MathCS Seminar 2011

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Seminar Organizers: Peter Jipsen and Cyril Rakovski

Contents

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.

Poster




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.

Poster




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

Poster




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.

Poster




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.

Poster




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.

Poster




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.

Poster




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.

Poster




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.

Poster



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
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