# MathCS Seminar

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

*Seminar Organizers:* Mihaela Vajiac and Justin Dressel

## Fall 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). Sometimes there will be a change of venue and the announcement will reflect this change.

See [http://www.chapman.edu/discover/maps-directions/index.aspx Maps and directions], Von Neumann Hall is Building 48 on the Campus Map [https://www.chapman.edu/about/_files/maps-and-directions/current-maps/campus-map.pdf Campus map]

### Friday, December 15th at 4:00pm (Argyros Forum 212)

#### *Speaker:* ** Dr. Andrew Jordan from University of Rochester, and Kater Murch from Washington University in St. Louis **

*Title:* ** The arrow of time in quantum mechanics**

*Abstract: * The question of why we perceive time moving from past to future is perplexing, especially in light of the fact that microscopic laws of motion are the same running forward or backwards in time. Some have thought that the answer to this puzzle may lay in quantum wavefunction collapse. We will discuss how recent experiments have shown that quantum measurement may not be as irreversible as commonly thought, and discuss deep questions relating to the flow of time and quantum physics. We will approach the topic from three points of view, that of the experimentalist, the quantum theorist, and the philosopher.

### Thursday, December 14th at 4:00pm (Argyros Forum 212)

#### *Speaker:* ** Dr. Kater Murch from Washington University in St. Louis **

*Title:* ** Measurement and control in superconducting qubits: from the quantum Zeno effect to quantum enhanced metrology **

*Abstract: * The quantum Zeno effect is a feature of measurement-driven quantum evolution where frequent measurement inhibits the decay of a quantum state. We will explore how the opposite effect; the anti-Zeno effect - where frequent measurement accelerates decay - can also emerge from frequent measurement. The emergence of one effect or the other elucidates the nature of quantum measurement and the role measurement plays in controlling quantum evolution. In a second experiment, we investigate how control over a single qubit can be used achieve a quantum speedup in the precision of frequency measurements, demonstrating a frequency sensitivity that improves as 1/T^2, where T is the duration of the experiment.

### Thursday, December 14th at 1:00pm (Argyros Forum 212)

#### *Speaker:* ** Dr. Alyssa Ney from UC Davis**

*Title:* ** Physics and Fundamentality **

*Abstract: * What justifies the allocation of funding to research in physics when many would argue research in the life and social sciences may have more immediate impact in transforming our world for the better? Many of the best justifications for such spending depend on the claim that physics enjoys a kind of special status vis-a-vis the other sciences, that physics or at least some branches of physics exhibit a form of fundamentality. The goal of this paper is to articulate a conception of fundamentality that can support such justifications. I argue that traditional conceptions of fundamentality in terms of dynamical or ontic completeness rest on mistaken assumptions about the nature and scope of physical explanations.

### Monday, December 11th at 3:00pm (tea and cookies at 2:30pm, in VN)

#### *Speaker:* ** Dr. Gabriel Uzquiano (USC)**

*Title:* ** Cantorian arguments and the limits of thought **

*Abstract: * Cantor’s theorem has often been thought to play a central role in Kaplan’s paradox for possible worlds semantics for intensional logic. Though we argue that Cantorian reasoning doesn't ultimately reach at the root of the problem, we suggest that some versions of the theorem may in fact be used as a heuristic for arriving at more informative results in the vicinity of Kaplan’s observation.

### Friday, December 1st at 3:00pm (tea and cookies at 2:30pm, in VN)

#### *Speaker:* **Sarah Alexander and Nadiya Upegui, under the supervision of Prof. Peter Jipsen, Chapman University**

*Title:* ** Partial Algebras and their Applications in Generalizations of Effect Algebras **

*Abstract: * A partial algebra is an algebra where at least one of its operations is partial. Like total algebras, partial algebras have their own concepts of homomorphisms, subalgebras, products, and congruences that account for partiality. An effect algebra is a partial algebra with a partial binary operation + that satisfies associativity and commutativity and induces a natural partial order with a bottom element 0 and a top element 1. It also contains a total unary operation ’, behaving in such a way that for all x there exists an x’ such that x+x’=1. Effect algebras can be generalized or specialized by adding or removing axioms to change their structure. A much more general class related to effect algebras is the class of generalized pseudo-effect algebras (GPEAs). GPEAs maintain associativity but are no longer commutative and need not have a unary operation ’ nor a single maximum element. It can be proven that the existence of the unary operation ’ in effect algebras in conjunction with some of the other axioms implies that + is cancellative and that 0 acts as an identity for +. These results cannot be deduced when working with GPEAs, so these properties are adopted in the GPEA axiomitization. This class of algebras also assumes a weaker notion of commutativity called conjugation. Taking a maximal element and adding it to or removing it from a GPEA creates another GPEA, and this result is the basis for a more efficient version of a Python program used to generate and enumerate these structures. Moreover, studying how changes can be made to the structure of these classes of algebras makes them easier to understand and work with. We describe two such processes, unitization and totalization, which produce involutive residuated partially ordered monoids as the resulting structures.

### Conference: Tuesday to Sunday, November 14th to November 19th in Sandhu Conference Center

==== *Speaker:* **Mathematics, Signal Processing and Linear Systems:**
New Problems and Directions** ====**

*Title:* **Mathematics, Signal Processing and Linear Systems:**
New Problems and Directions

*Abstract: * For directions, schedule, and book of abstracts, see CECHA Webpage: CECHA Webpage, or the Conference Webpage: Conference Webpage

### CECHA Workshop: Monday to Friday, November 6th to November 10th in VN Hall

#### *Speaker:* **Professor David Walnut, GMU**

*Title:* **An Overview of Wavelet Theory,**
With an eye on Superoscillations

*Abstract: * For a schedule, see CECHA Webpage: CECHA Webpage, or the Workshop Webpage: Wavelet Workhsop

### Friday, November 3rd at 3:00pm (tea and cookies at 2:30pm, in VN)

#### *Speaker:* **Dr. Marcy Robertson, Lecturer, School of Mathematics and Statistics, University of Melbourne**

*Title:* ** TBA**

*Abstract: * TBA

Dr. Robertson's Webpage: Dr. Marcy Robertson

### Thursday, November 2nd at 4:00pm (tea and cookies at 3:30pm, in VN)

#### *Speaker:* **Dr. Alain Yger (University of Bordeaux)**

*Title:* ** Transposing (p,q) - Calculus from the Complex to the Real setting, Applications**

*Abstract: * I will discuss in this talk the various presentations of the so-called theory of super (p,q) currents in the real setting (initiated by A. Lagerberg), together the relations it induces between the Fourier and Legendre-Fenchel dualities. Some applications towards number theory will be also discussed. It seems also that such ideas could be explored in other settings than the complex versus real one (for example quaternionic versus complex). I will in particular speak about the work of my student Robert Gualdi about getting closed formulaes for the logarithmic arithmetic height of an algebraic hypersurface in a toric variety; I will, if time permits, present a certain number of tools that could be necessary in order to extend the codimension one result to the higher codimension case.

### Friday, October 27th at 3:00pm (tea and cookies at 2:30pm, in VN)

#### *Speaker:* ** Dr. Ahmed Sebbar, University of Bordeaux **

*Title:* **Harmonic series, Harmonic sums and the Riemann Hypothesis**

*Abstract: * The harmonic series
$$1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\cdots $$ is one the most known series. It diverges.
We comment on why the series
$ \sum \frac{1}{p} $
(the sum is over all the primes) also diverges, however the sum $ \sum^{'} \frac{1}{n} $ where the sum is over all the integers whose decimal expansion has no nines converges.

In the second part of the talk we comment on a beautiful result of G. Robin (Limoges University) and J. Lagarias (University of Michigan) concerning the harmonic numbers $\displaystyle H_n= 1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\cdots+ \frac{1}{n} $ saying that if $\displaystyle \sigma_1(n) $ is the sum of the divisors of $n$ and if $\displaystyle \sigma_1(n) \leq e^{H_n} \log H_n +H_n$ then the Riemann hypothesis is true. This is a very elementary formulation of a Millennium Problem.

### Wednesday, October 25th at 2:30pm (Argyros Forum 208)

#### *Speaker:* ** George Csicsery, Visiting Documentary Filmmaker**

*Title:* ** Finding the hook **

*Abstract: * Filmmaker George Csicsery will discuss how he presents mathematics and mathematicians to lay audiences in a society with attention deficit disorder. Excepts from his films “N is a Number: A Portrait of Paul Erdös” (1993), “Julia Robinson and Hilbert’s Tenth Problem” (2008,” “Hard Problems” (2008), “Counting from Infinity” (2015), and “Navajo Math Circles” (2016), will be used to illustrate his main theme: that scientific ideas and math can be smuggled into a story when there is suspense, drama or genuine human interest at the foundation.

### Saturday, October 21st at 3:00pm (tea and cookies at 2:30pm, in VN)

#### *Speaker:* **Mathematical Association of America Meeting**

*Title:* ** MAA Fall 2017 SoCal-Nevada Section Meeting **

*Abstract: * See the MAA Webpage: SoCal-Nevada MAA Webpage

### Friday, October 20th at 3:00pm (tea and cookies at 2:30pm, in VN)

#### *Speaker:* ** Guillaume Jeremie Massas (UCI)**

*Title:* **Constructive representations of Heyting algebras and semantics for Intuitionistic Predicate Logic**

*Abstract: * Model Theory and the theory of Boolean algebras are two fruitful tools for investigating classical logic. Gödel’s completeness theorem and Stone’s representation theorem are regarded as cornerstones of each field respectively. The celebrated Rasiowa-Sikorski lemma allows one to construct term models out of ultrafilters on Boolean algebras, thus bridging the gap between the two: Rasiowa and Sikorski’s original proof relied on Stone’s representation theorem, and their lemma was used to give a purely algebraic proof of Gödel's completeness theorem. However the general methods behind this approach, because of their appeal to ultrafilters, are highly non-constructive.
Over the years, various generalizations of Stone’s theorem, of Gödel’s theorem, and of the Rasiowa-Sikorski lemma have taken either of the two orthogonal directions:

generalizing the results to wider classes of logics and algebras (in particular to intuitionistic logic and Heyting algebras); working under restricted forms of the axiom of choice and providing more constructive proofs. In this talk, I will present how to combine the two programs. I will first provide more constructive versions of the Rasiowa-Sikorski Lemma and constructive representation theorems for distributive lattices and Heyting algebras. I will then combine those two results to define a constructive semantics for intuitionistic predicate logic, and show how it generalizes existing semantics.

### Friday, October 6th at 3:00pm (tea and cookies at 2:30pm, in VN)

#### *Speaker:* ** David Wallace (USC)**

*Title:* **What is orthodox quantum mechanics?**

*Abstract: * What is called ``orthodox* quantum mechanics, as presented in standard foundational discussions, relies on two substantive assumptions --- the projection postulate and the eigenvalue-eigenvector link --- that do not in fact play any part in practical applications of quantum mechanics. I argue for this conclusion on a number of grounds, but primarily on the grounds that the projection postulate fails correctly to account for repeated, continuous and unsharp measurements (all of which are standard in contemporary physics) and that the eigenvalue-eigenvector link implies that virtually all interesting properties are maximally indefinite pretty much always. I present an alternative way of conceptualising quantum mechanics that does a better job of representing quantum mechanics as it is actually used, and in particular that eliminates use of either the projection postulate or the eigenvalue-eigenvector link, and I reformulate the measurement problem within this new presentation of orthodoxy.*

### Friday, September 29th at 3:00pm (tea and cookies at 2:30pm, in VN)

#### *Speaker:* ** Dr. Justin Dressel, Chapman University**

*Title:* ** Watching a quantum system: How to continuously measure a superconducting qubit**

*Abstract: * It has recently become experimentally possible to monitor the energy levels of a superconducting transmon qubit continuously in time using microwave fields. Such measurements weakly perturb the qubit per unit time, lead to a competition between unitary Hamiltonian dynamics and non-unitary collapse dynamics. I review several subtleties about modeling this measurement process, and discuss several recent achievements made in collaboration with the Siddiqi laboratory at UC Berkeley. Topics include simultaneous measurements of multiple non-commuting observables, the active use of the quantum Zeno effect with a moving measurement basis for qubit control, and subtle aspects about the information content contained in the collected stochastic readout.

Bio: Justin Dressel received his Ph.D. in quantum physics from U Rochester in 2013, was a visiting researcher at RIKEN Wako-shi in Saitama, Japan in 2013, and was a postdoctoral scholar at UC Riverside between 2013-2015, after which he started as an Assistant Professor in Physics and Computational Science at Chapman University. He researches quantum information, computing, and foundations, which is a natural intersection point between physics, mathematics, and computer science. His recent research has focused on algebraic approaches to generalized quantum measurements, quantum computation with superconducting transmon quantum bits using circuit quantum electrodynamics, and Clifford algebraic approaches to relativistic field theory. Though the bulk of his work is theoretical in nature, he works closely with experimental teams whenever possible.

### Friday, September 22nd at 2:00pm (tea and cookies at 1:30pm, in VN)

#### *Speaker:* **Dr. Ahmed Sebbar, University of Bordeaux**

*Title:* **Algebraic methods in analytic questions Second Example**

*Abstract: *The talk concerns some arithmetic and algebraic questions behind the paperfolding. One of its main objectives is to study the power series
\begin{equation}\label{new} \xi(z)= \sum_{n\geq 1} s_n z^n,\; \; |z|<1
\end{equation}
where $(s_n)$ is the paperfolding sequence defined by
\[s_{2n}= s_n,\quad s_{2n+1}= (-1)^n, \quad n\in \mathbb{Z}_+ .\]

This sequence arises as follows: A sheet of paper can be folded once right half over left or left half over right. During the talk we explore the dynamics of a continued fraction related to $ \xi(z) $, and also a relation of the problem with the modularity theorem(formerly the Taniyama-Shimura-Weil conjecture).

### Thursday, September 14th at 4:15pm (tea and cookies at 3:45pm, in VN)

#### *Speaker:* **Dr. Ahmed Sebbar, University of Bordeaux**

*Title:* **Algebraic methods in analytic questions First Example**

*Abstract: * For a fixed $x$, we consider the two functions in $h$, analytic near the origin:
\[\phi(h)= \log\left(1+3hx-h^3\right)†= \sum_{n=0}^{\infty} P_n(x) h^n†\]
and
\[\psi(h)= \log\left(1+3h^2x-h^3\right)†= \sum_{n=0}^{\infty} Q_n(x) h^n†\]
with polynomials in $x$, $P_n(x)$ and $Q_n(x)$. We wonder whether there is a relationship between $P_n(x)$ and $Q_n(x)$. The positive answer to this question depends on the peculiarities of the two quadratic extensions $\mathbb{Q}(i) $ and $\mathbb{Q}(\rho), \rho= \frac{-1+i\sqrt{3}}{2} $ and their relation with theory of complex multiplication for elliptic curves (Lemniscatomic and anharmonic cases). We give a brief history of Class Field Theory, Clausen hypergeometric function and the answer to the question. We show that the polynomials $P_n(x)$ and $Q_n(x)$ have some divisibillity propriety, similar to the one for Lucas sequences.

### Thursday, September 7th at 4:15pm (tea and cookies at 3:45pm, in VN)

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

*Title:* **On subspaces of pointfree bitopological spaces and their smallest dense subspace**

*Abstract: * With the purpose of finding a Stone duality for bitopological spaces, A. Jung and A. Moshier introduced in the category of d-frames in which objects are structures that comprise two frames, thought of as lattices of open sets, and two relations that connect both frames, as abstractions of the covering and disjointness relation. The aim of this talk is to explore an approach to the notion of parts of a space in this pointfree bitopological setting.

### Friday, September 1st at 5:00pm (tea and cookies at 4:30pm, in VN)

#### *Speaker:* **Dr. Razieh Mohseninia, Sharif University of Technology**

*Title:* **Topological quantum computation and the stability of topological memories**

*Abstract: * Quantum computers are necessary to simulate quantum systems. The fragility of qubits in presence of decoherence and external noise is the biggest obstacle in realizing a quantum computer. To overcome such problems, topological quantum computation has been introduced by Kitaev that combines the main quantum feature of the quantum world, namely, superposition of states, with the robustness of classical bits which is the result of a macroscopic number of very small entities, comprising each bit. In this way, topological features which are robust against local perturbations are used for storing information. It is well-known that the $Z_3$ Kitaev model can be used to perform universal quantum computation. Due to unwanted interactions in the system, perturbations may be added to a possible realization of the system. In the first part of my talk, stability of the $Z_3$ Kitaev model in the presence of external perturbations in the form of Potts interaction is studied. Our study relies on two high-order series expansions based on continuous unitary transformations in the limits of small and large Potts couplings as well as mean-field approximation. Our analysis reveals that the topological phase of the $Z_3$ Kitaev model breaks down to the Potts model through a first-order phase transition. Another example of topological memories is the Topological Color Code, which in 2 dimensions can be used for implementing the Clifford group in a fully topological and transversal manner. In the second part of my talk, I study thermal stability of this model in presence of a thermal bath. The auto-correlation functions of the observables are used as a figure of merit for the thermal stability. I show that all of the observables auto-correlation functions decay exponentially in time. By finding a lower bound on the decay rate, which is a constant independent of the system size, I show that the Topological Color Code is unstable against thermal fluctuations from the bath at finite temperature, even though it is stable at $T=0$ against local quantum perturbations.

## Summer 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). Sometimes there will be a change of venue and the announcement will reflect this change.

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]

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

#### *Speaker:* **Dr. Angelyn R. Lao, Department of Mathematics, De La Salle University, Philippines**

*Title:* **Systems Approaches in Making Sense of Data and Providing Meaning to Biological Models**

*Abstract: * Systems biology is an interdisciplinary approach that aims at understanding the dynamic interactions between components of biological system. It is also an approach by which biological questions are addressed through integrating experiments in iterative cycles with computational modelling, simulation and theory. This approach is best applied when there is synergistic usage of the models and data. As such that the models established are meaningful and make sense to the collected data. Depending on the types, quality, and amount of data and the purpose of the model, the kind of modeling approaches will also vary. Modelling is not the final goal, but is a tool to increase understanding of the system, to develop more directed experiments and finally allow predictions.
Why model? By modeling, we aim to guide data collection, discover new questions, formulate hypotheses, and reveal simplicity in complexity. There are different ways of modeling. One of the most popular approaches is the use of ordinary differential equations (ODE) to model the interactome among the components of a system. Other modeling approaches include stochastic modeling, Boolean modeling, agent-based modeling and etc.
Systems biology is a cross talk between different disciplines. It requires interdisciplinary collaboration between fields like biology, physics, computer science and engineering, with the goal of having a deeper insight of the functional bases of life.

## 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). Sometimes there will be a change of venue and the announcement will reflect this change.

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]

### Friday, May 26th at 2:00pm (tea and cookies at 1:30pm, in AF 212)

#### *Speaker:* **Dr. Eleanor Rieffel, Quantum Artificial Intelligence Laboratory, NASA Ames Research Center**

*Title:* **A NASA Perspective on Quantum Computing: Opportunities and Challenges**

*Abstract: * The success of the abstract model of computation, in terms of bits, logical operations, algorithms, and programming language constructs makes it easy to forget that computation is a physical process. Our cherished notions of computation and information are grounded in classical mechanics, but the physics of our universe is quantum. A natural question to ask is how computation would change if we adopted a quantum mechanical, instead of a classical mechanical, model of computation.
In the early 80s, Richard Feynman, Yuri Manin, and others recognized that certain quantum effect could not be simulated efficiently on conventional computers. This observation led researchers to speculate that perhaps such quantum effect could be used to speed up computation more generally. Slowly, a new picture of computation arose, one that gave rise to a variety of faster algorithms, novel cryptographic mechanisms, and alternative methods of communication.

For most computational problems, however, it is currently unknown whether quantum algorithms can provide an advantage, and if so by how much, or how to design quantum algorithms that realize such advantages. Many of the most challenging computational problems arising in the practical world are tackled today by heuristic algorithms that have not been mathematically proven to outperform other approaches but have been shown to be effective empirically. While quantum heuristic algorithms have been proposed, empirical testing becomes possible only as quantum computation hardware is built. The next decade promises to be exciting emerging hardware makes empirical testing of quantum heuristic algorithms more and more feasible.

In the first part of the talk, I will introduce key concepts underlying quantum computing and correct common misconceptions. In the second half of the talk, I will discuss applications of quantum computing, known advantages and limitations, including work at NASA on quantum heuristics. I will briefly touch on the current state-of-the-art in building quantum computers, quantum error correction and fault tolerance, and the many open research questions that remain.

Bio: Eleanor G. Rieffel leads the Quantum Artificial Intelligence Laboratory
at the NASA Ames Research Center. She joined NASA Ames Research Center in 2012 to work on their expanding quantum computing effort, after working at FXPAL where she performed research in diverse fields including quantum computation, applied cryptography, image-based geometric reconstruction of 3D scenes, bioinformatics, video surveillance, and automated control code generation for modular robotics.
Her research interests include quantum heuristics, evaluation and utilization of near-term quantum hardware, fundamental resources for quantum computation, quantum error suppression, and applications for quantum computing. She received her Ph.D. in mathematics from the University of California, Los Angeles. She is best known for her 2011 book Quantum Computing: A Gentle Introduction with coauthor Wolfgang Polak and published by MIT press.

### Friday, May 19th at 2:00pm (tea and cookies at 1:30pm)

*Title:* **Random variables, entanglement and nonlocality in infinite translation-invariant systems**

*Abstract: * We consider the problem of certifying entanglement and nonlocality in one-dimensional translation-invariant (TI) infinite systems when just averaged near-neighbor correlators are available. Exploiting the triviality of the marginal problem for 1D TI distributions, we arrive at a practical characterization of the near-neighbor density matrices of multi-separable TI quantum states. This allows us, e.g., to identify a family of separable two-qubit states which only admit entangled TI extensions. For nonlocality detection, we show that, when viewed as a vector in R^n, the set of boxes admitting an infinite TI classical extension forms a polytope, i.e., a convex set defined by a finite number of linear inequalities. Using DMRG, we prove that some of these inequalities can be violated by distant parties conducting identical measurements on an infinite TI quantum state. Both our entanglement witnesses and our Bell inequalities can be used to certifyentanglement and nonlocality in large spin chains (namely, finite, and not TI chains) via neutron scattering.

Our attempts at generalizing our results to TI systems in 2D and 3D lead us to the virtually unexplored problem of characterizing the marginal distributions of infinite TI systems in higher dimensions. In this regard, we show that, for random variables which can only take a small number of possible values (namely, bits and trits), the set of nearest (and next-to-nearest) neighbor distributions admitting a 2D TI infinite extension forms a polytope. This allows us to compute exactly the ground state energy per site of any classical nearest-neighbor Ising-type TI Hamiltonian in the infinite square or triangular lattice. Remarkably, some of these results also hold in 3D. In contrast, when the cardinality of the set of possible values grows (but remaining finite), we show that the marginal nearest-neighbor distributions of 2D TI systems are not described by a polytope or even a semi-algebraic set. Moreover, the problem of computing the exact ground state energy per site of arbitrary 2D TI Hamiltonians is undecidable.

### Wednesday, May 17th at 2:30pm (tea and cookies at 2:00pm)

#### *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, May 12th at 2:00pm (tea and cookies at 1:30pm)

#### *Speaker:* **Prof. Abhijit Banerjee, Krishnath College, Berhampore, Calcutta**

*Title:* **Mathematical Formulations of PT Symmetric Bicomplex Quantum Mechanics**

*Abstract: * With a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one-dimensional Schrodinger equation corresponding to a PT-symmetric bicomplex Hamiltonian H we present the mathematical formulations of an analogous version of the Schrodinger equation. Since in such a setting three different types of conjugates of bicomplex numbers appear, each is found to define, in a natural way, a separate class of time reversal operator. However, the induced parity (P)-time (T)-symmetric models turn out to be mutually incompatible, except for two of them which could be chosen uniquely. The later models are then explored by working within an extended phase space. Applications to the problems of harmonic oscillator and isotonic oscillator are considered and many new interesting properties are uncovered for the new types of PT symmetries.

### Wednesday, May 10th at 1:00pm (tea and cookies at 2:00pm, after the seminar)

#### *Speaker:* **Dr. Purbita Jana, Department of Pure Mathematics, University of Calcutta, India **

*Title:* **Graded Frame and related Mathematical structures**

*Abstract: * In this talk the main focus will be the notion of graded frame and its connection with graded fuzzy topological system, fuzzy topological
space with graded inclusion and fuzzy geometric logic with graded consequence. As a ground work we will first discuss the topic of Topology
via Logic" written by S. Vickers, where the notion of topological system and its usefulness in duality theory as well as study of topological
space via logic is mentioned. Generalizing Vickers's work step by step we will reach to the concept of graded frame, related structures and
their utilities.

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

#### *Speaker:* **Prof. Alberto 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.

### Friday, May 5th at 12:00pm (tea and cookies at 1:30pm) in collaboration with the Philosophy Department

#### *Speaker:* **Professor Pierre Wagner, Sorbonne**

*Title:* **The normative character of logic and pluralism**

*Abstract: * Logic is traditionally regarded as normative and the justification of such a view has often been that the laws of logic do not have any descriptive content: they do not describe how we do think but provide prescriptions about how we ought to think. This presupposes that logic is about thought or at least that it has something to do with reasoning. A philosophical contemporary debate about the normativity of logic questions the exact nature of the connection between logic and thought, and tries to assess the reasons for regarding logic as having a normative character. In this talk, we shall discuss this point and its essential connection with pluralism in logic.

### 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.