MathCS Seminar

This is the homepage of the Chapman University Mathematics, Physics, and Computation Seminar

Seminar Organizers: Mihaela Vajiac and Justin Dressel

Fall 2016

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/about/_files/maps-and-directions/current-maps/campus-map.pdf Campus Map]

Thursday, September 15th at 4pm (tea and cookies at 3:30pm)

Speaker: Prof. Ali Nayeri, Chapman University

Title: Truly quantum Gibbs: Thermal state of a system whose charges don’t commute

Abstract: I present a detailed analysis for the classical stability of $4$ dimensional Anti-de Sitter spacetime (AdS$_4$) by decomposing the first-order perturbations of a spherical symmetric gravitational field into so called tensor harmonics which transform as irreducible representative of the rotation group (Regge-Wheeler decomposition). It is shown that there is no nontrivial stationary perturbation for the angular momentum $l < 2$. The stability analysis forces the frequency of the gravitational modes to be constrained in a way that the frequency of scalar modes are constrained..

CECHA Workshop: Monday to Monday, August 22nd to August 29th at 9am (tea and cookies at 8:30am)

Speaker: Professor Takashi Aoki, Kindai University

Title: Operators of infinite order and exact WKB analysis

Abstract: Contents and Description: Part I Differential operators, microdifferential operators and pseudodifferential operators of infinite order 1. Introduction 2. Algebraic definitions of pseudodifferential operators in complex analytic category 3. Kernel functions 4. Symbols of pseudodifferential operators and symbolic calculus 5. Exponential calculus 6. Applications

Part II Exact WKB analysis 1. Introduction 2. WKB solutions of ODE of second order with a large parameter 3. Borel sums of WKB solutions and connection formulas 4. WKB solutions and microdifferential operators 5. Higher-order equations and infinite-order equations 6. Applications to special functions

Prerequisites for this lecture are complex function theory and ordinary differential equations in the complex domain.

Friday, August 26th at 11am (tea and cookies at 8:30am)

Speaker: Nicole Yunger Halpern (Institute for Quantum Information and Matter, California

Title: Truly quantum Gibbs: Thermal state of a system whose charges don’t commute

Abstract: The grand canonical ensemble lies at the core of statistical mechanics. A small system thermalizes to this state while exchanging heat and particles with a bath. A quantum system may exchange quantities, or “charges,” represented by operators that fail to commute. Whether such a system thermalizes, and what form the thermal state has, concerns truly quantum thermodynamics. I characterize this state in three ways: First, I generalize the system-and-bath microcanonical ensemble. Tracing out the bath yields the system’s thermal state. Second, this thermal state is expected to be the fixed point of typical dynamics. Finally, the thermal state is completely passive (unable to output thermodynamic work) in a resource-theory model for thermodynamics. This study opens new avenues into equilibrium in the presence of quantum noncommutation. References: Yunger Halpern et al. Nature Communications 7, 12051 (2016). Yunger Halpern arXiv:1409.7845 (2014). This work was conducted with Philippe Faist, Jonathan Oppenheim, and Andreas Winter.

Tuesday, August 23rd at 11am (tea and cookies at 8:30am)

Speaker: Professor Howard Wiseman of Griffith University, and the Centre for Quantum Computation and Communication Technology

Title: Quantum State Smoothing - what does an open quantum system do when it is only partly observed? (by Howard Wiseman and Ivonne Guevara)

Abstract: Under noisy observations, estimation theory allows one to infer the state of the measured system, if its a priori statistics are given. In the continuous time situation, three different types of estimation can be distinguished: filtering, which is estimating of the state at time t from earlier records; retro-filtering, which is estimating the state at time t from later records; and smoothing, which is estimating the state at time t from both earlier and later records. Of the three, smoothing allows the greatest precision. This theory has been well developed in classical systems, but its application to quantum systems has only recently begun to be explored. Previous works have used the term “quantum smoothing” to mean estimating classical parameters, [Tsang, Phys. Rev. Lett. 102, 250403 (2009)], which is essentially classical smoothing in which the noisy observation of the classical parameters is mediated by a quantum system. Here we introduce quantum state smoothing, where the state of a partially observed open quantum system itself is smoothed [Guevara and Wiseman, Phys. Rev. Lett. 115, 180407 (2015).]. We achieve this by applying classical smoothing to a hypothetical unobserved noisy measurement record correlated with the stochastic dynamics ("quantum trajectories") of the system, induced by that hypothetical measurement. Using the formalism of linear quantum trajectories, we simulate quantum state smoothing for a qubit, and quantify how well the unobserved results can be estimated. Our investigations shed new light on the nature of the quantum state.

Thursday, August 18th at 4:00pm (tea and cookies at 3:30pm)

Speaker: Professor Fabrizio Colombo, Politecnico di Milano

Title: Quaternionic spectral theory

Abstract: In this talk we give an overview of the quaternionic spectral theory based on the notion of S-spectrum. We present the state of the art of the quaternionic version of the various functional calculi associated with slice hyperholomorphic functions. Moreover we discuss the spectral theorem for quaternionic (unbounded) normal operators using the notion of S-spectrum. The proof consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With the spectral theorem we complete the foundation of spectral analysis of quaternionic operators. An important motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which plays a crucial role in the quaternionic quantum mechanics.

Spring 2016

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]

Thursday, May 19th at 4pm (tea and cookies at 3:30pm)

Speaker: Professor Marco Panza, Professor of Philosophy at Universite Paris 1 (Sorbonne)

Title: Platonisms (in Philosophy of Mathematics)

Abstract: Platonism is a very often mentioned option in the discussion about the foundation and methodology of mathematics, but it encompasses quite different conceptions. In my talk, I will try to present an overview of different platonist views in contemporary (and less contemporary) philosophy of mathematics.

Wednesday, April 27th at 12noon (tea and cookies at 11:30am)

Speaker: Alessandra Palmigiano, TU Delft, The Netherlands

Title: Algorithmic correspondence and canonicity for non-distributive logics

Abstract: Since the 1970s, correspondence theory has been one of the most important items in the toolkit of modal logicians. Unified correspondence [6] is a very recent approach, which has imported techniques from duality, algebra and formal topology [10] and exported the state of the art of correspondence theory well beyond normal modal logic, to a wide range of logics including, among others, intuitionistic and distributive lattice-based (normal modal) logics [8], non-normal (regular) modal logics [17], substructural logics [9, 7, 5], hybrid logics [13], and mu-calculus [2, 4, 3].

The breadth of this work has stimulated many and varied applications. Some are closely related to the core concerns of the theory itself, such as the understanding of the relationship between different methodologies for obtaining canonicity results [16, 7], or of the phenomenon of pseudo-correspondence [11]. Other, possibly surprising applications include the dual characterizations of classes of finite lattices [14], the identification of the syntactic shape of axioms which can be translated into analytic rules of a proper display calculus [15], and the design of display-type calculi for the logics of capabilities and resources, and their applications to the logical modelling of business organizations [1]. Finally, the insights of unified correspondence theory have made it possible to determine the extent to which the Sahlqvist theory of classes of normal DLEs can be reduced to the Sahlqvist theory of normal Boolean expansions, by means of Gödel-type translations [12]. It is interesting to observe that, through the development of applications such as [16, 15, 11], the algorithm ALBA acquires novel conceptual significance, which cannot be reduced exclusively to its original purpose as a computational tool for correspondence theory.

The most important technical tools in unified correspondence are: (a) a very general syntactic definition of the class of Sahlqvist formulas, which applies uniformly to each logical signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives; (b) the algorithm ALBA, which effectively computes first-order correspondents of input term-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which, like the Sahlqvist class, can be defined uniformly in each mentioned signature, and which properly and significantly extends the Sahlqvist class.

In this talk, these technical tools will be illustrated in the setting of normal lattice expansions [9]. Time permitting, constructive canonicity will be also discussed [7, 3], as well as the epistemic interpretation of modalities on RS-frames [5].

References

Monday, April 18th at 2:30pm (tea and cookies at 2:00pm)

Speaker: Isabel M. Serrano (CSUF)

Title: Geometry in the Dark Ages

Abstract: Isidore's Etymologies enjoyed a wide audience during the medieval period. We examine the structure of mathematics, as it is described in the Etymologies, and we discuss the sources on which Isidore relied when he collected his etymological definitions. We remark that for Isidore, mathematics is described as the science of learning, and among his sources there have been the classical Greek authors, most likely available in Boethius' and Cassiodorus' Latin translations performed in the early 6th century. These translations are today lost. That's why the authors writing in the middle ages had to start from scratch in many of their investigations. We will illustrate this idea with one example, namely the discovery of curvature. In a paper published in 1952, J. L. Coolidge points out that the first writer to give a hint of the definition of curvature was the fourteenth century writer Nicolas Oresme". Coolidge writes further: Oresme conceived the curvature of a circle as inversely proportional to the radius; how did he find this out?" Tractatus de configurationibus qualitatum et motuum, written by Orseme sometime between 1351 and 1355, contains the key.

Wednesday, April 13th at 4pm (tea and cookies at 3:30pm)

Speaker: Matt Pusey (Perimeter Institute)

Title: From the Kochen-Specker theorem to robust noncontextuality inequalities

Abstract: Published in 1967, the Kochen-Specker theorem shows that quantum measurements do not simply reveal pre-existing values (satisfying a natural requirement). A different result along these lines, Bell's theorem, has had a much larger impact on quantum information. I will argue that this is because Bell's theorem has a clear operational meaning, independent of the quantum formalism and directly relevant to experiment. This is the motivation for various attempts to "operationalize" the Kochen-Specker theorem, and I will describe the approach to this I find most compelling. To the extent that this operationalization has succeeded, the audience should not require any knowledge of quantum theory to understand it!

Wednesday, April 13th at 11am (tea and cookies at 10:30am)

Speaker: Luca Spada, University of Salerno, Italy

Title: A general algebraic approach to dualities

Abstract: In this talk I will show how several dualities in mathematics e.g., the ones of Gelfand, Pontryagin, Stone, etc. can be seen as the manifestation of a general framework in which one develops the algebraic geometry of structures different from fields. This is a joint work with O. Caramello (University Paris Diderot) and V. Marra (University of Milan).

Monday, February 15th at 4:00pm (tea and cookies at 3:30pm)

Speaker: Mircea Pitici, Ph.D. in Mathematics Education, Cornell University

Title: Interpreting Mathematics, Counterfactuals, and the Paradox of Reward

Abstract: I will describe how he uses the vast literature on mathematics in my Writing in Mathematics seminar, how it relates to The Best Writing on Mathematics series I edit for Princeton, and how it matters to my teaching of mathematics and worldview.

Friday, January 8th at 10:00am (tea and cookies at 9:30am)

Speaker: Professor Paula Cerejeiras, Departamento de Matematica, Universidade de Aveiro, Portugal

Title: Applications of the monogenic signal processing to radiological images

Abstract: Medical ultrasonography imaging for nodule detection is a non-invasive diagnostic test, which combines low cost, short acquisition time, and sensitivity to the number and size of abnormal nodules. However, a chief problem is that ultrasound images have low contrast, making it hard for the experts to interpret and classify the nodules detected. In this talk we discuss techniques based on the concepts of monogenic signal which aims to enhance the edges of abnormalities. Hereby, we use a combination of Riesz transforms and monogenic curvelets in order to determine the phase and phase angle of a given image. Riesz transforms have remarkable properties: they are shift- and scale-invariant, preserve $L^2$ inner-product, and are steerable. Based on this approach, one is able to determine size and position of abnormalities present in images.

Thursday, January 7th at 4pm (tea and cookies at 3:30pm)

Speaker: Professor Uwe Kahler, Departamento de Matematica, Universidade de Aveiro, Portugal

Title: Compressed sensing for quaternionic representation of color images

Abstract: In the last decade a new paradigm has taken hold in signal and image processing: compressed sensing. The possibility of reconstructing a signal by only a few measurements under the condition that the representation in a given basis or frame is sparse has allowed to look at new methods and algorithms. Although sparsity constraints are directly connected only with non-convex optimization the uniqueness property shown by Candes, Rhomberg, and Tao allows the application of simple convex algorithms, such as linear programming. In parallel, during the last 15 years quaternion-valued functions have been used to represent color images, in particular RGB images. Hereby, representations using the discrete and continuous quaternionic Fourier transforms play a particular important role. In this talk we will show that it is possible to combine both approaches, i.e. to use sparse sampling methods in the quaternionic representation of color images. This is a priori not so evident due to the non-commutative structure of the quaternions. For instance, it is not clear that quaternionic sampling matrices will fulfil the RIP condition as the traditional condition for compressed sensing. Therefore, we intend to go back to the origins of compressed sensing and follow the original approach by Rauhut to show that quaternionic color images allow sparse reconstruction by means of an $l_1$-minimization with high probability.

Fall 2015

Thursday, December 10th 2015 at 4pm (tea and cookies at 3:30pm)

Speaker: Dr. Justin Dressel, Chapman University

Title: Violating a Hybrid Bell-Leggett-Garg Inequality with Weak Quantum Measurements

Abstract: We discuss both the theoretical background and the experimental violation of a hybrid Bell-Leggett-Garg inequality using four superconducting Xmon qubits. The algorithm uses sequential weak measurements of a Bell state in the form of high-fidelity partial projections, realized by entangling an ancilla qubit to each data qubit using a controlled-Z two-qubit gate. After calibration of the ancilla readout, these partial projections indirectly measure qubit expectation values with a tunable amount of state disturbance. For sufficiently weak disturbance, the hybrid inequality can be violated using all data prepared in a single experimental configuration, thus avoiding both the fair sampling and the disjoint sampling loopholes that often appear in traditional Bell inequality implementations.

Wednesday, November 4th 2015 at 4pm in (tea and cookies at 3:30pm)

Speaker: Dr. Joshua Sack, California State University Long Beach

Title: Quantum Logic and Structure

Abstract: This talk presents logics for reasoning about properties of quantum systems and quantum algorithms. One logic, developed by Birkhoff and von Neumann, is used to reason about testable properties of a quantum system; the formal setting is the Hilbert lattice (the lattice of closed subspaces of a Hilbert space). Another logic is the logic of quantum actions, developed more recently to reason about the dynamics of a quantum system; the formal setting here is a quantum dynamic frame, a kind of labelled transition system often used in computer science to reason about classical programs. This talk also explains how these settings are essentially the same via a categorical duality between the lattices and the frames, and how a decidable probabilistic extension of the logic of quantum actions can be used to reason about quantum algorithms such as Grover's search algorithm.

Friday, October 23rd 2015 at noon, Beckman Hall room 107 (no tea and cookies this time, we will be taking the speaker to lunch after)

Speaker: Prof. Glen van Brummelen, Quest University, Canada

Title: The Mercurial Tale of Spherical Trigonometry

Abstract: The trigonometry we see in high school is merely a pale reflection of the creative, exciting subject that students learned only decades ago. Born of the desire to predict the motions of the heavenly bodies, the trigonometry of ancient astronomers took place not on a flat sheet of paper, but on the celestial sphere. This led to a theory with some of the most beautiful results in all of mathematics, and applications that led to the birth of major modern developments like symbolic algebra and logarithms. Until the subject dropped off radar screens after World War II, it continued to enjoy vitality through applications in navigation and crystallography. The mathematical path we now travel through high school and college, heavily emphasizing calculus, unfortunately has deprived students of other mathematical gems. In this talk, we shall polish some of the tarnish off one of the brightest of those jewels.

Friday, October 9th 2015, 2pm (tea and cookies 1:30pm)

Speaker: George J. Herrmann, Ph.D. student at University of Denver, Website: http://cs.du.edu/~herrmann

Title: A tour of Noncommutative Metric Geometry

Abstract: This talk will be an introduction to the area of Noncommutative Metric Geometry. We will start with discussing deformation quantization: the history and observations that motivate our continued interest. We will then shift gears slightly and discuss some results of Connes and Rieffel in Noncommutative Geometry that lead to Quantum (Compact) Metric Spaces and quickly introduce a few nontrivial objects in this category. Then we will end with the work of Latremoliere in establishing a metric on the category of Quantum Compact Metric Spaces.

Wednesday, September 30th, 2015, at 3-5pm

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

Title: Lecture Series by Professor Daniel Alpay, Lectures 5 and 6

Abstract: Rational functions are quotient of polynomials, or meromorphic functions on the Riemann sphere. Here we consider matrix-valued rational functions. A number of new aspects (a point can be at the same time a zero and a pole of the function) and new notions and methods appear (in particular the state space method. A key role is played by the realization of a matrix-valued rational function $M$, say analytic at the origin, that is, its representation in the form $M(z)=D+zC(I-zA)^{-1}B$, where $A,B,C,D$ are matrices of appropriate sizes.

We will discuss matrix-valued rational functions, and their connections with topics such as complex analysis, interpolation theory of analytic functions contractive in the open unit disk (Schur functions), the theory of linear systems (signal processing) and matrix theory.

Lecture 5, 3:00pm-4:00pm

1) Wavelet filters.

2) Convex invertible cones.

3) A new kind of realization.

Lecture 6, 4:00pm-5:00pm

1) Several complex variables.

2) The non commutative case.

3) Rational functions on a compact Riemann surface, theta functions.

4) Quaternionic setting.

Tuesday, September 29th, 2015, at 3:30-5:30pm

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

Title: Lecture Series by Professor Daniel Alpay, Lectures 3 and 4

Abstract: Rational functions are quotient of polynomials, or meromorphic functions on the Riemann sphere. Here we consider matrix-valued rational functions. A number of new aspects (a point can be at the same time a zero and a pole of the function) and new notions and methods appear (in particular the state space method. A key role is played by the realization of a matrix-valued rational function $M$, say analytic at the origin, that is, its representation in the form $M(z)=D+zC(I-zA)^{-1}B$, where $A,B,C,D$ are matrices of appropriate sizes.

We will discuss matrix-valued rational functions, and their connections with topics such as complex analysis, interpolation theory of analytic functions contractive in the open unit disk (Schur functions), the theory of linear systems (signal processing) and matrix theory.

Lecture 3, 3:30pm-4:30pm

1) Realization and geometry: $J$-unitary rational functions.

2) Applications to interpolation problems.

3) Inverse scattering problem (Krein and Marchenko).

Lecture 4, 4:30pm-5:30pm

1) First order degree systems.

2) Smith-McMillan local form.

3) Zero-pole structure.

4) Applications to inverse problems.

Monday, September 28th 2015 at 4:00pm (tea and cookies at 3:30pm)

Speaker: Prof. Ahmed Sebbar, Bordeaux University, France

Title: Capacities and Jacobi Matrices

Abstract: Given a system of intervals of the real line, we construct a Jacobi matrix (tridiagonal and periodic) whose spectrum is this given system of intervals. We discuss the underlying conditions and techniques, as well as possible applications.

Friday, September 25th, 2015, at 1-3pm

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

Title: Lecture Series by Professor Daniel Alpay, Lectures 1 and 2

Abstract: Rational functions are quotient of polynomials, or meromorphic functions on the Riemann sphere. Here we consider matrix-valued rational functions. A number of new aspects (a point can be at the same time a zero and a pole of the function) and new notions and methods appear (in particular the state space method. A key role is played by the realization of a matrix-valued rational function $M$, say analytic at the origin, that is, its representation in the form $M(z)=D+zC(I-zA)^{-1}B$, where $A,B,C,D$ are matrices of appropriate sizes.

We will discuss matrix-valued rational functions, and their connections with topics such as complex analysis, interpolation theory of analytic functions contractive in the open unit disk (Schur functions), the theory of linear systems (signal processing) and matrix theory.

Lecture 1, 1pm-2pm:

1)Preliminaries on rational functions. Notion of realization.

2) Transfer functions. Link with linear systems.

3) Resolvent operators.

4) Proof of the realization theorem: { The backward-shift realization}.

5) Various characterizations of rational functions.

6) The Wiener algebra.

Lecture 2, 2pm-3pm:

1) Main properties of the realization.

2) Another proof of the realization theorem.

3) Minimal realization.

4) Minimal factorizations.

5) Spectral factorizations.

6) Reproducing kernel spaces.

Thursday, September 17th, 2015 at 3pm (tea and cookies at 2:30pm)

Speaker: Prof. Ahmed Sebbar, Bordeaux University, France

Title: The Frobenius determinant theorem and applications.

Abstract: In this first talk, we will discuss the celebrated determinant Frobenius theorem and how it arised naturally in the study of a hierarchy of hypersurfaces, of partial differential operators and metrics.

The first elements of this hierarchy are the cubic $x^3 + y ^3 + z^3 - 3xyz = 1$ (so called Jonas hexenhut) and the partial differential operator $\Delta_3 = \frac{\delta^3}{\delta_{x^3}} + \frac{\delta^3}{\delta_{y^3}} + \frac{\delta^3}{\delta_{z^3}} -3 \frac{\delta^3}{\delta_x \delta_y \delta_z}$, introduced by P.Humbert in 1929 in another context. We explain why this operator is a good extension to ${\rm I\!R}^3$ of the Laplacian in two dimensions $\Delta_2 = \frac{\delta^2}{\delta_{x^2}} + \frac{\delta^2}{\delta_{y^2}}$ We discuss its links with Spectral theory, Elliptic functions, number theory and a sort of Finsler geometry.

This is a part of a large project conducted in collaboration with Daniele Struppa, Adrian Vajiac and Mihaela Vajiac.

Thursday, September 3rd, 2015 at 4pm (tea and cookies at 3:30pm)

Speaker: Prof. Yasushi Kondo, Kinki University, Osaka, Japan

Title: Composite Quantum Gates with Aharanov–Anandan phases.

Abstract: Unitary operations acting on a quantum system must be robust against systematic errors in control parameters for reliable quantum computing. Composite pulse technique in nuclear magnetic resonance realizes such a robust operation by employing a sequence of possibly poor-quality pulses. We show that composite pulses that compensate for a pulse length error in a one-qubit system have a vanishing dynamical phase and thereby can be seen as geometric quantum gates with Aharanov-Anandan phases.

Spring 2015

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:

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:

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:

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

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.