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A Symposium at the Intersection of Mathematics and Biology

Keynote Speaker: Dr. Simon Levin
J.S. McDonnell Distinguished University Professor in Ecology and Evolutionary Biology
Princeton University

To celebrate the conferral of UVic Honorary Doctor of Science to Dr. Levin

To register: Contact Kristina McKinnon, pims@uvic.ca, before May 15, 2019.

Contributed Presentations: Oral presenters will be selected from submitted abstracts. To submit an abstract for a talk: Please submit a title, list of authors, abstract (200 words max) and affiliation by e-mail before May 1, 2019 to: Kristina McKinnon, pims@uvic.ca

Sponsors:
PIMS
CAIMS/SCMAI
University of Victoria

Download poster (PDF)

PIMS - UVic Distinguished Lecture

(pre-lecture refreshments at 3:30 pm)

Abstract: Range expansion is a crucial population response to climate change. Genetic consequences are coupled to ecological dynamics that, in turn, are driven by shifting climate conditions. We model a population with a reaction–diffusion system, coupled to a heterogeneous environment that shifts with time due to climate change. We decompose the resulting traveling wave solution into neutral genetic components to analyze the spatio-temporal dynamics of its genetic structure. Our analysis shows that range expansion under slow climate change preserves genetic diversity. However, diversity is diminished when climate change occurs too quickly. We show that populations with intermediate dispersal ability are best for maintaining genetic diversity. Our study also provides new insight regarding traveling wave solutions in heterogeneous environments. This is joint with Jimmy Garnier (CNRS).

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Abstract: Assuming stationarity is unrealistic in many time series applications. A more realistic alternative is to allow for piecewise stationarity, where the model is allowed to change at given time points. In this talk, we propose a three-stage procedure for consistent estimation of both structural change points and parameters of high- dimensional piecewise vector autoregressive (VAR) models. In the first step, we reformulate the change point detection problem as a high-dimensional variable selection one, and propose a penalized least square estimator using a total variation penalty. We show that the proposed penalized estimation method over- estimates the number of change points. We then propose a backward selection criterion in conjunction with a penalized least square estimator to tackle this issue. In the last step of our procedure, we estimate the VAR parameters in each of the segments. We prove that the proposed procedure consistently detects the number of change points and their locations. We also show that the procedure consistently estimates the VAR parameters. The performance of the method is illustrated through several simulation scenarios and real data examples. This is a joint work with Ali Shojaie.

Jesse Johnson will talk about the paper "On Abelian 1-avoidable binary patterns”, by James D. Currie and Terry I. Visentin.

Abstract: In this talk I will present a next generation model for the spread of sea lice on a network of salmon farms. Sea lice are marine parasites with a free living larval stage and multiple attached stages where the parasites are directly attached to salmon on a farm. The larval stage can spread between salmon farms, and this spread is modelled by an advection diffusion equation. Survival and maturation through the attached stages are modelled using arbitrary functions, which can accommodate any maturation or survival delays in a stage. A next generation matrix is then used to examine the effect of different environmental variables on the source-sink distribution and persistence of the population.

Abstract: The Wright-Fisher diffusion and the Feller diffusion are among the first models of diffusion processes and were originally constructed from diffusion approximations of one-dimensional Markov chains (Wright, 1945; Feller, 1951). To this day, the method has been extensively developed for questions including establishing universality classes of stochastic differential equations, or stochastic partial differential equations (SPDEs), through scaling limits of discrete spatial models.

In this talk, I will start with an introduction of the Wright-Fisher diffusion and the Feller diffusion. Then I will discuss the progress in the last decade for two questions arising from diffusion approximations: (1) uniqueness in certain spatial generalizations of the Feller diffusion as solutions to SPDEs, and (2) spatial universality of the Wright-Fisher diffusion for interacting particle systems defined by deaths and births. This talk will be concluded with questions for future research.

Abstract: Non-Euclidean data that are indexed with a scalar predictor such as time are increasingly encountered in data applications, while statistical methodology and theory for such random objects are not well developed yet. To address the need for new methodology in this area, we develop a total variation regularization technique for nonparametric Fréchet regression, which refers to a regression setting where a response residing in a generic metric space is paired with a scalar predictor and the target is a conditional Fréchet mean. We show that the resulting estimator is representable by a piece-wise constant function and investigate the convergence rate of the proposed estimator for data objects that reside in Hadamard spaces. The method can also be applied to the problem of estimating multiple change-points in a sequence of non-Euclidean data. This is illustrated via the application to modeling the dynamics of brain networks and the study of evolving mortality distributions endowed with the Wasserstein distance.

Abstract: Types of interactions between entities of a system limit the hopes for the stability of the system, regardless of the level of interaction. In this talk, I will briefly introduce a dynamical system, namely the Kuramoto model, and then restrict it to a graph where attraction and repulsion define positive and negative edges of the graph, respectively. Then I'll present some results on how some graph-theoretic properties of the graph put bounds on the number of positive and negative eigenvalues of the Jacobian at a fixed point. These results are mainly from the 2014 paper by Bronski and DeVille, "Spectral theory for dynamics on graphs containing attractive and repulsive interactions", SIAM Journal of Applied Mathematics, 74 (1) 83–105.

Abstract: As tropical storms go, you have probably heard of Hurricanes, Tropical Cyclones, El Niño, and La Niña. But you probably haven't heard of the Madden-Julian Oscillation (MJO). It is the major contributor to rainfall in tropical regions and influences the climate in Victoria regularly. Unlike Hurricanes and El Niño, the MJO is still not well understood. In an effort to understand the mechanisms of the MJO, I will describe a model building from a dynamically stationary "background" tropical rainfall model and coupling that to a tropical wave model. These models use Stochastic Differential Equations (SDE) and Stochastic Partial Differential Equations (SPDE) as the building blocks. In the "background" model, an SDE model is used which leads to characteristics of criticality and phase transitions. For the full model with waves we use a continuous, one-dimensional, SPDE. Because of the simplicity of the models, we are able to solve many statistics exactly, or run fast numerical experiments.

Abstract: For a compact manifold M, the Atiyah-Singer index map is the map from the K-theory of the cotangent bundle of M to the integers induced by collapsing T*M to a point. The construction of this map makes use of the Thom isomorphism and wrong-way funtoriality for open embeddings. In this talk I will discuss the Thom isomorphism in topological K-theory, and show how given a map f:M->N between smooth manifolds which satisfies a certain orientability assumption, called "K-orientability", one can construct a map f!:K*(M)->K*(N), which generalizes the index map of Atiyah and Singer. I will then define topological correspondences, which consist in part of a K-orientable map, and show one can define bivariant K-theory for smooth manifolds using topological correspondences in a way which agrees with Kasparov's definition of KK-theory.

I will discuss some models for the shape of liquid droplets on rough solid surfaces. These are elliptic free boundary problems with oscillatory coefficients. The framework of homogenization theory allows to study the large scale effects of small scale surface roughness, including interesting physical phenomena such as contact line pinning, hysteresis, and formation of facets.

Abstract: As the title suggests, I'll be presenting a construction given by R.M. Wilson in 1973. This construction implies that N(mt+u)\geq min{N(m),N(m+1),N(t)-1,N(u)} whenever 0<u<t (where N(n) is the maximum size of a set of MOLS(n)). In the next talk (on Feb 28), I'll use this to present a proof that N(n)\geq n^{1/(14,8)} for large enough n.

This Alternative Math Education event will present fun methods of teaching math and computer science concepts to children (and adults!) using games and art. 

Lots of hands-on activities!

Including: 

• A Sorting Network 

• An Impossible Balancing Act

• Bubbles 

• Mathematical Puzzles 

• The Guessing Game 

• Sudoku• The Penny Game• The Set Game• And many more!

For more information please contact the school or Kristina McKinnon, PIMS University of Victoria Site at:

T: 250-472-4271

E: pims@uvic.ca

In the first part of the talk, we saw how regular dilation on the right angled Artin monoids unified two classical dilation theorems. Right angled Artin monoids belongs to the class of right LCM semigroups. The operator algebras related to right LCM semigroups attracts a lot of attention since Xin Li's construction of semigroup C*-algebras (also Nica's covariant representations of quasi-lattice ordered semigroups).

In the part two of the talk, I present the generalized notion of regular dilation on right LCM semigroups and the connection between regular dilation and Xin Li's covariant representations. This provides another angle to look at the concept of regular dilation.

See abstract (PDF).

Abstract: Advances in technology have been generating data with increased complexity. Modern data objects are often high-dimensional and can also lay in 2D, 3D and even 4D Euclidean and sometimes non-Euclidean planes. One example are data generated from large-scale multi-site studies such as the Alzheimer's Disease Neuroimaging Initiative. In these scenarios, complexity comes in the form of brain images such as magnetic resonance imaging (MRI), a 3D object that is high-dimensional (number of voxels is around 900K for each subject), with measurements that are correlated in space. Another example are data coming from an instrument specifically designed to measure scleral strain in donor eyes. The device generates functional data with functions defined on a non-Euclidean 2D partial spherical domain. Associating these complex data with clinical, environmental, and genetic variables can be challenging and classical statistical tools need to evolve to keep up with the complexity of these new data types. In this talk, I address the challenges and solutions to some important statistical questions in this context.

Jesse Johnson will talk about the paper "Strongly non-repetitive sequences and progression-free sets" by F.M. Dekking F.M, published in J. Combin. Theory Ser. A 27., 1979.

We conduct a novel imaging genetics study of the Alzheimer’s Disease Neuroimaging Initiative based on resting-state fMRI (rs-fMRI) and genetic data obtained from 112 subjects, where each subject is classified as either cognitively normal (CN), as having mild cognitive impairment (MCI), or as having Alzheimer’s Disease (AD). A Dynamic Causal Model (DCM), a state space model for neuronal dynamics, is fit to the rs-fMRI time series in order to estimate a directed network representing effective brain connectivity within the default mode network (DMN), a key network commonly known to be active when the brain is at rest. These networks are then related to genetic data and Alzheimer’s disease in the first imaging genetics study to use DCM as a neuroimaging phenotype. Our proposed pipeline is comprised of four analyses linked together with the objective of shedding light on the relationship between brain connectivity and genetics in relation to disease. In the first analysis we examine differences in effective connectivity across disease groups. In the second analysis we relate the probability of disease to genetics and obtain a subset of priority single-nucleotide polymorphisms (SNPs) potentially related to disease. In the third analysis we investigate how effective brain connectivity is related to the subset of priority SNPs obtained in the second study. In the fourth and final analysis we examine longitudinal data on changes with respect to MCI and AD through a classifier in order to determine how well disease progression can be predicted from the combination of effective brain connectivity and genetic data. Our new pipeline for connectome genetics has general applicability and its specific application to the ADNI study motivates a number of future studies in this nascent area.

Dr. Farouk Nathoo is an Associate Professor and Tier 2 Canada Research Chair in the Department of Mathematics and Statistics at the University of Victoria, and an Adjunct Professor in the Department of Statistics and Actuarial Science at Simon Fraser University. His primary areas of research are the analysis of neuroimaging data, imaging genetics, statistical modelling, and Bayesian methods

Abstract: Dilation theory originated from Sz.Nagy's celebrated dilation theorem, where it states that every contractive operator can be nicely embedded in an isometry. Generalizing Sz.Nagy's dilation to multi-variable setting has been an interesting area of research.

In the first part of the talk, I briefly give an overview of the study of the dilation theory. In particular, I focus on the concept of regular dilation, first originated from Brehmer's study on family of commuting contractions and since been generalized to contractive representations on various classes of semigroups. We will look into the case of right-angled Artin monoids and examine how regular dilation unified two seemingly unrelated dilation theorems.

Abstract: We use the idea of ground states and excited states in nonlinear dispersive equations (e.g. Klein-Gordon and Schr\"odinger equations) to characterize solutions in the N-body problem with strong force under some energy constraints. Indeed, relative equilibria of the N-body problem play a similar role as solitons in PDE. We introduce the ground state and excited energy for the N-body problem. We are able to give a conditional dichotomy of the global existence and singularity below the excited energy, the proof of which seems original and simple. This dichotomy is given by the sign of a threshold function $K_\omega$. The characterization for the two-body problem in this new perspective is non-conditional and it resembles the results in PDE nicely. For $N\geq3$, we give some refinements of the characterization, in particular, we examine the situation where there are infinitely transitions for the sign of the threshold function.

This is joint work with Slim Ibrahim.

Department of Psychology Cognition and Brain Sciences Seminar

Abstract: We conduct an imaging genetics study of the Alzheimer's Disease Neuroimaging Initiative based on resting-state fMRI (rs-fMRI) and genetic data obtained from 112 subjects, where each subject is classified as either cognitively normal (CN), as having mild cognitive impairment (MCI), or as having Alzheimer's Disease (AD). A Dynamic Causal Model (DCM; Li et al., 2011; Friston et al., 2014; Razi et al., 2015), a state-space model for coupled neuronal fluctuations, is fit to the rs-fMRI time series in order to estimate a directed network representing effective brain connectivity within the default mode network (DMN), a key network commonly known to be active when the brain is at rest. We consider four important regions of the DMN, and use DCM to estimate a 16-parameter graph representing effective connectivity. These graphs are then related to genetic data and disease in four analyses that: (i) examine differences in effective connectivity networks across disease groups; (ii) relate the probability of disease to genetics; (iii) investigate how effective brain connectivity in the DMN is related to genetics; (iv) examine longitudinal data on the changes to cognitive impairment in order to determine how well disease progression can be predicted from the combination of brain connectivity networks and genetic data. Our four studies together lead to a novel pipeline for connectome genetics data analysis and motivate a number of future studies in this nascent area. The talk will have a strong focus on statistical methods used in our analyses and will also include a technical review of DCM.

Karoubi introduced a notion of relative K-theory for C*-algebras: given a C*-algebra, A, and a C*-subalgebra, A', there is a relative group K(A';A). I will give the definition (along with a quick definition of the non-relative groups) and discuss some basic properties. An excision theorem is a result that says that this group depends only on the difference, in some sense, between A' and A. As an example, if A' is actually an ideal then K(A';A) is naturally isomorphic to the K-theory of the quotient algebra A/A'. I will present another theorem of this type with some applications to groupoid C*-algebras arising from dynamical systems.

In this talk, we study a minimization problem associated to ground state solutions of combined power-type nonlinear Schr\"{o}dinger equations with Sobolev critical exponent. In four and higher space dimensions, it is known that for any given positive frequency, there exists always a minimizer. Contrary to that, we shall show that in three space dimensions, the minimization problem has no minimizer when the frequency is high and the exponent of the subcritical power nonlinearity is less than or equal to three. In addition, we determine a frequency threshold that classifies the existence and non-existence of minimizers.

To prove our results, we employ a resolvent expansion as in Coles and Gustafson (preprint). The resolvent expansion has a singularity because of a resonance in three space dimensions. The frequency threshold appears thanks to the presence of this singularity.

This talk is based on a joint work with T. Akahori(Shizuoka), S. Ibrahim (Victoria) and H. Nawa (Meiji)

Mark your calendars! On Thursday December 6th we will be having a day of activities to celebrate the works of Emeritus Professor Ahmed Sourour. Laurent Marcoux and Heydar Radjavi from the University of Waterloo will be giving research talks covering Sourour's scientific contributions. Sourour's former PhD student Maria Inez Cardoso Gonçalves from the Federal University of Santa Catarina, Brazil will also be joining us. We will also have a problem-solving session in recognition of Sourour's contributions to the annual UVic Math Contest and Putnam competitions. Please see the Ahmed Fest website for more detailed information.

Abstract: Fell bundles may be viewed as generalised partial group actions. Under mild assumptions, a Fell bundle comes from a partial group action, and if it is saturated, the action is global. Amenability for Fell bundles is defined so that it generalises amenability of groups. In this talk, we will introduce a notion of Fell bundles extended from positive cones of quasi-lattice orders. That is, they arise from certain actions of semigroups. This includes semi-saturated Fell bundles over $\mathbb{Z}$ and semi-saturated and orthogonal Fell bundles over free groups as considered by Exel. We will describe cross sectional $\mathrm{C}^*$-algebras of Fell bundles extended from free semigroups and from Baumslag--Solitar semigroups as relative Cuntz--Pimsner algebras. We then use uniqueness theorems for relative Cuntz--Pimsner algebras to prove amenability for such Fell bundles.

Abstract:
The Enskog-Vlasov equation extends the Boltzmann equation by accounting for the attractive forces between, and the finite volume of, the gas particles. Hence, it gives a van-der-Waals-like description of a non-ideal gas, including liquid-vapor phase change. Specifically, the equation describes the liquid phase, the vapor phase, and a diffusive transition region connecting both phases. While not the most accurate model, solutions of the Enskog-Vlasov equation exhibit all relevant phenomena occurring in the evaporation and condensation of rarefied or dense vapors.

Our interest here is to derive macroscopic transport equations—moment equations—from the Enskog-Vlasov equation, which describe liquid vapor and transition region in terms of a few macroscopic properties. To this end, we use Grad’s moment method to derive a closed set of 26 moment equations. In the appropriate limits, these reduce to the Navier-Stokes-Fourier system for liquid and vapor. Our main interest is to study non-hydrodynamic effects, in particular transport in and across the transition region, and the interplay between the transition region and Knudsen layers. We will present first results of this program, including the closed transport equations for 26 moments, discussion of the limits, and solutions in simple geometries.

Title: A lower bound for the gap between the first and second Lyapunov
exponents for a class of cocycles of Perron-Frobenius operators preserving
a cone.

Abstract: Lyapunov exponents provide a generalization to cocycles of
operators (“random” compositions of operators) of the spectrum for single
operators. This spectral data allows us to understand expansion and decay
rates of parts of the space on which the operators act; when applied in
the case of Perron-Frobenius operators related to dynamics on a space, it
helps us to understand how mass evolves under the dynamics and decay rates
of transients in terms of spectral gaps.  The problem of computing
Lyapunov exponents is generally acknowledged to be a hard one.

In this series of three talks, we will see how we can find a lower bound
on the gap between the first and second Lyapunov exponents for certain
Perron-Frobenius operator cocycles: those that preserve and contract
(somehow) a nice cone of functions. The proof begins with a general
theorem about cocycles of operators and cones where the hypotheses are
checkable and the numbers are, in principle, computable. Applications to
concrete examples require specific tools if they are not “toy” problems.
We will develop one such tool, a specialized Lastota-Yorke inequality, and
use it to estimate the gap for our specific example.  We will also use
this analysis to study the situation where the system passes under
perturbation from multiple equilibria to a unique equilibrium with a small
spectral gap.

In this second talk, we will describe a new Lasota-Yorke inequality that
is 
well-adapted for cocycles of paired tent maps, and we will explain why we
need 
this new inequality, recalling some facts about cones and the Cocycle
Perron-Frobenius theorem.

Please join us for a meet and greet at 3 pm in DTB A514, refreshments will be served.  The talk will take place at 3:30 pm after the meet and greet. 

ABSTRACT:

The problem of source-sink dynamics in mathematical ecology originated with a 1969 paper of Levins. The idea is that a species lives in an environment that is heterogeneous: some locations are favourable to the species persistence (sources), while others are unfavourable and can only be populated if there is an inflow of individuals to these locations (sinks). This setup has been studied extensively in the case of so-called Levins-type metapopulation models, which count the number of locations (patches) in various states and couple them implicitly. The case of metapopulation models with explicit movement, where individuals move between locations, is less known.  

In this talk, I will discuss the solution to a simple problem set in the latter context: is there a critical number of the number of patches that are sources such that the population persists in the entire system when the number of sources is above the threshold? An existence result is provided that relies heavily on matrix analysis, illustrating the power of linear algebra in this type of large scale problem. In a particular case, the principle of equitable partitions allows us to obtain an explicit form for the threshold. This is joint work with Nicolas Bajeux (University of Manitoba) and Steve Kirkland (University of Manitoba).

Download poster (PDF)

One of the commonly used approaches to modeling extremes is the peaks-overthreshold (POT) method. The POT method models exceedances over a threshold that is sufficiently high so that the exceedance has approximately a generalized Pareto distribution (GPD). This method requires the selection of a threshold that might affect the estimates. Here we propose an alternative method, the Log-Histospline (LHSpline), to explore modeling the tail behavior and the remainder of the density in one step using the full range of the data. LHSpline applies a smoothing spline model to a finely binned histogram of the log transformed data to estimate its log density. By construction, a LHSpline estimation is constrained to have polynomial tail behavior, a feature commonly observed in daily rainfall observations. We illustrate the LHSpline method by analyzing precipitation data collected in Houston, Texas.

A deterministic model with spatial consideration for a class of human disease-transmitting vectors is presented and analyzed. The model takes the form of a nonlinear system of delayed ordinary differential equations in a compartmental framework. Using the model, existence conditions of a non-trivial steady-state vector population are obtained when more than one breeding site and human habitat site are available. Model analysis confirms the existence of a non-trivial steady state, uniquely determined by a threshold parameter, R_0^j, whose value depends on the distribution and distance of breeding site j to human habitats. Results are based on the existence of a globally and asymptotically stable non-trivial steady-state human population.

This is where mathematics and economics meet. I will introduce and explain the terms given in the title, discuss their meaning and relations, and introduce models of kinetic type and their use. The classical Bass model is a special case.

Abstract: I will introduce a new class of semigroup C*-algebras associated with algebraic number fields. After explaining the construction, I will discuss several initial results, including an explicit description of the primitive ideals of these C*-algebras, and, time permitting, a phase transition theorem. 

ABSTRACT: I will describe methods for bounding instantaneous extreme values over attractors of dynamical systems governed by differential equations. The methods achieve tight bounds by using computer assistance, in particular using polynomial optimization and semidefinite programming. As an example, I will show results for the chaotic Lorenz attractor. The talk will be aimed at a broad audience.

Abstract: I will introduce a new class of semigroup C*-algebras associated with algebraic number fields. After explaining the construction, I will discuss several initial results, including an explicit description of the primitive ideals of these C*-algebras, and, time permitting, a phase transition theorem. 

Abstract: The ultrapower construction takes a C*-algebra $A$ and an ultrafilter $\mathcal{U}$, and produces a new C*-algebra $A^{\mathcal{U}}$ that shares many features with $A$, but also satisfies precise versions of some properties that $A$ only satisfies approximately.  While the construction appears to depend on both the C*-algebra and the ultrafilter, it is not easy to distinguish the ultrapowers obtained by using the same C*-algebra with different ultrafilters.

In the first talk I will describe the ultrapower construction and some of its properties, and also discuss the connection between ultrapowers and model theory.  In the second talk I will show that the question ``Does $A^{\mathcal{U}}$ depend on $\mathcal{U}$?” cannot be answered on the basis of the standard set-theoretic foundations of mathematics, by sketching proofs that if the Continuum Hypothesis holds then the answer is “no” (Ge-Hadwin, 2001) while if the Continuum Hypothesis fails then the answer is (in a very strong way) “yes” (Farah-Hart-Sherman, 2010).

Abstract: The ultrapower construction takes a C*-algebra $A$ and an ultrafilter $\mathcal{U}$, and produces a new C*-algebra $A^{\mathcal{U}}$ that shares many features with $A$, but also satisfies precise versions of some properties that $A$ only satisfies approximately.  While the construction appears to depend on both the C*-algebra and the ultrafilter, it is not easy to distinguish the ultrapowers obtained by using the same C*-algebra with different ultrafilters.

In the first talk I will describe the ultrapower construction and some of its properties, and also discuss the connection between ultrapowers and model theory.  In the second talk I will show that the question ``Does $A^{\mathcal{U}}$ depend on $\mathcal{U}$?” cannot be answered on the basis of the standard set-theoretic foundations of mathematics, by sketching proofs that if the Continuum Hypothesis holds then the answer is “no” (Ge-Hadwin, 2001) while if the Continuum Hypothesis fails then the answer is (in a very strong way) “yes” (Farah-Hart-Sherman, 2010).

The UVic Mathematics Competition is held annually in the fall and is open to all UVic undergraduate students. The competition will be held on Tuesday, September 25, 2018 between 4:30-6:30 pm in MacLaurin D010. There are monetary prizes. To participate, just show up.

Here are some recent question papers:  2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006.

ABSTRACT:  In the Pioneer 100 (P100) Wellness Project (Price and others, 2017), multiple types of data are collected on a single set of healthy participants at multiple time points, in order to characterize and optimize wellness. One way to do this is to identify clusters, or subgroups, among the participants, and then to develop personalized health recommendations tailored to each subgroup. It is tempting to cluster the participants using all of the data types and timepoints, in order to fully exploit the available information. However, clustering the participants based on multiple data views implicitly assumes that a single underlying clustering of the participants is shared across all data views. If this assumption does not hold, then clustering the participants using multiple data views will lead to spurious results. In this paper, we seek to evaluate the assumption that a single underlying clustering of the participants is shared across all data views. Specifically, we ask the question: are the clusters within each data view dependent or independent? We develop a new test for answering this question, which we then apply to clinical, proteomic, and metabolomic data, across two distinct time points, from the P100 study. We find that while the subgroups of the participants defined with respect to any single data type seem to persist across time, the clustering among the participants based on one data type (e.g. proteomic data) appears not to be associated with the clustering based on another data type (e.g. clinical data).

Abstract:  In this talk I will sketch some new constructions of mutually orthogonal latin squares and difference triangle sets that arise from intersecting classical finite-field difference sets with a subgroup.  Some well-studied Diophantine equations are encountered, yet it is an unexpected property of their solutions which leads to good constructions.