# MathCS Seminar 2011

*Seminar Organizers:* Peter Jipsen and Cyril Rakovski

## Fall 2011

### Thursday, December 8 at 4:00 pm in VN 116

*Speaker:* **Professor Yakir Aharonov, Chapman University and Center of Excellence in Quantum Studies**

*Title:* **Weak measurements and the discovery of a new kind of mathematics called superoscillations**

*Abstract:* We are taught that if waves with a variety of wavelengths are superposed, then we cannot create structures whose details are smaller than the shortest wavelength in the original superposition. But this is not correct. That is, even if the smallest wave has a wavelength of 1m, we can create arbitrarily detailed structures, say, with feature sizes of 1mm.

This miracle comes with a penalty: outside the superoscillatory region, the wave must be exponentially large. As an example, suppose we wanted to reproduce a Beethoven symphony with sound frequencies not larger than 1Hz, then somebody (possibly far away) would have to listen to sounds exp(10^19) louder.

The new kind of mathematics we’ve been developing here at Chapman (a/k/a Superoscillations) has had significant implications for signal processing and has led to new kinds of microscopes which can resolve sub-wavelength structures without the use of evanescent waves.

We were led to this discovery as a result of our work in quantum mechanics concerning the relationship between past, present and future and weak measurements.

In this talk I will describe how this came about along with some new exciting discoveries concerning the nature of superoscillations.

### Thursday, December 8 at 12:00 noon in VN 116

*Speaker:* **Dr. Joanne Walters-Wayland, Chapman University Center of Excellence in Computation, Algebra and Topology (CECAT)**

*Title:* **The wizard of Coz**

*Abstract:* The importance of considering topologies and topological properties via real-valued functions is indisputable – the well-known book by Gillman and Jerison, Rings of Continuous Functions, more than substantiates this claim. In a pointfree setting, this becomes, in my opinion, even more obvious. I shall endeavor to explain my fascination with the cozero part of a frame. By illustrating various aspects of “Coz”, highlighting a few of my favorite properties and explaining some of the problems I am attempting to understand, I hope to convey to the audience, albeit in a small and somewhat superficial way, why "Coz" is so magical and alluring.

### Friday, November 18 at 3:00 pm in VN 116

**First Chapman University and Cal State University Fullerton joint Mathematics Colloquium**

**Program**

**Dr. Bogdan Suceava (CSUF)** – 3:00 - 3:20

*Title:* A question of David Hilbert and its application to classes of metrics

**Lucy Odom (CSUF Undergraduate)** – 3:25 - 3:45

*Title:* Chasles Angle Invariance Property for the Helicoid

**Peter T. Ho (CSUF Undergraduate)** – 3:50 - 4:10

*Title:* Rotation Hypersurfaces in Lorentz-Minkowski Space with All Points Umbilics

**Dr. Peter Jipsen (Chapman)** – 4:15 - 4:35

*Title:* Computational Investigations of the Lattice of Lattice Varieties

### Thursday, November 17 at 4:00 pm in VN 116

*Speaker:* **Robert Campbell, University of California at Irvine**

*Title:* **Interplanetary Communication Made Possible With Algebraic Coding**

*Abstract:* What does it take to send a message across our solar system? Moreover, if I have trouble maintaining a connection on my cell phone calls how does NASA plan to communicate with its satellites in deep space? Algebraic Coding to the rescue! Combining the power of computer science with the theory of mathematics we can create messages that fix themselves when an error occurs.

### Thursday, November 3 at 4:00 pm in VN 116

*Speaker:* **Monroe Eskew, University of California at Irvine**

*Title:* **Definability and Infinity**

*Abstract:* What does it mean for a collection to be finite? On the one hand, we have our preschool notion that a collection is finite when it can be counted with natural numbers in a way that terminates. On the other hand, there is a definition due to Dedekind that a set is finite if and only if it cannot be put in one-to-one correspondence with a proper subset. Intuitively these two notions should be equivalent, but can we prove it? I will argue that to avoid a circular argument, one direction requires more care than one would initially think. Further, the other direction is true only by virtue of the Axiom of Choice. To outline the proof of this fact, we will examine formal notions of definiabilty and the set-theoretic technique of forcing.

### Thursday, October 6 at 3:00 pm in VN 116

*Speaker:* **Dr. Fred Dashiell, UCLA and Center of Excellence in Computation, Algebra and Topology (CECAT)**

*Title:* **Various descriptions of hyperstonian Boolean Algebras**

*Abstract:* The study of von Neumann algebras in the late 1940’s led Dixmier to identify the class of compact Hausdorff spaces L which occur when C(L) is a Banach dual space. Dixmier called these spaces “hyperstonian”, and they are totally disconnected, therefore they also represent certain Boolean algebras, which for the purposes of this talk I also call hyperstonian Boolean algebras. These Boolean algebras are complete as lattices, and obey a further condition as to the existence of certain measures on them. Every compact space has a unique “hyperstonian cover”, and this dualizes to the existence of a hyperstonian extension of every Boolean algebra. This talk discusses two approaches to an algebraic description of the hyperstonian cover.

### Thursday, September 29 at 3:00 pm in VN 116

*Speaker:* **Dr. Peter Jipsen, Chapman University and Center of Excellence in Computation, Algebra and Topology (CECAT)**

*Title:* **The finite congruence lattice representation problem in universal algebra**

*Abstract:* The central problem of this talk asks which finite lattices are congruence lattices of finite algebras. I will give all necessary background and show pictures of congruence lattices of small unary algebras. The problem is still open after 50 years, but recently, in collaboration with W. DeMeo, R. Freese, B. Lampe and J.B. Nation at the University of Hawaii, we have shown that all 7-element lattices, except possibly one of them, are congruence lattices of finite algebras. Students can make progress on this problem by searching for finite algebras with specific congruence lattices (either "by hand" or by doing some programming). I will also demonstrate Ralph Freese's Universal Algebra Calculator (uacalc.org) which makes it possible to easily calculate and display congruence lattices of small algebras.

### Thursday, September 22 at 3:00 pm in VN 116

*Speaker:* **Professor M. Andrew Moshier, Chapman University and Center of Excellence in Computation, Algebra and Topology (CECAT)**

*Title:* **Point-free duality for Archimedean l-groups with strong order unit**

*Abstract:* Classically, archimedean lattice ordered groups (l-groups) with strong order unit are representable as sub-l-groups of continuous real functions on compact regular spaces. This fact, however, hinges on understanding the real numbers non-constructively. For that matter, even the proof that archimedean l-groups are commutative is not trivial (without the aid of the representation theorem).

We establish an analogous representation of archimedean l-groups with strong order unit in a category of continuous sequent calculus. The construction of an l-group from a sequent calculus involves simple finitary proof rules using rational numbers. In particular, the group operations are captured by constructing certain entailments capturing informally how 'continuous real functions' are related by their behavior on rational numbers. As an illustration of the techniques, we can show that every archimedean l-group with strong order unit is commutative without recourse to non-constructive real numbers.

### Thursday, September 15 at 4:00 pm in VN 116

*Speaker:* **Professor Yakir Aharonov, Chapman University and Center of Excellence in Quantum Studies**

*Title:* **Can the future affect the present without violating causality?**

*Abstract:* I will discuss properties of pre- and post-selected ensembles in quantum mechanics and the way to observe these properties through the use of a new type of non-disturbing measurement called 'weak measurement'. A number of these new experiments have already been successfully performed.

I will also discuss another type of measurement which does not disturb the observed quantum system. The main result of this new approach is the realization that the basic difference between classical and quantum mechanics is the non-local aspects of quantum dynamics.

Theoretical analysis of the outcomes of these experiments have produced several very rich results. First, it has shed new light on the most puzzling features of quantum mechanics, such as interference, entanglement, etc. Secondly, it has uncovered a host of new quantum phenomena, which were previously hidden.

### Thursday, September 8 at 4:00 pm in VN 116

*Speaker:* **Professor Steven Krantz, Washington University in St. Louis**

*Title:* **A Matter of Gravity**

*Abstract:* We treat the concept of "center of gravity" from a new point of view. We derive
a number of original results about the stability of the center of gravity.

We also study asymptotic features of the center of gravity as the dimension of the ambient space tends to infinity.

These results are of interest to roboticists and to physicists who study quantum mechanics. But they also have intrinsic mathematical merit.

The talk will be accessible to anyone who has had multi-variable calculus.

## Spring 2011

### Tuesday, May 24 at 4:00 pm in VN 116

*Speaker:* **Dr. David Carfì, Researcher and Aggregate Professor, Faculty of Economics, University of Messina, Visiting Researcher at University of California Riverside**

*Title:* **Dirac Calculus in distribution spaces for Quantum Mechanics and Economics**

*Abstract:* In his famous treatise “Principles of Quantum Mechanics” (1930), the great mathematician and physicist Paul Dirac introduced several “manipulation rules” for vectors and operators of linear spaces, which together constitute the so-called “Dirac Calculus”. This Calculus is nothing more than a wide set of formal extensions of the basic properties of the finite-dimensional Linear Algebra to the case of infinite-dimensional vector spaces. The discourse is elegant and surprisingly efficient, but it is far from being a rigorous mathematical treatment. As mathematicians well know, the passage from the finite to the infinite dimensional case does not amount to a mere substitution of finite linear combinations with formal integrals!

The goal of the research introduced in this talk is to give a precise mathematical meaning and rigorous support to many analytic methods of Quantum Mechanics, starting from the fundamental Dirac Calculus, using the Weak Duality Theory of L. Schwartz and J. A. Dieudonné and the L. Schwartz Theory of Distributions. This approach will give a rigorous justification for the use of Dirac's tools, leaving them substantially “as they are” in Quantum Mechanics practice. Moreover, by providing a correct interpretation of these heuristic methods in terms of new solid and powerful mathematical entities and concepts, we will be helped in reaching a deeper understanding of the physical structures studied in Quantum Mechanics.

The new operations of continuous-superposition and that of the Dirac product allow us to build - in a mathematically rigorous way - the "extended Linear Algebra" of Dirac in the spaces of tempered distributions, via their natural topological linear structures. More precisely, we shall see that the algebraic-topological structure of tempered distribution spaces allows us to define - naturally - the linear combinations of a continuous family of vectors and operators and a scalar product (of a vector by such continuous families of vectors) which are absolutely necessary (and already formally introduced and used by Dirac himself) for the modern theoretical development of Quantum Mechanics.

### Monday and Tuesday, April 4 and April 5, at 2:30 pm in VN 116

*Speaker:* **Dr. Henry Stapp (Lawrence Berkeley National Laboratory)**

*Title:* **Discussions on quantum issues and their mathematical foundations**

*Abstract:* Discussions on quantum issues and their mathematical foundations.

### Thursday, March 7, at 4:30 pm in VN 116

*Speaker:* **Dr. Neil Donaldson (UCI)**

*Title:* **Isothermic surfaces - an overview**

*Abstract:* In this talk we consider some of the structure of classical isothermic surfaces; those for which there exist conformal curvature line co-ordinates. We moreover consider two modern approaches/generalizations of the classical theory which place the discussion firmly in the realm of integrable systems.

### January 10 - January 19, 2011 in VN 116

**THIRD ANNUAL CECAT WORKSHOP IN POINTFREE TOPOLOGY & ANALYSIS**

(see also the CECAT workshops page)

Monday 10th: |
12pm-2pm | Bernhard Banaschewski (McMaster University, Canada) |

"Essential completions - update" | ||

3pm-5pm | Olaf Klinke (University of Birmingham, UK) | |

"Pointfree bitopological compactification" | ||

Tuesday 11th: |
10am-12pm | Bernhard Banaschewski |

"Essential completions - update" | ||

2pm-4pm | Open session | |

Wednesday 12th: |
12pm-2pm | Bernhard Banaschewski |

"Essential completions - update" | ||

3pm-5pm | Andrew Moshier (Chapman University) | |

"Another Duality for Commutative C* Algebras" | ||

Thursday 13th: |
No scheduled talks | |

Friday 14th: |
11am-1pm | Bernhard Banaschewski |

"Extended real-valued functions" | ||

2pm-4pm | Open session | |

5pm - 7pm | Drinks and snacks at Von Neumann Hall | |

Saturday 15th: |
10am | Coffee at Café Lucas |

11am-1pm | Bernhard Banaschewski | |

"Extended real-valued functions" | ||

2pm-4pm | Andrew Moshier | |

"Integration in Logical Form" | ||

Sunday 16th: |
10am-12pm | Rick Ball (University of Denver) |

"Integration ideas in a pointfree context" | ||

1pm-3pm | Open session | |

Monday 17th: |
12pm-2pm | Bernhard Banaschewski |

"Extended real-valued functions" | ||

3pm-5pm | Open session | |

Tuesday 18th: |
10am-12pm | Bernhard Banaschewski |

"To be announced" | ||

2pm-4pm | Open session | |

Wednesday 19th: |
12pm-2pm | Bernhard Banaschewski |

"To be announced" | ||

3pm-5pm | Open session |