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Vakhtang Putkaradze, Variational Methods for Active Porous Media

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Many biological organisms are comprised of deformable porous media, with additional complexity of an embedded muscle. Using geometric variational methods, we derive the equations of motion for the dynamics of such active porous media. The use of variational methods allows to incorporate both the muscle action and incompressibility of the fluid and the elastic matrix…

Kailash Misra, NC State, Affine Lie Algebras and Crystals

SAS 1102

Affine Lie algebras, also sometimes called current algebras, are infinite-dimensional analogs of finite-dimensional semisimple Lie algebras. The representation theory of affine Lie algebras has applications in many areas of mathematics (number theory, combinatorics, group theory, geometry, topology, etc.) and physics (conformal field theory, integrable systems, statistical mechanics, etc.). To study the combinatorial properties of affine Lie algebra…

Ilse Ipsen, NC State Randomized Algorithms for Least Squares/Regression Problems

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We review randomized algorithms for the numerical solution of least squares/regression problems, with a focus on algorithms that row-sketch from the left, or column-sketch from the right. These algorithms tend to be efficient and accurate on matrices that have many more rows than columns. We present probabilistic bounds for the amount of sampling required to…

Peter McGrath, NC State, Calculus of Variations and the Bending Energy of Surfaces

SAS 1102

Beginning with the solution of the classical Plateau problem—the problem of finding an area-minimizing disk whose boundary is a prescribed simple closed curve in Euclidean 3-space—we will survey some applications of  Calculus of Variations to solve geometric extremal problems. Particular emphasis will be placed on the problem of finding a smooth surface in 3-space with…

Ailana Fraser, University of British Columbia, Geometries That Optimize Eigenvalues

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When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases…

Gregory Forest, UNC Chapel Hill, Modeling Insights Into SARS-CoV-2 Respiratory Tract Infections

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I and many collaborators, postdocs, and students from many disciplines have explored lung mechanics and disease pathology for over 2 decades in a pan-university effort called the UNC Virtual Lung Project. In the last decade we have explored how viruses “traffic” in mucosal barriers, including the human respiratory tract (RT), in the presence of antibodies.…

Gloria Mari Beffa, University of Wisconsin, Discrete Geometry of Polygons and Soliton Equations

SAS 4201

 In this talk we will discuss the connection between invariant evolutions of polygons and completely integrable discrete systems via polygonal geometric invariants. We will give examples and show how some open problems for bi-Hamiltonian structures of discrete systems were made easier and solved using this correspondence. If time allows we will discuss some open problems.…

Semyon Tsynkov, NC State, Imaging With Synthetic Aperture Radar

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Synthetic aperture radars (SAR) use microwaves to obtain images of the Earth's surface from airplanes or satellites. SAR images can be taken during nighttime and prove insensitive to the clouds or dust in the atmosphere. Therefore, SAR complements the aerial or spaceborne photography, even though there are fundamental differences between the two technologies. For example,…

RESCHEDULED Thomas J. R. Hughes, University of Texas at Austin, Isogeometric Analysis: Mathematical and Engineering Perspectives

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The vision of Isogeometric Analysis (IGA) was first presented in a paper published October 1, 2005 . Since then it has become a focus of research within both the fields of Finite Element Analysis (FEA) and Computer Aided Design (CAD) and has become a mainstream analysis methodology and provided a new paradigm for geometric design…

Colloquium: Thomas J.R. Hughes, University of Texas at Austin, The Finite Element Method and Computational Mechanics: Past, Present, and a Vision of the Future

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I will begin by probing into the past to discover the origins of the Finite Element Method (FEM), and then trace the evolution of those early developments to the present day in which the FEM is ubiquitous in science, engineering, mathematics, and medicine, and the most important discretization technology in Computational Mechanics. However, despite its…

Colloquium: Naihuan Jing, NC State, Quantum Linear Algebra

SAS 1102

In linear algebra we know that the Pfaffian of an antisymmetric matrix is a square root of the determinant of matrix. In this talk I will explain how one does the quantum linear algebra, a recent popular area that can be traced back to Gauss and is well connected with many areas of mathematics such as…

Colloquium: Alison Marsden, Stanford University, Multi-Physics Modeling of Flow and Cardiac Function in Pediatric Cardiology

SAS 2203

Congenital heart disease affects 1 in 100 infants and is the leading cause of infant mortality in the US. Among the most severe forms of congenital heart disease is single ventricle physiology, in which the heart develops with only one functional pumping chamber. These patients typically undergo three open chest surgeries, culminating in the Fontan…

Colloquium: Eitan Tadmor, University of Maryland, Swarm-Based Random Descent Method for Non-Convex Optimization

SAS 4201

We introduce a Swarm-Based Random Descent (SBRD) method for non-convex optimization. The swarm consists of agents, identified with positions, x, and masses, m.  There are three key aspects to the SBRD dynamics: (i) persistent transition of mass from high to lower ground; (ii) marching along the gradient descent: an m-dependent random choice of marching direction…

Colloquium: Moody Chu, NC State, Optimal Hamiltonian Synthesis for Quantum Computing

SAS 4201

Simulating the time evolution of a Hamiltonian system on a classical computer is hard—the computational power required to even describe a quantum system scales exponentially with the number of its constituents, let alone integrating its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge. Assuming that a quantum…

Colloquium: Christopher K.R.T. Jones, University of North Carolina at Chapel Hill, Do We Need to Adapt to a Changing Climate, or to the Rate at Which it is Changing?

SAS 4201

The climate is changing due to the heat trapping caused by the rapid increase in greenhouse gases, mainly carbon dioxide, in the atmosphere. One way to state the issue is that we cannot, as a species, adapt to the new conditions quickly enough. This is an example of rate-induced tipping for which the mathematics has…

Colloquium: Gadi Fibich, Tel Avis University, Effects of Network Structure on Spreading of Innovations

SAS 4201

Spreading (diffusion) of new products is a classical problem. Traditionally, it has been analyzed using the compartmental Bass model, which implicitly assumes that all individuals are homogeneous and connected to each other. To relax these assumptions, research has gradually shifted to the more fundamental Bass model on networks, which is a particle model for the…

Colloquium: Nikolaos Kapouleas, Brown University, Minimal Surface Doublings and Their Geometry

SAS 4201

Minimal surfaces are fundamental geometric objects which have been studied intensively since the 1700's. Classes of minimal surfaces of particular interest are the complete embedded ones in Euclidean space, closed (compact boundaryless) embedded in the round three-sphere, free boundary compact embedded ones in the unit Euclidean three-ball, and self-shrinkers of the mean curvature flow. Since…

Colloquium: Bojko Bakalov, NC State, Dynamical Lie algebras

SAS 4201

Quantum computers are physical machines that process information using the principles of quantum mechanics, which in turn is underpinned by linear algebra. The talk will start with a review of Lie algebras (consisting of matrices under the operation of commutator) and their role in quantum mechanics. The dynamical Lie algebra (DLA) of a quantum system…

Colloquium: Stefan Steinerberger, University of Washington, Growth Models in the Plane

SAS 4201

We'll study the growth of (two-dimensional) things. Think about lichen growing on a tree (tends to be sort of round). Another fun example is electricity propagating through wood (tends to be sort of fractal). A famous and still very mysterious model is called DLA: it forms the most beautiful fractal patterns (pictures will be provided).…