Arindam Roy

Speaker: Tuoc Phan, Professor of Mathematics, University of Tennessee at Knoxville

Title: Recent results on linear elliptic and parabolic equations with singular or degenerate coefficients

Abstract: The talk is about the analysis theory for second order linear elliptic and parabolic equations. We particularly focus on the regularity estimates, existence, and uniqueness of solutions to classes of equations whose coefficients are just measurable. Several classical results when the coefficients are uniformly elliptic and bounded will be reviewed and explained. Motivations to study the equations whose coefficients are neither uniformly elliptic nor bounded will be addressed. Some recent results and developments will be reported. Important open research directions will be mentioned. The main results discussed in talk are based on several papers which are the joint work with Hongjie Dong (Brown University) and with Hung Tran (University of Wisconsin at Madison).

Speaker:  Zixuan Cang , Assistant Professor of Mathematics, NC State University

Title: Spatiotemporal analysis of single-cell and spatial genomics data

Abstract: The emerging single-cell and spatial genomics techniques allow us to elucidate the governing rules of multicellular systems with unprecedented resolution and depth. These datasets are often high-dimensional, complex, and heterogeneous. Mathematical tools are needed to extract biological insights from such data. In this talk, we will discuss several mathematical and machine learning methods for exploring the tissue structures, temporal signatures, and cell-cell communication processes on single-cell and spatial genomics data. We will also discuss supervised optimal transport which is motivated by these biological applications where application-induced constraints are enforced in the optimal transport problem.

Speaker: Vladimir Panov, Associate Professor of Statistics and Data Analysis, HSE University (Zoom Talk)

Title: Mellin Transform Techniques for Statistical Inference

Abstract: My talk is devoted to the application of the Mellin transform techniques to various statistical problems arising in the context of  the mixture models.  For  the variance-mean mixtures, I will present a new semiparametric approach based on the properties of the superposition of Mellin and Laplace transforms.  Later, we will show an adaptation of this method to more complicated model of moving average Levy processes, which is closely related to a wide (and rather popular) class of models known as the ambit fields. Also we will present some new theoretical facts concerning the Mellin transform (e.g., the analogue of the Berry-Esseen inequality), which yield some fresh ideas for the statistical estimation in the classical model of multiplicative mixtures.

Speaker: Lorena Bociu, Professor of Mathematics, NC State University

Title: Analysis and Control in Poroelastic Systems with Applications to Biomedicine

Abstract: Fluid flows through deformable porous media are relevant for many applications in biology, medicine and bio-engineering, including tissue perfusion, fluid flow inside cartilages and bones, and design of bioartificial organs. Mathematically, they are described by quasi-static nonlinear poroelastic systems, which are implicit, degenerate, coupled systems of partial differential equations (PDE) of mixed parabolic-elliptic type. We answer questions related to tissue biomechanics via well-posedness theory, sensitivity analysis, and optimal control for the poroelastic PDE coupled systems mentioned above. One application of particular interest is perfusion inside the eye and its connection to the development of neurodegenerative diseases.

Speaker: Christian Kümmerle, Assistant Professor of Computer Science, UNC Charlotte

Title: Learning of Transition Operators From Sparse Space-Time Samples

Abstract: We present a framework for solving the nonlinear inverse problem of learning a transition operator A from partial observations across different time scales. This problem is motivated by the recent interest in learning time-varying graph signals driven by graph operators which depend on the underlying topology. We show that its non-linearity can be addressed computationally by reformulating it as a matrix completion problem utilizing a low-rank property of a suitable block Hankel embedding matrix. 

For a uniform and an adaptive random space-time sampling model, we quantify the recoverability of the transition operator via incoherence parameters of block Hankel embedding matrices, whose behaviors depend on the interplay between the dynamics and the graph topology for graph transition operators. We show local quadratic convergence of a suitable Iteratively Reweighted Least Squares algorithm under the two observation models from a minimal amount of samples, and present how our analysis informs a suitable adaptive sampling strategy based on a fixed budget of spatio-temporal samples. Finally, we present numerical experiments which confirm that the theoretical findings accurately track empirical phase transitions.

Speaker: TBA

Title: TBA

Abstract: TBA

Speaker: Rodney Keaton, Assistant Professor of Mathematics, East Tennessee State University

Title: Stacking Results in Graph Rubbling

Abstract: Graph pebbling was first presented as a technique for solving an elementary number theory problem by Lagarias and Saks. This resulted in a 1989 paper of Chung, where the notion of graph pebbling was formalized and used to solve the previously mentioned problem. A variant of graph pebbling, known as graph rubbling, was first introduced in 2009 by Belford and Sieben. In this talk we provide an introduction to graph pebbling and graph rubbling. Then, we will present several recent results in this area which make connections between graph rubbling, stacking, and graph domination.

Speaker: Khai Nguyen, Professor of Mathematics, NC State University

Title: Shocks interaction for the Burgers-Hilbert Equation

Abstract: In 2009 J. Biello and J. Hunter derived a balance law modeling nonlinear waves with constant frequency, obtained from Burgers’ equation by adding the Hilbert transform as a source term.  For general L^2(R) initial data, the global existence of entropy weak  solutions was proved by Bressan and Nguyen in 2014, together with a partial uniqueness result. Recently, unique piecewise continuous solutions with a single shock and the shock formation have been recently studied. This talk will describe a further type of local generic singularities for solutions, namely, points where two shocks interact. 

Speaker: Zhen Li, Assistant Professor of Mechanical Engineering, Clemson University

Title: Memory Effects in Coarse-Grained Modeling of Soft Matter

Abstract: Elimination of degrees of freedom from a complex dynamic system often introduces non-negligible memory effects, resulting in a non-Markovian, generalized Langevin equation (GLE) for the coarse-grained (CG) system in the context of the Mori-Zwanzig (MZ) formalism. For the conservation of momentum of the CG system, GLE can be reformulated into its pairwise version, i.e., non-Markovian dissipative particle dynamics (DPD) upon a pairwise decomposition. In this talk, I will introduce how to apply the rigorous theoretical approach of MZ to derive new governing equations for CG dynamics of polymer systems. I will demonstrate the coarse-graining procedure by running a molecular dynamics simulation of polymer melts and constructing the MZ-guided CG model directly from atomistic trajectories, as well as the computation of the memory kernel of dissipation in CG dynamics. Unlike ad hoc coarse-graining procedures, MZ-guided coarse-graining generates accurate and efficient CG model that reproduces the correct static and dynamic properties of its corresponding atomistic system. Moreover, I will introduce different coarse-graining strategies with GLE and non-Markovian DPD models to incorporate the memory effects in practical CG simulations.