Patricia Hersh : Topology and combinatorics of regular CW complexes
- Algebraic Geometry ( 144 Views )Anders Björner characterized which finite, graded partially ordered sets (posets) are closure posets of finite, regular CW complexes, and he also observed that a finite, regular CW complex is homeomorphic to the order complex of its closure poset. One might therefore hope to use combinatorics to determine topological structure of stratified spaces by studying their closure posets; however, it is possible for two different CW complexes with very different topological structure to have the same closure poset if one of them is not regular. I will talk about a new criterion for determining whether a finite CW complex is regular (with respect to a choice of characteristic functions); this will involve a mixture of combinatorics and topology. Along the way, I will review the notions from topology and combinatorics we will need. Finally I will discuss an application: the proof of a conjecture of Fomin and Shapiro, a special case of which says that the Schubert cell decomposition of the totally nonnegative part of the space of upper triangular matrices with 1's on the diagonal is a regular CW complex homeomorphic to a ball.
Natasa Sesum : On the extension of the mean curvature flow and the Ricci flow
- Geometry and Topology ( 105 Views )In the talk we will discuss curvature conditions under which we can guarantee the existence of a smooth solution to the Ricci flow and the mean curvature flow equation. These are improvements of Hamilton's and Husiken's results on extending the Ricci flow and the mean curvature flow, under conditions that the norm of Riemannian curvature and the norm of the second fundamental form are uniformly bounded along the flow in finite time, respectively.
Thomas Witelski : Mean field models and transient effects for coarsening dynamics in fluid films
- Nonlinear and Complex Systems ( 100 Views )Motivated by the dewetting of viscous thin films on hydrophobic substrates, we study models for the coarsening dynamics of interacting localized structures in one dimension. For the thin films problem, lubrication theory yields a Cahn-Hilliard-type governing PDE which describes spinodal dewetting and the subsequent formation of arrays of metastable fluid droplets. The evolution for the masses and positions of the droplets can be reduced to a coarsening dynamical system (CDS) consisting of a set of coupled ODEs and deletion rules. Previous studies have established that the number of drops will follow a statistical scaling law, N(t)=O(t^{-2/5}). We derive a Lifshitz-Slyozov-Wagner-type (LSW) continuous model for the drop size distribution and compare it with discrete models derived from the CDS. Large deviations from self-similar LSW dynamics are examined on short- to moderate-times and are shown to conform to bounds given by Kohn and Otto. Insight can be applied to similar models in image processing and other problems in materials science. Joint work with M.B. Gratton (Northwestern Applied Math).
Laura Miller : Scaling effects in heart development: Changes in bulk flow patterns and the resulting forces
- Applied Math and Analysis ( 92 Views )When the heart tube first forms, the Reynolds number describing intracardial flow is only about 0.02. During development, the Reynolds number increases to roughly 1000. The heart continues to beat and drive the fluid during its entire development, despite significant changes in fluid dynamics. Early in development, the atrium and ventricle bulge out from the heart tube, and valves begin to form through the expansion of the endocardial cushions. As a result of changes in geometry, conduction velocities, and material properties of the heart wall, the fluid dynamics and resulting spatial patterns of shear stress and transmural pressure change dramatically. Recent work suggests that these transitions are significant because fluid forces acting on the cardiac walls, as well as the activity of myocardial cells which drive the flow, are necessary for correct chamber and valve morphogenesis.
In this presentation, computational fluid dynamics was used to explore how spatial distributions of the normal forces and shear stresses acting on the heart wall change as the endocardial cushions grow, as the Reynolds number increases, and as the cardiac wall increases in stiffness. The immersed boundary method was used to simulate the fluid-structure interaction between the cardiac wall and the blood in a simplified model of a two-dimensional heart. Numerical results are validated against simplified physical models. We find that the presence of chamber vortices is highly dependent upon cardiac cushion height and Reynolds number. Increasing cushion height also drastically increases the shear stress acting on the cushions and the normal forces acting on the chamber walls.
Miles Crosskey : Spectral bounds on empirical operators
- Graduate/Faculty Seminar ( 99 Views )Many machine learning algorithms are based upon estimating eigenvalues and eigenfunctions of certain integral operators. In practice, we have only finitely many randomly drawn points. How close are the eigenvalues and eigenfunctions of the finite dimensional matrix we construct in comparison to the infinite dimensional integral operator? In what way can we say these two are close if they do not even operate on the same spaces? To answer these questions, I will be showing some results from a paper "On Learning with Integral Operators" by Rosasco, Belkin, and De Vito.
Leonid Berlyand : Flux norm approach to finite-dimensional homogenization approximation with non-separated scales and high contrast
- Applied Math and Analysis ( 149 Views )PDF Abstract
Classical homogenization theory deals with mathematical models of strongly
inhomogeneous media described by PDEs with rapidly oscillating coefficients
of the form A(x/\epsilon), \epsilon → 0. The goal is to approximate this problem by a
homogenized (simpler) PDE with slowly varying coefficients that do not depend
on the small parameter \epsilon. The original problem has two scales: fine
O(\epsilon) and coarse O(1), whereas the homogenized problem has only a coarse
scale.
The homogenization of PDEs with periodic or ergodic coefficients and
well-separated scales is now well understood. In a joint work with H. Owhadi
(Caltech) we consider the most general case of arbitrary L∞ coefficients,
which may contain infinitely many scales that are not necessarily well-separated.
Specifically, we study scalar and vectorial divergence-form elliptic PDEs with
such coefficients. We establish two finite-dimensional approximations to the
solutions of these problems, which we refer to as finite-dimensional homogenization
approximations. We introduce a flux norm and establish the error
estimate in this norm with an explicit and optimal error constant independent
of the contrast and regularity of the coefficients. A proper generalization of
the notion of cell problems is the key technical issue in our consideration.
The results described above are obtained as an application of the transfer
property as well as a new class of elliptic inequalities which we conjecture.
These inequalities play the same role in our approach as the div-curl lemma
in classical homogenization. These inequalities are closely related to the issue
of H^2 regularity of solutions of elliptic non-divergent PDEs with non smooth
coefficients.
Michael Henry : Connections between existing Legendrian knot invariants
- Geometry and Topology ( 124 Views )In this talk, we will investigate existing Legendrian knot invariants and discuss new connections between the theory of generating families and the Chekanov-Eliashberg differential graded algebra (CE-DGA). The geometric origins of the CE-DGA are Floer theoretic in nature and come out of the Symplectic Field Theory developed by Eliashberg and Hofer. On the other hand, Legendrian invariants derived from the study of 1-parameter families of smooth functions (called generating families) are Morse theoretic in nature. In the last decade, connections have been found between the Legendrian invariants derived using these two methods. In this talk, I will try to provide a clearer picture of the relationship between generating families and the CE-DGA.
Dan Lee : Black hole uniqueness and Penrose inequalities
- Geometry and Topology ( 113 Views )I will tell two stories. The first is the story of static spacetimes with black hole boundaries and the attempt to classify them. The second is the story of the Penrose inequality. I will then weave these two stories together in the setting of negative curvature. This last part is a report on joint work-in-progress with A. Neves.
Hongqiang Wang : Non-equipartition in a binary granular system and measurement of velocity distribution in a 3D vibrated granular system
- Nonlinear and Complex Systems ( 111 Views )Fluidized granular systems with inelastic inter-particle collisions exhibit distinguishing behavior from it's elastic counterpart. Two species of particles in a binary granular system typically do not have the same mean kinetic energy, in contrast to the equipartition of energy required in equilibrium. It is found that not only the mechanical properties of these two types of particles, but also the heating mechanism plays an important role in affecting the extent of nonequipartition of kinetic energy, even in the bulk of the system. An experimental measurement of the velocity distribution of a 3D vibration fluidized granular medium by spatial resolved high speed video particle tracking is also reported. It is found that the distribution is wider than a Gaussian and broadens continuously with increasing volume fraction.
Junping Wang : Mathematics and Computation of Sediment Transport for Open Channels
- Applied Math and Analysis ( 106 Views )The purpose of this presentation is to communicate some mathematical and computational issues in sediment transport for open channels. The main topics are: (1) mathematical simulation for surface and subsurface fluid flow, (2) mathematical modeling of sediment transport in open channels as a 2D problem, and (3) numerical methods for fluid flow and sediment transport.
Graham Cox : Unsolvable problems in geometry and topology
- Graduate/Faculty Seminar ( 112 Views )The resolution of Hilbert's tenth problem yields the following unsolvability result: there is no algorithm for determining whether or not a given polynomial equation p(x_1,...,x_n) = 0 with integer coefficients will admit an integer solution. After a few definitions and examples, I will discuss another well-known unsolvable problem: the word problem for finitely presented groups. It can be shown that there is no algorithm for determining when an arbitrary word in a finitely presented group is trivial. This has many remarkable topological consequences, including the result that there is no algorithm that will determine when two given manifolds are homeomorphic (provided the dimension is at least four). The unsolvability theorem also has significant geometric applications, allowing one to prove that certain manifolds admit an infinite number of contractible closed geodesics (regardless of the Riemannian structure).
Graeme Wilkin : Morse theory and stable pairs
- Geometry and Topology ( 133 Views )In the early 1980s Atiyah and Bott described a new approach to studying the cohomology of the moduli space of stable bundles: the equivariant Morse theory of the Yang-Mills functional. There are many other interesting moduli spaces that fit into a similar framework, however the catch is that the total space is singular, and it is not obvious how to construct the Morse theory of the appropriate functional. In this talk I will describe how to get around these difficulties for the moduli space of stable pairs, for which we prove a Kirwan surjectivity theorem and give a Morse-theoretic interpretation of the change in cohomology due to a flip. This builds upon earlier work with George Daskalopoulos, Jonathan Weitsman and Richard Wentworth for rank 2 Higgs bundles.
Badal Joshi : A coupled Poisson process model for sleep-wake cycling
- Applied Math and Analysis ( 114 Views )Sleep-wake cycling is an example of switching between discrete states in mammalian brain. Based on the experimental data on the activity of populations of neurons, we develop a mathematical model. The model incorporates several different time scales: firing of action potentials (milliseconds), sleep and wake bout times (seconds), developmental time (days). Bifurcation diagrams in a deterministic dynamical system gives the occupancy time distributions in the corresponding stochastic system. The model correctly predicts that forebrain regions help to stabilize wake state and thus modifies the wake bout distribution.
Aubrey HB : Persistent Homology
- Graduate/Faculty Seminar ( 171 Views )Persistent Homology is an emerging field of Computational Topology that is developing tools to discover the underlying structure in high-dimensional data sets. I will discuss the origins and main concepts involved in Persistent Homology in an accessible way, with illustrations and comprehensive examples. If time allows, I will also describe some current, as well as, future applications of Persistent Homology.
Gábor Székelyhidi : Greatest lower bounds on the RIcci curvature of Fano manifolds
- Geometry and Topology ( 110 Views )On a Fano manifold M we study the supremum of the possible t such that there is a Kähler metric in c_1(M) with Ricci curvature bounded below by t. We relate this to Aubin's continuity method for finding Kähler-Einstein metrics and we give bounds on it for certain manifolds.
Cécile Piret : Overcoming the Gibbs Phenomenon Using a Modified Radial Basis Functions Method
- Applied Math and Analysis ( 119 Views )The Radial Basis Functions (RBF) method is not immune from the disastrous effects of the Gibbs phenomenon. When interpolating or solving PDEs whose solutions are piecewise smooth functions, the RBF method loses its notorious spectral accuracy. In this talk, a new method will be presented, based on the RBF method, which incorporates singularities using Heaviside functions and which keeps track of their location using the level set method. The resulting sharp interface method will be shown to recover the lost spectral accuracy and thus overcome the Gibbs phenomenon altogether.
Benoit Charbonneau : Gauge theory and modern problems in geometry
- Graduate/Faculty Seminar ( 112 Views )I will survey some modern questions in geometry that were solved or that could be solved using tools of gauge theory. This talk should be accessible to first year grad students, and of interest to anyone who is curious about what happens in the field of geometry.
Andrew Christlieb : A high order adaptive mesh refinement algorithm for hyperbolic conservation laws based on weighted essentially non-oscillatory methods
- Applied Math and Analysis ( 137 Views )In this work, we combine the adaptive mesh refinement (AMR) framework with high order finite difference weighted essentially non-oscillatory (WENO) method in space and TVD Runge-Kutta (RK) method in time (WENO-RK) for hyperbolic conservation laws. Our goal is to realize mesh adaptivity in the AMR framework, while maintaining very high (higher than second) order accuracy of the WENO-RK method in the finite difference setting. To maintain high order accuracy, we use high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from the coarse to find grid, and at ghost points. The resulting scheme is high order accuracy, robust and efficient, due to the mesh adaptivity and has high order accuracy in both space and time. We have experimented the third and fifth order AMR-finite difference WENO-RK schemes. The accuracy of the scheme is demonstrated by applying the method to several smooth test problems, and the quality and efficiency are demonstrated by applying the method to the shallow water and Euler equations with different challenging initial conditions. From our numerical experiment, we conclude a significant improvement of the fifth order AMR - WENO scheme over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, which we think is due to the very low numerical diffusion of high order schemes. This work is in collaboration with Dr. Chaopeng Shen and Professor Jing-Mei Qiu.
Michael Gratton : Transient and self-similar dynamics in thin film coarsening
- Applied Math and Analysis ( 144 Views )Coarsening is the phenomenon where many objects (water drops, molecular islands, particles in a freezing liquid) becoming smaller in number but larger in size in an orderly way. This talk will examine modeling one such system, nanoscopic liquid drops, through three models: a PDE for the fluid, a coarsening dynamical system for the drops, and an LSW-type ensemble model for the distribution of drops. We will find self-similar solutions for the drop population valid for intermediate times and discuss transient effects that can delay the self-similar scaling.
Dave Rose : Categorification and knot homology
- Graduate/Faculty Seminar ( 102 Views )Categorification can be viewed as the process of lifting scalar and polynomial invariants to homology theories having those invariants as (graded) Euler characteristics. In this talk, we will discuss categorification in general and as manifested in specific examples (ie Khovanov homology and knot Floer homology). Examples will be given showing how the categorified invariants are stronger and often more useful than the original invariants. I will motivate categorification using familiar constructions from (very basic) topology. It is my hope that this will make the discussion accessible to a wide audience. No prior knowledge of knot theory or category theory needed!
David M. Walker : Contact Network Analysis of Granular Media
- Nonlinear and Complex Systems ( 102 Views )The particles in a deforming assembly of a granular material continually rearrange themselves when subject to loading. This rearrangement can be usefully represented by an evolving (complex) contact network reflecting the changing connectivity. The tools of complex networks summarize the properties of these contact networks and changes in the physical material manifest in changes to these properties. We consider two different DEM systems, a biaxial compression test and a second system which allows for particle breakage, and discuss how different properties of the contact networks help to reveal different aspects of the materials'response to loading. (Joint work with Antoinette Tordesillas)
Pierre Degond : Asymptotic-Preserving numerical methods for variable-scale problems. Examples from fluids and plasma dynamics
- Applied Math and Analysis ( 100 Views )Multiscale problems are often treated via asymptotic of homogenization techniques: one first determines the asymptotic limit and then finds an appropriate numerical methods to solve it. Variable scale problems which exhibit a continuous variation of the perturbation parameter from a finite to an infinitesimal value cannot be solved by this method alone. They require the coupling of the asymptotic problem to the original one across the region of scale variation. This coupling is often quite complex and lacks robustness. Asymptotic-Preserving methods represent an alternative to the coupling strategy and provide a way to resolve the original problem without resorting to its asymptotic limit. They provide a systematic methodology to resolve multiscale problems even in situations where the asymptotic limit is quite complex. We will provide examples of this methodology for the treatment of the low-Mach number regime, of quasineutrality in plasmas, large magnetic fields or strong anisotropy in diffusion equations.
Tom Witelski : Perturbation analysis for impulsive differential equations: How asymptotics can resolve the ambiguities of distribution theory
- Graduate/Faculty Seminar ( 169 Views )Models for dynamical systems that include short-time or abrupt forcing can be written as impulsive differential equations. Applications include mechanical systems with impacts and models for electro-chemical spiking signals in neurons. We consider a model for spiking in neurons given by a nonlinear ordinary differential equation that includes a Dirac delta function. Ambiguities in how to interpret such equations can be resolved via perturbation methods and asymptotic analysis of delta sequences.
Oliver R Diaz : Long wave expansions for water waves over random bottom
- Probability ( 99 Views )We introduce a technique, based on perturbation theory for Hamiltonian PDEs, to derive the asymptotic equations of the motion of a free surface of a fluid over a rough bottom (one dimension). The rough bottom is described by a realization of a stationary mixing process which varies on short length scales. We show that the problem in this case does not fully homogenize, and random effects are as important as dispersive and nonlinear phenomena in the scaling regime. We will explain how these technique can be generalized to higher dimensions
Pengzi Miao : On critical metrics on compact manifolds with boundary
- Geometry and Topology ( 96 Views )It is known that, on closed manifolds, Einstein metrics of negative scalar curvature are critical points of the usual volume functional constrained to the space of metrics of constant scalar curvature. In this talk, I will discuss how this variational characterization of Einstein metrics can be localized to compact manifolds with boundary. I will present the critical point equation and focus on geometric properties of its general solutions. In particular, when a solution has zero scalar curvature and its boundary can be isometrically embedded into the Euclidean space, I will show that the volume of this critical metric is always greater than or equal to the Euclidean volume enclosed by the image of the isometric embedding and two volumes are the same if and only if the critical metric is isometric to the Euclidean metric on a standard round ball. This is a joint work with Prof. Luen-Fai Tam.
Hubert Bray : Voting Rules for Democracy without Institutionalized Parties
- Graduate/Faculty Seminar ( 107 Views )This talk will be a fun discussion of the mathematical aspects of preferential ballot elections (in which voters are allowed to express their rankings of all of the candidates). After describing how single vote ballots can lead to an institutionalized two party system by discouraging third party candidates, we will then discuss the various vote counting methods for preferential ballot elections and the characteristics, both good and bad, that these various methods have. We will also touch on Arrow's Paradox, one of the most over-rated "paradoxes" in mathematics, and explain how it is much less relevant to discussions of vote counting methods than is sometimes believed.