Michael Grabchak

Abstract: With the advance of high-throughput sequencing technologies, it has become feasible to investigate the influence of the entire spectrum of sequencing variations on complex human diseases. Although association studies utilizing the new sequencing technologies hold great promise to unravel novel genetic variants, especially rare genetic variants that contribute to human diseases, the statistical analysis of high-dimensional sequencing data remains a challenge. Advanced analytical methods are in great need to facilitate high-dimensional sequencing data analyses. In this talk, we will introduce a generalized genetic random filed (GGRF) method for association analyses of sequencing data in case-control studies. We will then further extend GGRF method to a family-based GGRF (FB-GGRF) method for family-based association studies. Both GGRF and FB-GGRF methods are compared with other existing methods through simulation studies and real data applications for investigating the genetic etiology of complex diseases/traits.

Abstract: Asymptotic behavior of the tuning parameter selection in the standard cross-validation methods is investigated for the high-dimensional variable selection problem. It is shown that the shrinkage effect of the Lasso penalty is not always the true reason for the over-selection phenomenon in the cross-validation based tuning parameter selection. After identifying the potential problems with the standard cross-validation methods, we propose a new procedure, Consistent Cross-Validation (CCV), for selecting the optimal tuning parameter. CCV is shown to enjoy the tuning parameter selection consistency property under certain technical conditions. Extensive simulations and real data analysis support the theoretical results and demonstrate that CCV also works well in terms of prediction.

Mikhail Malioutov,  Department of Mathematics, Northeastern University

Abstract:  Modeling applications such as literary texts as stationary time series is a firmly established tool in computer linguistics. A popular problem is an authorship attribution of texts which in statistical terms is called homogeneity testing which must be nonparametric with respect to the multivariate distribution of time series since data will never suffice to find their parametric model. An adequate tools turn out to be Variable memory Length MC and Conditional Complexity of  Compression. The latter was initially inspired by analogy with non-computable Kolmogorov Complexity. A review of the underlying theory and many applications will be presented.

Michael Klibanov,  Department of Mathematics and Statistics, UNC Charlotte

Abstract:  This is the first solution of a long standing problem which was posed in chapter 10 of the book of K. Chadan and P. Sabatier “Inverse Problems in Quantum Scattering Theory”, Springer-Verlag, New York, 1977. 

In quantum scattering one measures only scattering cross section. This means that the modulus of the complex valued scattering wave field is measured. But phase is not measured. In the meantime the entire theory of inverse quantum scattering is based on the assumption that both modulus and phase are measured. Thus, the question was posed in the above book whether it is possible, at least in principle, to uniquely reconstruct the scattering potential if only the modulus of the scattered field is known for all frequencies. This question was addressed positively for the first time in the paper of the author published in SIAM J. Applied Mathematics, 74, #2, 392-410, 2014.

Dr. Alexander Bendikov,  Department of Mathematics, Cornell University

Abstract:  Let (X,d) be a locally compact separable metric space. We assume that (X,d) is proper and totally disconnected. Given a measure m on X and a choice-function C(B) defined on the set of all non-singleton balls B of X we consider the hierarchical Laplacian L=L_{C}. The operator L acts in L(X,m), is essentially self-adjoint and has a purely point spectrum. Choosing a sequence of i.i.d. random variables we define a perturbated choice-function and a perturbated hierarchical Laplacian. We study asymptotic behaviour of the arithmetic means of the eigenvalues of the perturbated hierarchical Laplacian. We prove that under some mild assumptions the arithmetic means are asymptotically normal whereas without these conditions the limited distribution is not normal. We prove existence of the integrated density of states (i.d.s.) associated with the operator perturbated hierarchical Laplacian – the quantitative characteristics of the phase transition in the Dyson’s model. Assuming that the i.d.s is continuous we prove that the number of eigenvalues which fall in a small interval is approximately Poissonian. As an example we consider random perturbations defined by the i.i.d. Bernoulli random variables of the Vladimirov-Taibleson operator acting on the ring of p-adic numbers.

Dr. Chun Liu,  Department of Mathematics, Pennsylvania State University

Abstract:  In the talk, I will explore the underlying mechanism governing various diffusion processes.  We will employ a general framework of energetic variational approaches, consisting of, in particular, Onsager’s Maximum Dissipation Principles, and their specific applications to biology and physiology. We will discuss the roles of different stochastic integrals (Ito’s form, Stratonovich’s form and other possible forms), and the procedure of optimal transport in the context of general framework of theories of linear responses.

Dr. Marc Ryser,  Department of Mathematics, Duke University

Abstract: In the first part of my talk I will discuss the well-posedness of the 2D stochastic Allen-Cahn equation, a commonly used stochastic partial differential equation in physics and material sciences. I will tell the whole tale, from numerical investigations and formulation of a conjecture, to heuristic arguments and the conjecture-turned-theorem. In the second part of my talk I will present ongoing work at the interface of mathematics and cancer biology. I will introduce a stochastic model of carcinogenesis, and show how a combination of analysis and simulations are used to address clinically relevant problems. To finish, I will talk about an ongoing collaboration with cancer surgeons at Duke, where we develop a modeling framework to make patient-specific predictions.