# MathCS Seminar

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

*Seminar Organizers:* Mihaela Vajiac and Justin Dressel

## 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 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.