of Mathematics, at Ben-Gurion University of the Negev

Title: Bochner and Bochner-Minlos theorem. Hida’s white noise space and Kondratiev’s spaces of stochastic distributions, Stationary increments stochastic processes. Linear stochastic systems.

Abstract: We discuss the Bochner-Minlos theorem and build Hida’s white noise space. We build stochastic processes in this space with derivative in the Kondratiev space of stochastic distributions. This space is an algebra with the Wick product, and its structure of tallows to define stochastic integrals.

More details at:

http://blogs.chapman.edu/scst/2015/02/02/daniel-alpay/


Tuesday, February 10th, 10:00 a.m. to noon

Speaker: Professor Daniel Alpay, Earl Katz Chair in Algebraic System Theory, Department of Mathematics, at Ben-Gurion University of the Negev

Title: Positive definite functions, Countably normed spaces, their duals and Gelfand triples

Abstract: We survey the notion of positive definite functions and of the associated reproducing kernel Hilbert spaces. Examples are given relevant to the sequel of the talks. We also define nuclear spaces and Gelfand triples, and give as examples Schwartz functions and tempered distributions.

More details at:

http://blogs.chapman.edu/scst/2015/02/02/daniel-alpay/



Wednesday, January 14th, 2015 at 3pm (tea and cookies at 2:30pm)

Speaker: Prof. Richard N. Ball, University of Denver

Title: Pointfree Pointwise Suprema in Unital Archimedean L-Groups (joint work with Anthony W. Hager, Wesleyan University, and Joanne Walters-Wayland, Chapman University)

Abstract: When considering the suprema of real-valued functions, it is often important to know whether this supremum coincides with the function obtained by taking the supremum of the real values at each point. It is therefore ironic, if not surprising, that the fundamental importance of pointwise suprema emerges only when the ideas are placed in the pointfree context.

For in that context, namely in $\mathcal{R}L$, the archimedean $\ell$-group of continuous real valued functions on a locale $L$, the concept of pointfree supremum admits a direct and intuitive formulation which makes no mention of points. The surprise is that pointwise suprema can be characterized purely algebraically, without reference to a representation in some $\mathcal{R}L$. For the pointwise suprema are precisely those which are context-free, in the sense of being preserved by every $W$-morphism out of $G$.

(The algebraic setting is the category $W$ of archimedean lattice-ordered groups (`$\ell$-groups) with designated weak order unit, with morphisms which preserve the group and lattice operations and take units to units. This is an appropriate context for this investigation because every $W$-object can be canonically represented as a subobject of some $\mathcal{R}L$.)

Completeness properties of $\mathcal{C}X$ with respect to (various types of) bounded suprema are equivalent to (various types of) disconnectivity properties of $X$. These are the classical Nakano-Stone theorems, and their pointfree analogs for $\mathcal{R}L$ are the work of Banaschewski and Hong. We show that every bounded (countable) subset of $\mathcal{R}^+L$ has a join in $\mathcal{R}L$ iff $L$ is boolean (a $P$-frame). More is true: every existing bounded (countable) join of an arbitrary $W$-object $G$ is pointwise iff the Madden frame $\mathcal{M}G$ is boolean (a $P$-frame).

Perhaps the most important attribute of pointwise suprema is that density with respect to pointwise convergence detects epicity. We elaborate. Of central importance to the theory of $W$ is its smallest full monoreflective subcategory $\beta{}W$, comprised of the objects having no proper epic extensions. That means each $W$-object $G$ has a largest epic extension $G \to \beta G$, and this extension is functorial. It turns out that a $W$-extension $A \leq B$ is epic iff $A$ is pointwise dense in $B$. Thus the epireflective hull $\beta G$ of an arbitrary $W$-object $G$ can be constructed by means of pointwise Cauchy filters.


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.





Previous Seminar talks

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