Dave Rose : The EilenbergMazur swindle
- Graduate/Faculty Seminar ( 133 Views )At some point in every mathematician's life they have seen the paradoxical 'proof' that 1=0 obtained by different groupings of the infinite sum 1-1+1-1+... As we learn, the issue is that this series does not converge. The Eilenberg-Mazur swindle is a twist on this argument which shows that A+B+A+B+... = 0 implies that A=0=B in certain situations where we can make sense of the infinite sum. In this talk, we will explore these swindles, touching on many interesting areas of mathematics along the way.
Ming Fang : Miscellany on branching random walks
- Probability ( 99 Views )Branching random walk can be viewed as particles performing random walks while branching at integer time. We review some of the existing results on the maximal (or minimal) displacement, when each particle moves and branches independently according the same step distribution and the same branching law. Then we will compare them with similar but different models. Roughly speaking, in one variation, we will consider the asymptotic behavior of the particle at time n, whose ancestors location are consistently small. In another variation, we will consider the maximal displacement for the model, where the step distributions vary with respect to time.
Lek-Heng Lim : Multilinear Algebra and Its Applications
- Applied Math and Analysis ( 109 Views )In mathematics, the study of multilinear algebra is largely limited to properties of a whole space of tensors --- tensor products of k vector spaces, modules, vector bundles, Hilbert spaces, operator algebras, etc. There is also a tendency to take an abstract coordinate-free approach. In most applications, instead of a whole space of tensors, we are often given just a single tensor from that space; and it usually takes the form of a hypermatrix, i.e.\ a k-dimensional array of numerical values that represents the tensor with respect to some coordinates/bases determined by the units and nature of measurements. How could one analyze this one single tensor then? If the order of the tensor k = 2, then the hypermatrix is just a matrix and we have access to a rich collection of tools: rank, determinant, norms, singular values, eigenvalues, condition number, etc. This talk is about the case when k > 2. We will see that one may often define higher-order analogues of common matrix notions rather naturally: tensor ranks, hyperdeterminants, tensor norms (Hilbert-Schmidt, spectral, Schatten, Ky Fan, etc), tensor eigenvalues and singular values, etc. We will discuss the utility as well as difficulties of various tensorial analogues of matrix problems. In particular we shall look at how tensors arise in a variety of applications including: computational complexity, control engineering, mathematical biology, neuroimaging, quantum computing, signal processing, spectroscopy, and statistics.
Anne Shiu : Multiple steady states in chemical reaction systems
- Graduate/Faculty Seminar ( 110 Views )In a chemical reaction system, the concentrations of chemical species evolve in time, governed by the polynomial differential equations of mass-action kinetics. This talk provides an introduction to chemical reaction network theory, and gives an overview of algebraic and combinatorial approaches to determining whether a chemical reaction network admits multiple steady states. In general, establishing the existence of (multiple) steady states is challenging, as it requires the solution of a large system of polynomials with unknown coefficients. However, for networks that have special structure, various easy criteria can be applied. This talk will highlight results from Deficiency Theory (due to Feinberg), criteria for multistationarity for chemical reaction systems whose steady states are defined by binomial equations (in joint work with Carsten Conradi, Mercedes Pérez Millán, and Alicia Dickenstein), and a classification of small multistationary chemical reaction networks (in joint work with Badal Joshi).
Gitta Kutyniok : Frames and Sparsity
- Applied Math and Analysis ( 121 Views )Frames are nowadays a standard methodology in applied mathematics, computer science, and engineering when redundant, yet stable expansions are required. Sparsity is a new paradigm in signal processing, which allows for significantly reduced measurements yet still highly accurate reconstruction. In this talk, we will focus on the main two links between these exciting, rapidly growing areas. Firstly, the redundancy of a frame promotes sparse expansions of signals, thereby strongly supporting sparse recovery methods such as Compressed Sensing. After providing an overview of sparsity methodologies, we will discuss new results on sparse recovery for structured signals, in particular, which are a composition of `distinct' components. Secondly, in very high dimensions, frame decompositions might be intractable in applications with limited computing budget. This problem can be addressed by requiring sparsity of the frame itself, and we will show how to derive optimally sparse frames. Finally, we will discuss how some of the presented results generalize to the novel notion of fusion frames, which was introduced a few years ago for modeling distributed processing applications.
Andrei Zelevinsky : Cluster algebras via quivers with potentials
- Gergen Lectures ( 268 Views )This lecture ties together the strands developed in the first two lectures. We discuss a recent proof (due to H. Derksen, J. Weyman, and the speaker) of a series of conjectures on cluster algebras by means of the machinery of quivers with potentials. An important ingredient of our argument is a categorification of cluster algebras using quiver Grassmannians, a family of projective algebraic varieties that are a far-reaching generalization of ordinary Grassmannians. Generalizing an idea due to P. Caldero, F. Chapoton and B. Keller, we show that the Euler characteristics of these varieties carry crucial information about the structure of cluster algebras.
Andrei Zelevinsky : Quivers with potentials, their representation and mutations
- Gergen Lectures ( 258 Views )A quiver is a finite directed graph. A quiver representation assigns a finite-dimensional vector space to each vertex, and a linear map between the corresponding spaces to each arrow. A fundamental role in the theory of quiver representations is played by Bernstein-Gelfand-Ponomarev reflection functors associated to every source or sink of a quiver. In joint work with H. Derksen and J. Weyman (based on an earlier joint work with R. Marsh and M. Reineke) we extend these functors to arbitrary vertices. This construction is based on a framework of quivers with potentials; their representations are quiver representations satisfying relations of a special kind between the linear maps attached to arrows. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, and cluster algebras. However, no special knowledge will be assumed in any of these subjects, and the exposition aims to be accessible to graduate students.
Elizabeth Bouzarth : Using Regularized Stokeslets to Model Immersed Biological Fibers
- Mathematical Biology ( 117 Views )The behavior of inextensible fibers immersed in a fluid is of interest in a variety of applications ranging from polymer suspensions to actin filament transport. In these cases, the dynamics of an immersed fiber can play a large role in the observed macroscale fluid dynamics. The method of regularized Stokeslets provides a way to calculate fluid velocities in the Stokes fluid flow regime due to a collection of regularized point-forces without computing fluid velocities on an underlying grid. In this discussion, the method of Regularized Stokeslets will be used to model the dynamics of an inextensible flexible fiber immersed in a two-dimensional cellular background flow in comparison with results found in the experimental and mathematical literature. Studying this scenario with regularized Stokeslets provides insight into the documented stretch-coil transition and macroscale random walk behavior supported by mathematical models and experimental results.
Markos Katsoulakis : Accelerated Kinetic Monte Carlo methods: hierarchical parallel > algorithms and coarse-graining
- Probability ( 101 Views )In this talk we present two intimately related approaches in speeding-up molecular simulations via Monte Carlo simulations. First, we discuss coarse-graining algorithms for systems with complex, and often competing particle interactions, both in the equilibrium and non-equilibrium settings, which rely on multilevel sampling and communication. Second, we address mathematical, numerical and algorithmic issues arising in the parallelization of spatially distributed Kinetic Monte Carlo simulations, by developing a new hierarchical operator splitting of the underlying high-dimensional generator, as means of decomposing efficiently and systematically the computational load and communication between multiple processors. The common theme in both methods is the desire to identify and decompose the particle system in components that communicate minimally and thus local information can be either described by suitable coarse-variables (coarse-graining), or computed locally on a individual processors within a parallel architecture.
Sam Payne : Boundary complexes and weight filtrations
- Algebraic Geometry ( 116 Views )The boundary complex of an algebraic variety is the dual complex of the boundary divisor in a compactification of a log resolution. I will present recent work showing that the homotopy type of this complex is independent of the choice of resolution and compactification, and give relations between these complexes and Deligne's weight filtration on singular cohomology.
Mark Levi : Arnold diffusion in a chain of coupled pendula
- Applied Math and Analysis ( 127 Views )A chain of pendula connected by nearest neighbor coupling is a near-- integrable system if the coupling is weak. As a consequence of KAM, for most initial data the energy of each pendulum stays near its initial value for all time. We show that these KAM motions coexist with ``diffusing" motions for which the energy can leak from any pendulum to any other pendulum, and it can do so with a prescribed itinerary. This is joint work with Vadim Kaloshin.
Chris O'Neill : An Introduction to Ehrhart Theory and Lattice Point Enumeration
- Graduate/Faculty Seminar ( 110 Views )A polytope is a subset of R^d which is the convex hull of a finite set of vertices. Given a polytope P, we can consider integer dilations of P, and ask how many integer points are contained in each dilation, as a function of the dilation factor. A theorem by Eugene Ehrhart tells us that, under the right conditions, this counting function is a polynomial, with some very interesting and unexpected properties. To demonstrate the usefulness of these results, we will give alternative proofs to some well known results from far outside the realm of geometry, including some basic facts about the chromatic polynomial of a graph. This talk will contain a little geometry, a little analysis, a little algebra, and a little combinatorics, and will be accessible to anyone who enjoys at least one of these topics.
Lisa Fauci : Waving rings and swimming in circles: some lessons learned through biofluiddynamics
- Mathematical Biology ( 88 Views )Dinoflagellates swim due to the action of two eucaryotic flagella - a trailing, longitundinal flagellum that propagates planar waves, and a transverse flagellum that propagates helical waves. The transverse flagellum wraps around the cell in a plane perpendicular to the trailing flagellum, and is thought to provide both forward thrust along with rotational torque. Motivated by the intriguing function of this transverse flagellum, we study the fundamental fluid dynamics of a helically-undulating ring in a viscous fluid. We contrast this biofluiddynamic study, where the kinematics of the waveform are taken as given, with a model of mammalian sperm hyperactivated motility. Here, our goal is to examine how the complex interplay of fluid dynamics, biochemistry, and elastic properties of the flagellum give rise to the swimming patterns observed.
Brian Rider : Log-gases and Tracy-Widom laws
- Probability ( 190 Views )The now ubiquitous Tracy-Widom laws were first discovered in the context of the Gaussian Orthogonal, Unitary, and Symplectic Ensembles (G{O/U/S}E) of random matrix theory. The latter may be viewed as logarithmic gases with quadratic (Gaussian) potential at three special inverses temperatures (beta=1,2,4). A few years back, Jose Ramirez, Balint Virag, and I showed that that one obtains generalizations of the Tracy-Widom laws at all inverse temperatures (beta>0), though still for quadratic potentials. I'll explain how similar ideas (and considerably more labor) extends the result to general potential, general temperature log-gases. This is joint work with Manjunath Krishnapur and Balint Virag.
Gadi Fibich : Aggregate Diffusion Dynamics in Agent-Based Models with a Spatial Structure
- Applied Math and Analysis ( 115 Views )The diffusion or adoption of new products (such as fax machines, skype, facebook, Ipad, etc.) is one of the key problems in Marketing research. In recent years, this problem was often studied numerically, using agent-based models (ABMs). In this talk I will focus on analysis of the aggregate diffusion dynamics in ABMs with a spatial structure. In one-dimensional ABMs, the aggregate diffusion dynamics can be explicitly calculated, without using the mean-field approximation. In multidimensional ABMs, we introduce a clusters-dynamics approach, and use it to derive an analytic approximation of the aggregate diffusion dynamics. The clusters-dynamics approximation shows that the aggregate diffusion dynamics does not depend on the average distance between individuals, but rather on the expansion rate of clusters of adopters. Therefore, the grid dimension has a large effect on the aggregate adoption dynamics, but a small-world structure and heterogeneity among individuals have only a minor effect. Our results suggest that the one-dimensional model and the fully-connected Bass model provide a lower bound and an upper bound, respectively, for the aggregate diffusion dynamics in agent-based models with "any" spatial structure. This is joint work with Ro'i Gibori and Eitan Muller
Gabor Szekelyhidi : Extremal Kahler metrics and the Yau-Tian-Donaldson conjecture
- Graduate/Faculty Seminar ( 108 Views )I will first explain with a few simple examples a beautiful picture in geometric invariant theory which relates quotient constructions in symplectic and algebraic geometry. Then we will naively apply this picture in a suitable infinite dimensional setting, leading us to the notion of extremal Kahler metrics and the Yau-Tian-Donaldson conjecture, which is an important problem in Kahler geometry today.
Suzanne Lenhart : Optimal Harvesting in Fishery Models
- Mathematical Biology ( 99 Views )We discuss two types of partial differential equation models of fishery harvesting problems. We consider steady state spatial models and diffusive spatial-temporal models. We characterize the distribution of harvest effort which maximizes the harvest yield, and in the steady state case, also minimizes the cost of the effort. We show numerical results to illustrate various cases. The results inform ongoing debate about the use of reserves (regions where fishing is not allowed), and are increasingly relevant as technology enables enforcement of spatially structured harvest constraints.
Scott Schmidler : Mixing times for non-stationary processes
- Probability ( 190 Views )Markov chain methods for Monte Carlo simulation of complex physical or statistical models often require significant tuning. Recent theoretical progress has renewed interest in "adaptive" Markov chain algorithms which learn from their sample history. However, these algorithms produce non-Markovian, time-inhomogeneous, irreversible stochastic processes, making rigorous analysis challenging. We show that lower bounds on the mixing times of these processes can be obtained using familiar ideas of hitting times and conductance from the theory of reversible Markov chains. The bounds obtained are sufficient to demonstrate slow mixing of several recently proposed algorithms including adaptive Metropolis kernels and the equi-energy sampler on some multimodal target distributions. These results provide the first non-trivial bounds on the mixing times of adaptive MCMC samplers, and suggest a way of classifying adaptive schemes that leads to new hybrid algorithms. Many open problems remain.
Dongho Chae : On the presure conditions for the regularity and the triviality in the 3D Euler equations
- Applied Math and Analysis ( 116 Views )In this talk we present some observations regarding the pressure conditions leading to the vanishing of velocity in the Euler and the Navier-Stokes equations. In the case of axisymmetric 3D Euler equations with special initial data we find that the unformicity condition for the derivatives of the pressure is not consistent with the global regularity.
Mainak Patel : Temporal Binding Emerges as a Rapid and Accurate Encoding Tool Within a Network Model of the Locust Antennal Lobe
- Mathematical Biology ( 125 Views )The locust olfactory system interfaces with the external world through antennal receptor neurons (ORNs), which represent odors in a distributed, combinatorial manner. ORN axons bundle together to form the antennal nerve, which relays sensory information centrally to the antennal lobe (AL). Within the AL, an odor produces a stimulus-specific temporal progression of neuronal spiking, inspiring the hypothesis that the AL encodes odors through dynamically evolving ensembles of active cells. Such a coding strategy, however, requires higher olfactory centers to integrate a prolonged dynamic profile of AL signals prior to stimulus assessment, a process that is likely to be slow and inconsistent with the generation of quick behavioral responses. Our modeling work has led us to propose an alternate hypothesis: the dynamical interplay of fast and slow inhibition within the locust AL induces transient correlations in the spiking activity of an odor-dependent neural subset, giving rise to a temporal binding code and allowing rapid stimulus detection by downstream elements.
Franziska Hinkelmann : Analysis of discrete models of biological systems using computer algebra
- Algebraic Geometry ( 115 Views )Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, bounded Petri nets, and agent-based models. Simulation is a common practice for analyzing discrete models, but many systems are far too large to capture all the relevant dynamical features through simulation alone. We convert discrete models into algebraic models and apply tools from computational algebra to analyze their dynamics. The key feature of biological systems that is exploited by our algorithms is their sparsity: while the number of nodes in a biological network may be quite large, each node is affected only by a small number of other nodes. In our experience with models arising in systems biology and random models, this structure leads to fast computations when using algebraic models, and thus efficient analysis. All algorithms and methods are available in our package Analysis of Dynamic Algebraic Models (ADAM), a user friendly web-interface that allows for fast analysis of large models, without requiring understanding of the underlying mathematics or any software installation. ADAM is available as a web tool, so it runs platform independent on all systems.
Franziska Michor : Evolutionary dynamics of cancer
- Mathematical Biology ( 122 Views )Cancer emerges due to an evolutionary process in somatic tissue. The fundamental laws of evolution can best be formulated as exact mathematical equations. Therefore, the process of cancer initiation and progression is amenable to mathematical investigation. Of special importance are changes that occur early during malignant transformation because they may result in oncogene addiction and represent promising targets for therapeutic intervention. Here we describe a mathematical approach, called Retracing the Evolutionary Steps in Cancer (RESIC), to deduce the temporal sequence of genetic events during tumorigenesis from crosssectional genomic data of tumors at their fully transformed stage. When applied to a dataset of 70 advanced colorectal cancers, our algorithm accurately predicts the sequence of APC, KRAS, and TP53 mutations previously defined by analyzing tumors at different stages of colon cancer formation. We further validate the method with glioblastoma and leukemia sample data and then apply it to complex integrated genomics databases, finding that high-level EGFR amplification appears to be a late event in primary glioblastomas. RESIC represents the first evolutionary mathematical approach to identify the temporal sequence of mutations driving tumorigenesis and may be useful to guide the validation of candidate genes emerging from cancer genome surveys.
Jasmine Foo : Modeling diversity in tumor populations and implications for drug resistance
- Probability ( 102 Views )In this talk I will discuss a branching process model developed to study intra-tumor diversity (i.e. the variation amongst the cells within a single tumor). This variation arises due to heritable (epi)genetic alterations which can confer changes in cellular fitness and drug response. In the asymptotic (t-> infinity) regime, we study the growth rates of the population as well as some ecological measures of diversity in the tumor. In the second half of the talk I will discuss applications of this model to studying the emergence of drug resistant populations in Chronic Myeloid Leukemia (CML). (Joint work w/K. Leder, J. Mayberry, R. Durrett, F. Michor)
Joshua Goldwyn : Analysis of stochastic auditory nerve models with applications to cochlear implant psychophysics
- Mathematical Biology ( 111 Views )Cochlear implants are neural prostheses that restore a sense of hearing to individuals with severe to profound deafness. Two fundamental theoretical questions that we face are: How does the auditory nerve respond to electrical stimulation? And how is sound information represented in the spike trains of auditory nerve fibers? We will discuss model-based efforts to investigate these questions. I will focus on the development of reduced models that incorporate essential biological features of this complicated system, and remain useful tools for analyzing neural coding.
Using a point process model of the auditory nerve, I simulate amplitude modulation detection, a common test of temporal resolution. I find that the temporal information in the simulated spike trains does not limit modulation sensitivity in cochlear implant users, and discuss how the point process framework can be extended to include additional biophysical mechanisms. Next, I illustrate how spatial spread of excitation and neural degeneration can lead to of within- and across-patient variability in listening outcomes. This points toward an important goal of computational modeling: to develop patient-specific models that can be used to optimize stimulation strategies for individual cochlear implant users.
Matt Bowen : A numerical method for cardiac cell models
- Graduate/Faculty Seminar ( 131 Views )The prevailing numerical methods for solving the reaction-diffusion systems in models of cardiac electrical activity currently use second-order adaptive mesh refinement, refining the spatial and temporal meshes only near the traveling wavefront(s). However, in two and three spatial dimensions under biologically relevant initial conditions and forcing, these wavefronts can constitute a relatively high percentage of the computational domain, limiting the effectiveness of the scheme. In this talk, I will present a numerical scheme based on higher order finite elements and spectral deferred correction designed to improve the efficiency in computing for domains of cardiac cells.
Aaron Fogelson : Two Examples of Chemical Modulation of the Properties and Dynamics of Physiological Gels: Fibrin Formation and Mucin Swelling
- Mathematical Biology ( 120 Views )Gels formed from mixtures of polymers and solvent are ubiquitous in physiological systems. The structure and properties of a gel can change dramatically in response to chemical modulation. Two examples of the influence of chemistry on gel properties will be discussed. The structure of fibrin gels that form during blood clotting is strongly influenced by the concentration of the enzyme thrombin that produces the fibrin monomers from which the gel is built. Presumably at higher thrombin concentrations, fibrin monomers are produced more rapidly from the precursor fibrinogen molecules. I will present an analysis of a possible mechanism of fibrin branching that can explain the sensitivity of fibrin structure to the rate of supply of monomers. Mucin gel is released from vesicles in goblet cells. During this exocytotic process, the polyelectrolyte mucin gel swells to many times its original volume at a very rapid rate. I will present a model in which this swelling is triggered by an exchange of divalent calcium ions in the vesicle and monovalent sodium ions in the extracellular space, and in which the ion concentrations and the rheological properties of the mucin gel determine its equilibrium size and the dynamics of its swelling.