Departmental Colloquium
Spring 2014
Regular Day: Thursday
Regular Time: 4:00PM - 5:00PM
Regular Location: JWB 335
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January 30 |
William R Holmes
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In this talk I will give an overview of two research lines in mathematical cell biology. First I will discuss the development of new tools for understanding the spatio-temporal dynamics of complex biochemical networks. Currently there is an extensive toolbox for mapping and understanding the non-linear dynamics of 1) large, complex dynamical systems and 2) smaller, simplified spatial systems. However numerous biological questions require probing the spatial dynamics of large complex biochemical networks, for which these tools are less well suited. I will describe a new non-linear stability technique that helps fill this void and demonstrate its utility in the context of understanding the symmetry breaking events that lead to cell polarity. Second, I will discuss a joint modeling / experimental investigation of early embryonic development. A vital first step in this process (in mammalian embryos) is the formation of an early placenta prior to implantation. This requires proper spatial localization of different genetic factors in the presence of numerous challenges (e.g. cell movements, division, noise). I will discuss an experimentally driven model of this developmental event and show that systemic noise, rather than being a hindrance, is both necessary and sufficient for this process to occur robustly.
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February 4 |
Ting Zhou
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TAT is an example of a coupled-physics modality, which combines the high contrast of a physical phenomenon (here the electrical properties of tissues) with the high resolution of another phenomenon (here ultrasound). Thermo-acoustic imaging may be decomposed into two steps. The first step aims at reconstructing an amount of electromagnetic radiation absorbed by tissues from boundary measurements of ultrasound signals generated by these radiations. We assume this first step done. Quantitative thermo-acoustics then consists of reconstructing the conductivity coefficient in the equation from the now known absorbed radiation. This second step is the problem of interest in this work. Mathematically, quantitative thermo-acoustics consists of reconstructing the conductivity in time-harmonic Maxwell’s equations from available internal data that are linear in the conductivity and quadratic in the electric field. We consider inverse problems of this type with applications in thermo-acoustics. In this framework, we obtain uniqueness and stability of the reconstruction for a scalar model of time-harmonic wave propagation, by choosing appropriate illuminations known as complex geometric optics (CGO) solutions for the equation. At last but not least, we consider the full system models of Maxwell’s equations and see a different flavor of analysis of uniqueness and stability using CGO solutions. These are joint works with Guillaume Bal, Kui Ren and Gunther Uhlmann.
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February 6 |
Braxton Osting
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Several geometric methods for graph partitioning have been introduced in the past few years, with wide applications in clustering, community detection, and image analysis. These methods, which I'll review, are built on graph-based analogues of total variation, motion by mean curvature, the Ginzburg-Landau functional, and the Merriman-Bence-Osher threshold dynamics. In this talk, I'll discuss a new graph partitioning method where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. The resulting eigenvalue optimization problem can be solved by a rearrangement algorithm, which we show to converge in a finite number of iterations to a local minimum of a relaxed objective function. The method compares well to state-of-the-art approaches when applied to clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations. The model has a semi-supervised extension and provides natural representatives for the clusters as well.
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February 13 |
Karl Schwede
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We will consider solution sets to polynomial equations. For example, consider y^2 = x^3, the solution set has a singularity at the origin (a cusp). I will talk about different ways to measure singularities such as this one. First I will discuss the multiplier ideal, a way to measure singularities via integration. Second I will discuss the test ideal, a way to measure singularities if one considers them over a finite field. It has been known for nearly two decades that these two independently introduced notions are closely related. I will discuss recent joint work with Manuel Blickle and Kevin Tucker which shows that these two objects are in fact aspects of the same phenomenon.
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March 20 |
Timo Seppäläinen
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This talk begins with a reminder of classic random walk and then proceeds to models of random paths currently studied in probability and statistical mechanics. In particular, we discuss directed percolation and directed polymer models. Subadditive ergodic theory gives deterministic large scale limits for these models, but properties of these limits have remained a challenge for decades. We describe some new variational formulas that characterize these limits and connections with other features of the models such as fluctuation exponents. This talk is based in part on joint work with Nicos Georgiou, Firas Rassoul-Agha and Atilla Yilmaz.
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March 27 |
John Brady
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What do corn starch, swimming spermatozoa, DNA and self-assembling nanoparticles have in common? They are all (or can be modeled as) ‘particles’ dispersed in a continuum suspending fluid where hydrodynamic interactions compete with thermal (Brownian) and interparticle forces to set structure and determine properties. These systems are ‘soft’ as compared to molecular systems largely because their number density is much less and their time scales much longer than atomic or molecular systems. In this talk I will describe the common framework for modeling these diverse systems and the essential features that any hydrodynamic modeling must incorporate in order to capture the correct behavior. Actually computing the hydrodynamics in an accurate and efficient manner is the real challenge, and I will illustrate past successes and current efforts with examples drawn from the diffusion and rheology of colloids to the ‘swimming’ of catalytic nanomotors.
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April 3 |
Mitchell Luskin
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Predictive computational models for materials must include the interactions of multiple spatial and temporal scales. These scales can be classified as microscopic, mesoscopic, and macroscopic for simplicity, but there are typically structures at many scales with varying degrees of separation. The promise of technologies based on engineered high performance materials make materials modeling an increasing focus of research and development. Temporal and spatial multiscale challenges appear because the material response of crystalline solids is characterized by the nucleation, dynamics, and pattern formation of defects such as point defects (vacancies, interstitials, impurities), line defects (dislocations), and surface defects (grain boundaries, surfaces, cracks, etc.). To overcome these challenges and reach mesoscopic scales, new mathematical foundations and computational algorithms are needed for spatial coarse-graining (atomistic-to-continuum coupling methods) and temporal coarse-graining (accelerated dynamics methods), as well as methods combining spatial and temporal coarse-graining.
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April 10 |
David Damanik
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A subset of the real line is called a Cantor set if it is compact, perfect, and nowhere dense. Cantor sets arise in many areas; in this talk we will discuss their relevance in the spectral theory of Schröodinger operators. We discuss several results showing that the spectrum of such an operator is a Cantor set, from the discovery of the first example by Moser to a genericity result by Avila, Bochi, and Damanik. A Cantor measure is a probability measure on the real line whose topological support is a Cantor set. A primary example in the spectral theory context is the density of states measure in situations where the spectrum is a Cantor set. A conjecture of Simon claims a strict inequality between the dimensions of the set and the measure for the Fibonacci potential. If time permits, we will discuss a recent result of Damanik, Gorodetski, and Yessen, which establishes this conjecture in full generality.
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April 15 |
Martin Bridson
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There are many situations in geometry and group theory where it is natural or convenient to study infinite groups via their finite quotients and finite-index groups. If a group G is residually finite (ie every element survives in some finite quotient) then one might hope to recover a lot of information about the group from the totality of its finite quotients -- equivalently, its pro-finite completion. But precisely which properties of G can be detected in this way, and which cannot? To what extent is a residually-finite group determined by its pro-finite completion or representation theory? In this lecture I'll survey the recent activity around questions such as these and explain how it connects to other areas of mathematics, particularly geometry. Fall 2013
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