# MathCS Seminar 2015

## 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.