# MathCS Seminar

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

Seminar Organizers: Mihaela Vajiac and Justin Dressel

## Spring 2017

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]

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### Friday, May 5th at 2:00pm (tea and cookies at 1:30pm) in collaboration with the Philosophy Department

#### Speaker: Prof. A. Naibo, Sorbonne

Title: Harmony, Stability, and Identity: An intensional account in proof-theoretic semantics

Abstract: Proof-theoretic semantics are usually conceived in opposition to truth-theoretic semantics. In truth-theoretic semantics, truth is considered as a primitive (non-analyzed) notion, and meaning is then explained in terms of it. On the other hand, in proof-theoretic semantics, meaning is explained in terms of (our) inferential abilities, and truth is then explained in terms of proofs. In order to avoid any possible trivialization of proof-theoretic semantics — boiling it down to truth-theoretic semantics — an intensional, rather than an extensional approach, should be adopted. In particular, the semantic value of a sentence A should not be defined in terms of the simple existence of a proof of A, but in terms of the way in which A is proved, i.e. in terms of the inferential structure of proof of A. In order to specify this structure some properties are asked to be satisfied. The most known of them is the property of harmony, which corresponds to the reduction of local complexity peaks (detours) in a proof. However, as Dummett claims, this property is “an excessively modest demand”, and it should be complemented by another property, that of stability. It will be shown that this property can be captured by a more fundamental operation, that of expansion, allowing one to generate local complexity riffs within a proof. Even if this operation could seem very natural to add, it will be shown in fact how it is destructive for the intensional account proper to proof-theoretic semantics. In particular, when this operation of expansion is used in presence of negation and identity, it leads to the collapse of the set of proofs of negative and identity sentences, respectively. Finally, the case of identity is analyzed in details in the framework of Martin-Löf’s type theory. It is shown, in particular, that a possible way of avoiding the collapse between identity proofs can be found in the works of M. Hoffman and T. Streicher, where the operation of expansion is lifted from the level of proof-objects to the level of the sentences which speak about these proof-objects.

### Wednesday, April 26th at 4:00pm (tea and cookies at 3:30pm)

#### Speaker: Prof. Daniel Alpay, Chapman University

Title: Non-commutative Brownian Motion and a New Class of Topological Algebras

Abstract: We study a family of free stochastic processes whose covariance kernels K may be derived as a transform of a tempered measure σ. These processes arise, for example, in consideration non-commutative analysis involving free probability. Hence our use of semi-circle distributions, as opposed to Gaussians. In this setting we find an orthonormal bases in the corresponding non-commutative L2 of sample-space. We define a stochastic integral for our family of free processes.

Joint work with Palle Jorgensen and Guy Salomon

### Friday, April 21st at 2:00pm (tea and cookies at 1:30pm)

#### Speaker: Professor H. Turgay Kaptanoglu, Department of Mathematics, Bilkent University, Ankara

Title: Precise Inclusion Relations Among Bergman-Besov and Bloch-Lipschitz Spaces and H^\infty on the Unit Ball of C^n

Abstract: We describe exactly and fully which of the spaces of holomorphic functions in the title are included in which others. We provide either new results or new proofs. More importantly, we construct explicit functions in each space that show our relations are strict and best possible.

Joint work with A. Ersin Ureyen of Anadolu University, Eskisehir, Turkey

### Tuesday, April 11th at 1:00pm (tea and cookies at 12:30pm), please note the change in date/time

#### Speaker: Professor David Shoikhet, Holon Institute of Technology, The Technion Institute of Technology of Israel

Title: Old and New in Complex Dynamical Systems

Abstract: Historically, complex dynamics and geometrical function theory have been intensively developed from the beginning of the twentieth century. They provide the foundations for broad areas of mathematics. In the last fifty years the theory of holomorphic mappings on complex spaces has been studied by many mathematicians with many applications to nonlinear analysis, functional analysis, differential equations, classical and quantum mechanics. The laws of dynamics are usually presented as equations of motion which are written in the abstract form of a dynamical system: ((dx)/(dt))+f(x)=0, where x is a variable describing the state of the system under study, and f is a vector-function of x. The study of such systems when f is a monotone or an accretive (generally nonlinear) operator on the underlying space has recently been the subject of much research by analysts working on quite a variety of interesting topics, including boundary value problems, integral equations and evolution problems. In this talk we give a brief description of the classical statements which combine the celebrated Julia Theorem of 1920, Carathéodory's contribution in 1929 and Wolff's boundary version of the Schwarz Lemma of 1926 with their modern interpretations for discrete and continuous semigroups of hyperbolically non-expansive mappings in Hilbert spaces. We also present flow-invariance conditions for holomorphic and hyperbolically monotone mappings. Finally, we study the asymptotic behavior of one-parameter continuous semigroups (flows) of holomorphic mappings. We present angular characteristics of the flows trajectories at their Denjoy-Wolff points, as well as at their regular repelling points (whenever they exist). This enables us by using linearization models in the spirit of functional Schroeder's and Abel's equations and eigen-value problems for composition operators to establish new rigidity properties of holomorphic generators which cover the famous Burns-Krantz Theorem and to solve a Nevanlinna-Pick type boundary interpolation problem for generators.

### Friday, April 7th at 2:00pm (tea and cookies at 1:30pm)

#### Speaker: Dr. Imanol Mozo, Chapman University

Title: The unit circle in pointfree topology

Abstract: Pointfree topology is an lattice-theoretic approach to topology that takes abstract lattices of open sets as the primitive notion. This approach is motivated by the fact that the lattice of open sets of a topological space contain almost all the information. Indeed, some lattices, namely, frames, are sufficiently similar to lattices of open sets of topological spaces in order to be considered as generalized spaces [5]. One of the main differences between pointfree topology and classical topology is that the category of frames is algebraic, while the dual of the category of topological spaces is not. Consequently, we can present its objects by generators and relations, as in an algebraic fashion. This is a very useful tool that was used by Joyal in order to introduce the pointfree counterpart of the real line [3], which was later studied by Banaschewski in [1]. Besides, this procedure offers a natural way to introduce variants by modifying the set of generators or the defining relations. For instance, one has the frame of extended reals and the lattice of extended real functions studied in [2], and the frame of partial reals and the lattice of continuous partial real functions introduced in [4] which arose naturally in the construction of the Dedekind completion of the lattice of continuous real functions on a frame. In this talk, after a brief introduction to pointfree topology, we will discuss how the topology of the unit circle fits in this family of frames.

[1] B. Banaschewski, The real numbers in pointfree topology, Textos de Matemática vol. 12, Departamento de Matemática da Universidade de Coimbra (1997). [2] B. Banaschewski, J. Gutiérrez García, J. Picado, Extended real functions in pointfree topology, J. Pure Appl. Algebra 216 (2012), 905–922. [3] A. Joyal, Nouveaux fondaments de l’analyse. Lectures Montréal 1973 and 1974 (unpublished).971), 161–167. [4] I. Mozo Carollo, J. Gutiérrez García and J. Picado, On the Dedekind completion of function rings, Forum Math. 27 (2015), 2551 -2585. [5] J. Picado and A. Pultr, Frames and locales: Topology without points, Frontiers in Mathematics, vol. 28, Springer, Basel (2012).

### Friday, March 31st at 2:00pm (tea and cookies at 1:30pm) join with the Philosophy Department

#### Speaker: Professor John Mumma, CalState San Bernardino

Title: Lewis's infinite regress, mathematical proof, and the act of diagramming

Abstract: In 'What the Tortoise Said to Achilles.' Carroll shows how an infinite regress can be generated from the demand that all premises in a deductive inference be made explicit. In my talk I discuss the connection of the regress to the question of how mathematical proofs are accepted as proofs. A mathematical proof does not succeed unless it can be seen how acceptance of its premises force acceptance of its conclusion. Carroll's piece can be understood, I argue, as illuminating the difficulties in providing a satisfactory account of this seeing. I then focus on a restricted class of elementary geometric inferences, and explore whether the act of diagramming them resolves (for this restricted class) the general difficulties Carroll's piece raises.

### Friday, March 24th at 2:00pm (tea and cookies at 1:30pm)

#### Speaker: Professor Dan Volok, Kansas State University, Department of Mathematics

Title: Non-stationary point evaluation in the multiscale setting

Abstract: It was demonstrated by D. Alpay, P. Dewilde and H. Dym that the Hilbert space of triangular Hilbert-Schmidt operators can be equipped with a reproducing kernel structure quite similar to that of the classical Hardy space of the unit disk. This fact has many applications in the theory of non-stationary dissipative systems. It turns out that a multivariate generalization of Alpay-Dewilde-Dym reproducing kernel Hilbert space arises naturally in the setting of linear systems indexed by homogeneous trees, as introduced by A. Benveniste, R. Nikoukhah and A. Willsky.

This talk is based on joint work with D. Alpay and A. Dijksma.

### Friday, March 17th at 2:00pm (tea and cookies at 1:30pm)

#### Speaker: Professor Palle Jorgensen, University of Iowa, Department of Mathematics

Title: Markov processes, endomorphisms, and measurable dynamics

Abstract: The structures of positive operators, endomorphisms, transfer operators, measurable partitions, and Markov processes arise in both pure and applied mathematics. The talk offers unified setting, as well as new applications. The general setting is that of dynamics in Borel measure spaces and Markov fields. Hence the corresponding linear structures to be studied are infinite-dimensional. Nonetheless, we prove a number of analogues of the more familiar finite-dimensional settings, for example, the Perron-Frobenius theorem in the case of positive matrices, and the corresponding Markov chains.

### Friday, March 10th at 2:00pm (tea and cookies at 1:30pm)

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

Title: What is Quantum Markovianity?

Abstract: Markovianity versus non-Markovianity is a well-established distinction for classical systems. The same cannot be said for quantum systems. Different communities and individuals use “quantum Markovianity” to mean very different things. We argue that, to avoid confusion, it is best to avoid attributing that term any definite meaning at this stage. However, that does not mean that there is nothing to say about Markovianity for open quantum systens. We discuss a large number of concepts that have been, or could logically be, used to define quantum (non-)Markovianity, and prove hierarchical relations between them. Some are existing concepts, including “factorisation”, “quantum regression formula”, “divisibility”, and “Lindblad”. Others we introduce, including “past-future independence”, and “composability”. We also prove relations between these and other properties of interest for open quantum systems, such as the applicability of dynamical decoupling to preserve quantum information, the existence of (quantum) information backflow from the environment, and the physical reality of stochastic pure-state trajectories. Finally, we discuss in which concept the closest analogue of classical Markovianity lies.

Joint work with: Li (Kenny) Li, and Michael Hall

### Tuesday, March 7th at 5:30pm (tea and cookies at 5:00pm)

#### Speaker: Professor Giuseppe Longo, CNRS, CREA, École Polytechnique, et CIRPHLES, ENS, Paris

Title: The Structuralist Roots of Mathematical Understanding Reconsidered: Poincaré’s heritage

Abstract: The theological origin of the physicomathematical spaces; the geometrization of time Abstract: There is no mathematical plane nor space in Euclid's geometry. Lines are traced, extended, intersected on a plane, an “apeiron” (it has no boundary), which is “practiced” but not mathematized. These lines have no thickness, they intersect in a point, that is a sign (“semeion”); they are objects of a mythical, ideal realm. Infinity is only potential: lines in the plane or endless sequences of numbers can be extended with no limit. Actual infinity will be fully conceptualized much later, in the theological debate of late middle age, as an attribute of God. How actual infinity relates to or how can it be represented in the finite? The Renaissance Italian painters will show that this is possible: the projective limit of the newly invented linear perspective, first used in XIV century paintings of the Annonciation, shows the infinite in the finite and joins the infinity of God to the bodily, three dimensional presence of a human being, the Madonna. We will critically analyze paintings from Giotto to Piero della Francesca that show this joint invention of actual infinity and of mathematical spaces, which allows as well the expression of a new, corporeal humanity. Later, this theological invention will become the mathematics of Descartes' and Desargues' spaces, the science of Newton's infinities. In the XIX century, physics will extend this mystical creation of infinite space and time to the mathematical “phase space” of its new scientific rigor, based on an increasing mathematical unity of space and time.

### Friday, February 17th at 2pm (tea and cookies at 1:30pm)

#### Speaker: Andrew Jordan, University of Rochester

Title: Postselection, Superconductors, and Quantum Information in Black Holes

Abstract: This talk will demonstrate how the quantum information entering black holes is analogous to quantum information entering a superconductor. The correspondence maps the interior of a black hole to a superconductor, and the exterior of the black hole to a normal metal. We show that the metal-superconductor interface can be thought of as an event horizon: The proximity effect in superconductor-metal interfaces (where Cooper pairs tend to form in the normal metal) is analogous to electron-positron creation at the event horizon in black-holes, which gives rise to Hawking radiation. Existing popular ideas of preserving quantum information entering black holes – the Preskill informational mirror, and the Horowitz-Maldacena mechanism for black-hole evaporation (which necessitates a unique final state for the black-hole), can be exactly incarnated as quantum information swapping or transfer using Andreev reflection processes. I will present mesoscopic physics analogs to wormholes and time loops using postselection on the superconducting ground state of a condensed gas of Cooper pairs – and conjecture that the BCS ground state also describes the final quantum state of a black hole.

### Wednesday, February 15th at noon (tea and cookies at 11:30am)

#### Speaker: Gerhard Heinzmann, Université de Lorraine/CNRS & Archives Henri-Poincaré (UMR 7117), Nancy, France

Title: The Structuralist Roots of Mathematical Understanding Reconsidered: Poincaré’s heritage

Abstract: This paper proposes a reconsideration of mathematical structuralism. It adopts the "practical turn" that owes much to Henri Poincare. By reconsstructing his group theoretic approach of geometry, it seems possible to explain the main difficulty of modern structuralism, inaugurated by the French collective Bourbaki around the middle of the XXth century: the unclear ontological status of ‘structures’ and ‘places’. For Poincaré, the formation of the group concept - a 'universal' - is suggested by a specific system of stipulated sensations and, read as a relational set, the general group concept constitutes a model of the group axioms, which are exemplified (in the Goodmanien sense) by the sensation system. In other words, the shape created in the mind leads to a particular type of platonistic universals, which is a model (in the model theoretical sens) of the mathematical axiom system of the displacement group. The elements of the displacement group are independent and complet entities with respect to the axiom system of the group. But, by analysing the subgroups of the displacement group (common to geometries with constant curvature) one transformes the variables of the axiom system in ‘places’ whose ‘objects’ lack any ontological commitment except with respect to the specified axioms. In general, a structure R is interpreted as a second order relation which is exemplified by (axiomatic) systems according to the pragmatic maxim of Charles Sanders Peirce.

### Friday, February 10th at 2pm (tea and cookies at 1:30pm)

#### Speaker: Natalie Paquette, Burke Fellow, Walter Burke Institute of Theoretical Physics at CalTech

Title: Moonshine: Old and New

Abstract: The whimsically-named Monstrous Moonshine is a mathematical story born in the late 1970's, which provided startling connections between two fundamental objects in mathematics. It eventually found an explanatory framework in the physics of an exotic solution of string theory. Starting in 2010, moonshine phenomena reemerged in the context of a more conceptually and physically central corner of string theory. In this talk, I will survey both old and new developments in moonshine with an emphasis on their physical meanings, and highlight the as-yet mysterious connections between the many beautiful mathematical and physical objects at play. I will summarize recent work clarifying these moonshine structures in string theory.

### Monday, February 6th to Saturday, February 11

#### Speaker: 9th ANNUAL CECAT WORKSHOP IN POINTFREE MATHEMATICS

Title: 9th ANNUAL CECAT WORKSHOP IN POINTFREE MATHEMATICS

Abstract: Talks in Von Neumann Hall throughout the week.

### Friday, February 3rd at 4pm (tea and cookies at 3:30pm)

#### Speaker: Dr. Alberto Fernandez-Nieves, Professor of soft condensed matter physics at Georgia Tech

Title: Active nematics on tori

Abstract: We will discuss our recent results with active nematics on toroidal surfaces. We will first briefly describe how we generate and stabilize an otherwise unstable toroidal drop. We use these droplets to study the interplay between nematic order, geometry and topology. We find defect unbinding and defect-curvature coupling, consistent with theoretical expectations for inactive ordered materials arranged on the surface a torus. In our experiments, however, the number of defects is far larger than what one would expect for inactive nematics. This brings about interesting analogies with what we could call the high-temperature limit of inactive nematic liquid crystals.

### Monday, January 9th at 4pm (tea and cookies at 2:30pm)

#### Speaker: Professor Uwe Kahler, Universidade de Aveiro, Portugal

Title: Riemann-Hilbert problems in Clifford analysis

Abstract: One of the classic topics in Complex Analysis is the question of boundary value problems for holomorphic functions, so-called Riemann-Hilbert problems. This is not only for pure inner mathematical reasons, but also for its many applications, ranging from Materials with Memory, Inverse Scattering problems, to Statistical Physics. Due to its easy applicability since the beginning the question of Riemann-Hilbert problems in higher dimensions has caught the interest of many mathematicians. But there are essential differences between the two-dimensional case and the higher-dimensional case. In this talk we will present the general framework and highlight the difference between the two cases. Several open problems in the framework of hypercomplex analysis are being discussed.

### Monday, January 9th at 3pm (tea and cookies at 2:30pm)

#### Speaker: Professor Paula Cerejeiras, Universidade de Aveiro, Portugal

Title: Applications of Monogenic Wavelets to Image Processing

Abstract: We present an overview of applications of Clifford analysis to problems in image processing. As Clifford analysis techniques are strongly linked to the geometry of the underlying space it has generate an increasing interest in its applications to analytic signals in the last decade. Motivated by the problem of edge detection we introduce the concept of monogenic signal and discuss appropriate wavelet frames for it. We will finalize with a discussion on the group theoretical approach.

## 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, December 15th at 4:00pm (tea and cookies at 3:30pm)

#### Speaker: Alexander Kurz, University of Leicester

Title: Reasoning in Applied Logics

Abstract: Reasoning is a fundamental challenge in many areas of Computer Science such as databases (query-answering), semantic web (ontologies), artifical ingelligence (planning), and software engineering (formal methods, verification). But reasoning in general purpose logics such as first-order predicate logic does not lend itself well to automatisation. Over the decades, this lead to the succesful development of a multitude of bespoke logics, each tailored to a specific application (time, knowledge, obligations, probabilities, dynamics, …). With the growing success of these logics applications are getting more ambitious and require reasoning methods for combinations of such logics. In this talk, we present an approach that aims at designing good proof systems for a wide variety of logics based on so-called multi-type display calculi. We will also report on our work of building tools that support reasoning about and in such calculi.

### Thursday, December 8th at 4:00pm (tea and cookies at 3:30pm)

#### Speaker: Dr. Michael Campbell, CSUF

Title: The Statistical Mechanics of Bounded-Rational Potential Games with Applications

Abstract: Frequently, real economic agents do not follow purely rational strategies. These individual non-rational behaviors (due to errors in judgment, incomplete information, emotional bias, etc.) can result in some fascinating organized large-scale structures, which depend on the degree of non-rational behavior.

We look at two such models for Potential Games [Shapley and Monderer]: a dynamical drift-diffusion model, and a static large deviation theory model based on Shannon information entropy and arbitrage. The equilibrium measure in both cases is the Gibbs measure found in statistical mechanics. We show that the variables that gauge non-rational behavior in both models are related to “temperature” by a fluctuation-dissipation relation.

A type of localized discrete Cournot oligopoly has a rich phase diagram with an "antiferromagnetic" checkerboard state, striped states and maze-like states with varying widths, and finally a "paramagnetic" unordered state. Such phases have economic implications as to how agents compete given various restrictions on how goods are distributed.

The theory is also applied to a Speculative and Hedging Model in Oil and U.S. Dollar Markets [Carfi and Musolino] for a single multinational “airline” and many “bank” players. Based on results for the Nash equilibrium (zero temperature) and preliminary results, there is a phase transition for which a single equilibrium exists at higher non-rational behavior (high temperature), and two equilibria at lower non-rational behavior (low temperature), when the “airline” makes no purchase of oil. The low temperature phase is in the spirit of the Sonnenschein–Mantel–Debreu theorem, with the extra insight of symmetry-breaking to explain multiple equilibria. Likewise, Huw Dixon’s result on the “inevitability of collusion” is shown to hold for a Cournot oligopoly with a Veblen good. Purely rational neoclassical theory (i.e., Nash equilibrium analysis) alone does not predict this, even though it is observed to occur in more general cases.

### Tuesday, December 6th at 4:00pm (tea and cookies at 3:30pm)

#### Speaker: Professor Alain Yger, IMB, Universit ́e de Bordeaux, Talence, France

Title: An arithmetic elimination theorem and bounds for multivariate residues

Abstract: I will present an elimination theorem inspired by a classical theorem of Oskar Perron, combined with the approach proposed in 2005 by Zbigniew Jelonek towards the sharp geometric effectiveness of Hilbert’s Nullstellensatz. I will show next how it can be used in order to get precise estimates (in terms of the geometric and arithmetic complexity of all the data, fitting with both geometric and arithmetic B ́ezout theorems) for total sums of multivariate residues related to polynomials maps defined over Q over an algebraic variety also defined over Q. Methods start with revisiting Euclid’s algorithm, together with Bergman-Weil developments. This is very recent joint work with Mart ́ın Sombra (ICREA and University of Barcelona).

### Thursday, December 1st at 4:00pm (tea and cookies at 3:30pm)

#### Speaker: Prof. Herbert W. Hamber, University of California at Irvine

Title: The Problem with Quantum Gravitation

Abstract: Of the four fundamental forces, Gravity is the one that has been studied the longest. Besides being an immediate fact of everyday life, it still presents us today with some of the deepest challenges in contemporary physics. Einstein’s (classical) relativistic Gravity is unique, in the sense that it influences both the very largest and the very smallest length scales. These include black holes, pulsars, quasars, the Big Bang, and the Universe as a whole, at one end of the spectrum, and the microscopic structure of space-time and unified theories at the other end. Moreover, one of its most basic predictions (gravitational waves) has recently been detected on earth. Recent attempts at a quantum theory of Gravity have tried to combine, in a consistent framework, what some have regarded as the two greatest achievements of 20-th century physics: General Relativity and Quantum Mechanics. A major challenge has been to develop specific predictions that might be tested by observation. The aim of my talk will be to give a very broad brush (and hopefully elementary) survey of our understanding of Gravity and its Quantum extension.

### Thursday, November 17th at 4:15pm (tea and cookies at 3:45pm)

#### Speaker: Prof. Daniel Alpay, Chapman University

Title: Linear stochastic systems, commutative and non commutative: a white noise space approach

Abstract: The Gelfand triple consisting of the Schwartz functions the Lebesgue space and tempered distributions play a key role in analysis, and in particular in the theory of partial differential equations. We describe related triples, where the Lebesgue space is replaced by the symmetric (resp. full) Fock space associated to the Lebesgue space. The term "white noise space" in the title refers to Hida's white noise space, which is a construction of the symmetric Fock space associated to the Lebesgue space using the Bochner-Minlos theorem. The tempered distributions are now replaced by spaces of stochastic distributions. These are instances of a new family of topological algebras, which generalizes the notion of Banach algebra. As applications we study stationary increments stochastic processes and their derivatives, stochastic calculus, and linear stochastic systems, where randomness is also in the parameters of the system.

The talk is based on joint works with Haim Attia, Palle Jorgensen, Alon Kipnis, David Levanony, Ariel Pinhas, and Guy Salomon.

### Thursday, November 10th at 4:15pm (tea and cookies at 3:45pm)

#### Speaker: Prof. Matthew Leifer, Chapman University

Title: Plausibility Measures on Test Spaces

Abstract: Plausibility measures are structures for reasoning in the face of uncertainty that generalize probabilities, unifying them with weaker structures like possibility measures and comparative probability relations. So far, the theory of plausibility measures has only been developed for classical sample spaces, but there are various reasons for wanting to apply them to quantum theory, as I shall explain. In this talk, I will generalize the theory to test spaces, so that plausibility measures can be applied to general operational theories, and to quantum theory in particular. Our main results are on when a plausibility measure agrees with a probability measure, i.e. when its comparative relations coincide with those of a probability measure. For strictly finite test spaces we obtain a precise analogue of the classical result that the Archimedean condition is necessary and sufficient for agreement between a plausibility and a probability measure. In the locally finite case, the Archimedean condition implies the weaker condition of almost agreement, and one needs a stronger version of the Archimedean condition to get agreement. This is the same as the condition needed in the classical measure-theoretic case, even though we are only dealing with tests with a finite number of outcomes.

This talk is based on joint work with Tobias Fritz (preprint available at: https://arxiv.org/abs/1505.01151 )

### CECHA Conference: Friday to Monday, November 4th to November 7th in Sandhu Conference Center

#### Speaker: CECHA: Celebrating Daniel Alpay’s 60th birthday

Title: International Conference on Complex Analysis and Operator Theory

Abstract: For directions, schedule, and book of abstracts, see CECHA Webpage: CECHA Webpage,

### Thursday, October 27th at 4pm (tea and cookies at 3:30pm)

#### Speaker: Prof. Roman Buniy, Chapman University

Title: Geometric invariants associated with linear transformations

Abstract: Invariant operators associated with linear transformations naturally lead to invariant differential forms and related geometric invariants. Complex analytic properties of transformations provide an efficient way to generate and compute the invariants.

### Thursday, October 20th at 4pm (tea and cookies at 3:30pm)

#### Speaker: Christopher Cantwell, USC

Title: Quantum Chess, Making Quantum Phenomena Accessible

Abstract: Quantum phenomena have remained largely inaccessible to the general public. There tends to be a scare factor associated with the word “Quantum”, with the usual responses being along the lines of “too complicated for me.” This is in large part due to the alien nature of phenomena such as superposition and entanglement. However, Quantum Computing is a very active area of research and one day we will have games that run on those quantum computers. Quantum phenomena such as superposition and entanglement will seem as normal as gravity. Is it possible to create such games today? Can we make games that are built on top of a realistic quantum simulation and introduce players of any background to quantum concepts in a fun and mentally stimulating way?

On of the difficulties with any quantum simulation run on a classical computer is that the Hilbert space grows exponentially, making simulations of an appreciable size physically impossible due largely to memory restrictions. Here we will discuss the conception and development of Quantum Chess, and how to overcome some of the difficulties faced. We can then ask the question, “What’s next?” What are some of the difficulties Quantum Chess still faces, and what is the future of quantum games?

### Thursday, October 13th at 4pm (tea and cookies at 3:30pm)

#### Speaker: N.D. Hari Dass, Tata Centre for Interdisciplinary Sciences, Tata Inst. for Fundamental Research (TIFR-TCIS), Hyderabad, India

Title: Three results on weak measurements

Abstract: I shall present three important results on weak measurements. They are: i) repeated weak measurements on a single copy can not provide any information on it and further that in the limit of very large such mea- surements, weak measurements have exactly the same characterstics as strong measurements.However, a number of interesting results can be obtained for joint probabilities for the random walks in the quantum state space under such repeated weak measurements, ii) the apparent non-invasiveness of weak measurements is no more advantageous than strong measurements in the spe- cific context of Leggett-Garg measurements when errors are properly taken into account and finally, iii) weak value measurements are optimal, in the precise sense of Wootters and Fields, when the post-selected states are mu- tually unbiased with respect to the eigenstates of the observable whose weak values are being measured. Furthermore, notion of weak value coordinates for state spaces are introduced and elaborated. It is shown that the metric on the state space in these coordinates is conformal.

### Thursday, October 6th at 4pm (tea and cookies at 3:30pm)

#### Speaker: Prof. Ahmed Sebbar, Bordeaux University

Title: Differentially algebraic functions

Abstract: We call an analytic function f(z), defined on some open subset Differentially Algebraic (DA) if it satisfies some differential equation of the form Q(z, f(z), f'(z),\cdots, f^{(n)}(z)) = 0 for all z in its domain, where Q is a nonzero polynomial of n+2 variables, with complex coefficients.The four functions: The exponential functions e^z, the Euler Gamma function Gamma(z), the Riemann Zeta function zeta(s) and the Jacobi theta function theta(tau) are all related by Mellin transformations.

We explain briefly why Gamma and zeta are not DA but the function theta verifies a nonlinear differential equation of the third order. We give various reasons (Geometry, Arithmetics, Dynamical Systems… ) as to why this equation must exist.

### Thursday, September 22nd at 4pm (tea and cookies at 3:30pm)

#### Speaker: Dr. Josh Mutus from Google, Santa Barbara

Title: What we do at the Google Quantum Hardware team

Abstract: We’re trying to build a quantum computer capable of serving Google’s billions of users worldwide. I’ll introduce why Google wants to build a quantum computer and outline our two major thrusts: quantum annealing and error-corrected universal quantum computation. I’ll describe how we’re building our quantum computer from the ground up, starting with the microfabrication techniques used to engineer our superconducting qubits. Also, I’ll share overview of our new Quantum Hardware lab, including specialized high-capacilty cryostats, custom built high-frequency electronics and a our stack of open-source experimental control software.

Josh works with the team of Dr. John Martinis, who heads a cutting-edge experimental program for realizing a quantum computer with superconducting quantum bits. You can read about some of their recent accomplishments here: http://web.physics.ucsb.edu/~martinisgroup/

### 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 10:30am)

#### 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 10:30am)

#### 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

[1] M. Bilkova, G. Greco, A. Palmigiano, A. Tzimoulis, and N. Wijnberg. The logic of resources and capabilities. In preparation, 2016. [2] W. Conradie and A. Craig. Canonicity results for mu-calculi: an algorithmic approach. Journal of Logic and Computation, forthcoming. ArXiv preprint 1408.6367. [3] W. Conradie, A. Craig, A. Palmigiano, and Z. Zhao. Constructive canonicity for lattice- based fixed point logics. Submitted. ArXiv preprint 1603.06547. [4] W. Conradie, Y. Fomatati, A. Palmigiano, and S. Sourabh. Algorithmic correspondence for intuitionistic modal mu-calculus. Theoretical Computer Science, 564:30–62, 2015. [5] W. Conradie, S. Frittella, A. Palmigiano, M. Piazzai, A. Tzimoulis, and N. Wijnberg. Categories: How I Learned to Stop Worrying and Love Two Sorts. Submitted. ArXiv preprint 1604.00777. [6] W. Conradie, S. Ghilardi, and A. Palmigiano. Unified Correspondence. In A. Baltag and S. Smets, editors, Johan van Benthem on Logic and Information Dynamics, volume 5 of Outstanding Contributions to Logic, pages 933–975. Springer International Publishing, 2014. [7] W. Conradie and A. Palmigiano. Constructive canonicity of inductive inequalities. Submitted. ArXiv preprint 1603.08341. [8] W. Conradie and A. Palmigiano. Algorithmic correspondence and canonicity for distributive modal logic. Annals of Pure and Applied Logic, 163(3):338 – 376, 2012. [9] W. Conradie and A. Palmigiano. Algorithmic correspondence and canonicity for non- distributive logics. Journal of Logic and Computation, forthcoming. ArXiv preprint 1603.08515. [10] W. Conradie, A. Palmigiano, and S. Sourabh. Algebraic modal correspondence: Sahlqvist and beyond. Submitted. [11] W. Conradie, A. Palmigiano, S. Sourabh, and Z. Zhao. Canonicity and relativized canonicity via pseudo-correspondence: an application of ALBA. Submitted. Arxiv preprint 1511.04271. [12] W. Conradie, A. Palmigiano, and Z. Zhao. Sahlqvist via translation. Submitted. ArXiv preprint 1603.08220. [13] W. Conradie and C. Robinson. On Sahlqvist theory for hybrid logic. Journal of Logic and Computation, DOI: 10.1093/logcom/exv045. [14] S. Frittella, A. Palmigiano, and L. Santocanale. Dual characterizations for finite lattices via correspondence theory for monotone modal logic. Journal of Logic and Computation, forthcoming. ArXiv preprint 1408.1843. [15] G. Greco, M. Ma, A. Palmigiano, A. Tzimoulis, and Z. Zhao. Unified correspondence as a proof-theoretic tool. Journal of Logic and Computation, forthcoming. ArXiv preprint 1603.08204. [16] A. Palmigiano, S. Sourabh, and Z. Zhao. J ́onsson-style canonicity for ALBA-inequalities. Journal of Logic and Computation, DOI:10.1093/logcom/exv041. [17] A. Palmigiano, S. Sourabh, and Z. Zhao. Sahlqvist theory for impossible worlds. Journal of Logic and Computation, forthcoming. ArXiv preprint 1603.08202.

### 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.

### Saturday, February 6th, 2014 in Von Neumann Hall

#### 8th Annual CECAT Workshop in Pointfree Mathematics

Hosted by the Center of Excellence in Computation, Algebra and Topology (CECAT)

Held at Chapman University, Von Neumann Hall (545 W. Palm Ave, Orange, CA 92866)

Program

10.00am-10.50am Andrew Moshier (Chapman University)

"Contexts that determine locales"

11:00 - 11:50am Ales Pultr (Charles University, Prague)

"An aspect of scatteredness in frames"

12:00 - 12:50pm Papiya Bhattacharjee (Pennsylvania State University, Erie)

"Complemented frames"

2:00 - 2:50pm Peter Jipsen (Chapman University)

"Duality for (residuated) lattices and correspondence theory"

3:00 - 3:50pm Joanne Walters-Wayland (CECAT)

"Smallest dense C and C*-quotients"

### 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.