Seminars and Colloquia by Series

Weak Galerkin Finite Element Methods for PDEs

Series
Research Horizons Seminar
Time
Wednesday, November 4, 2015 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chunmei WangDepartment of Mathematics, Georgia Institute of Technology

Please Note: Food and Drinks will be provided before the seminar.

Weak Galerkin (WG) is a new finite element method for partial differential equations where the differential operators (e.g., gradient, divergence, curl, Laplacian etc) in the variational forms are approximated by weak forms as generalized distributions. The WG discretization procedure often involves the solution of inexpensive problems defined locally on each element. The solution from the local problems can be regarded as a reconstruction of the corresponding differential operators. The fundamental difference between the weak Galerkin finite element method and other existing methods is the use of weak functions and weak derivatives (i.e., locally reconstructed differential operators) in the design of numerical schemes based on existing variational forms for the underlying PDE problems. Weak Galerkin is, therefore, a natural extension of the conforming Galerkin finite element method. Due to its great structural flexibility, the weak Galerkin finite element method is well suited to most partial differential equations by providing the needed stability and accuracy in approximation. In this talk, the speaker will introduce a general framework for WG methods by using the second order elliptic problem as an example. Furthermore, the speaker will present WG finite element methods for several model PDEs, including the linear elasticity problem, a fourth order problem arising from fluorescence tomography, and the second order problem in nondivergence form. The talk should be accessible to graduate students with adequate training in computational mathematics.

Polynomials and (Finite) Free Probability

Series
ACO Seminar
Time
Tuesday, November 3, 2015 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Adam MarcusMathematics and PACM, Princeton University
Recent work of the speaker with Dan Spielman and Nikhil Srivastava introduced the ``method of interlacing polynomials'' (MOIP) for solving problems in combinatorial linear algebra. The goal of this talk is to provide insight into the inner workings of the MOIP by introducing a new theory that reveals an intimate connection between the use of polynomials in the manner of the MOIP and free probability, a theory developed by Dan Voiculescu as a tool in the study of von Neumann algebras. I will start with a brief introduction to free probability (for those, like me, who are not operator theorists). In particular, I will discuss the two basic operations in free probability theory (the free additive and free multiplicative convolutions), and how they relate to the asymptotic eigenvalue distributions of random matrices. I will then show how certain binary operations on polynomials act as finite analogues of the free convolutions and how the MOIP is effectively transferring the asymptotic bounds obtained in free probability to bounds in the new theory (which can then be applied to finite scenarios). If time permits, I will show how such a theory gives far better intuition as to how one might apply the MOIP in the future, using recent results on restricted invertibility and the existence of Ramanujan graphs as examples.

The de Rham fundamental group, continued

Series
Nonabelian Chabauty Seminar
Time
Monday, November 2, 2015 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Douglas UlmerGeorgia Tech
I will finish my overview of the de Rham fundamental group by reviewing two explicit calculations: Deligne's completely concrete description of the unipotent fundamental group of the projective line minus three points in terms of the free nilpotent Lie algebra on two generators, and Chen's general calculation of the unipotent fundamental group of a manifold in terms of iterated integrals.

Rational curves on elliptic surfaces

Series
Algebra Seminar
Time
Monday, November 2, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Doug Ulmer Georgia Tech
Given a non-isotrivial elliptic curve E over K=Fq(t), there is always a finite extension L of K which is itself a rational function field such that E(L) has large rank. The situation is completely different over complex function fields: For "most" E over K=C(t), the rank E(L) is zero for any rational function field L=C(u). The yoga that suggests this theorem leads to other remarkable statements about rational curves on surfaces generalizing a conjecture of Lang.

Applications of number theory in hyperbolic geometry

Series
Geometry Topology Seminar
Time
Monday, November 2, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
BoGwang JeonColumbia University
In this talk, first, I'll briefly go over my proof of the conjecture that there are only afinite number of hyperbolic 3-manifolds of bounded volume and trace field degree. Then I'lldiscuss some conjectural pictures to quantitative results and applications to other similarproblems.

Co-dimension One Motion and Assembly

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 2, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor James von BrechtCal State University, Long Beach
In this talk, I will discuss mathematical models and tools for analyzing physical and biological processes that exhibit co-dimension one characteristics. Examples include the assembly of inorganic polyoxometalate (POM) macroions into hollow spherical structures and the assembly of surfactant molecules into micelles and vesicles. I will characterize when such structures can arise in the context of isotropic and anisotropic models, as well as applications of these insights to physical models of these behaviors.

Mining mesoscale physics from polycrystalline data sets

Series
CDSNS Colloquium
Time
Monday, November 2, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Siddharth MaddaliCarnegie Mellon
I present a formalism and an computational scheme to quantify the dynamics of grain boundary migration in polycrystalline materials, applicable to three-dimensional microstructure data obtained from non-destructive coarsening experiments. I will describe a geometric technique of interface tracking using well-established optimization algorithms and demonstrate how, when coupled with very basic physical assumptions, one can effectively measure grain boundary energy density and mobility of a given misorientation type in the two-parameter subspace of boundary inclinations. By doing away with any specific model or parameterization for the energetics, I seek to have my analysis applicable to general anisotropies in energy and mobility. I present results in two proof-of-concept test cases, one first described in closed form by J. von Neumann more than half a century ago, and the other which assumes analytic but anisotropic energy and mobility known in advance.

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