Harold Layton : Irregular Flow Oscillations in the Nephrons of Spontaneously Hypertensive Rats
- Graduate/Faculty Seminar ( 141 Views )The nephron is the functional unit of the kidney. The flow rate in each nephron is regulated, in part, by tubuloglomerular feedback, a negative feedback loop. In some parameter regimes, this feedback system can exhibit oscillations that approximate limit-cycle oscillations. However, nephron flow in spontaneously hypertensive rats (SHR) can exhibit highly irregular oscillations similar to deterministic chaos. We used a mathematical model of tubuloglomerular feedback (TGF) to investigate potential sources of the irregular oscillations and the associated complex power spectra in SHR. A bifurcation analysis of the TGF model equation was performed by finding roots of the characteristic equation, and numerical simulations of model solutions were conducted to assist in the interpretation of the analysis. Four potential sources of spectral complexity in SHR were identified: (1) bifurcations that produce qualitative changes in solution type, leading to multiple spectrum peaks and their respective harmonic peaks; (2) continuous lability in delay parameters, leading to broadening of peaks and their harmonics; (3) episodic lability in delay parameters, leading to multiple peaks and their harmonics; and (4) coupling of small numbers of nephrons, leading to broadening of peaks, multiple peaks, and their harmonics. We conclude that the complex power spectra in SHR may be explained by the inherent complexity of TGF dynamics, which may include solution bifurcations, variation in TGF parameters, and coupling between small numbers of neighboring nephrons.
Lenya Ryzhik : How an incompressible flow helps diffusion to mix things
- String Theory ( 219 Views )I will describe some recent results that concern various aspects of the mixing properties of a strong incompressible flow acting together with a diffusion. In particular, we will discuss the short-time decay of solutions of the corresponding initial value problem, asymptotics of the principle Dirichlet eigenvalue and the behavior of the explosion threshold in the Zeldovich problem when the incompressible flow is strong. When the flow is prescribed, the "enhancement" of these characteristics comes from the geometric properties of the flow. We will also show that flows arising from the Stokes-Bousisnesq problems possess these "enhancement" features.
Romyar Sharifi : A modular interpretation of a pairing on cyclotomic units
- Algebraic Geometry ( 162 Views )Class groups of cyclotomic fields have long been of central interest in number theory. We consider elements of these class groups that arise as values of a cup product pairing on cyclotomic units. These pairing values yield information on a wealth of algebraic objects, but any analytic interpretation of them was heretofore unknown. We will describe how, conjecturally, modular representations can be used to relate the pairing values to p-adic L-values of cusp forms.
Beatte Schmittmann : From asymmetric exclusion processes to protein synthesis
- Nonlinear and Complex Systems ( 159 Views )Asymmetric exclusion processes, with periodic or open boundaries, have been studied extensively in the mathematics and statistical physics communities, as paradigmatic models for stochastic particle transport far from equilibrium. Though significant progress was made only recently, the original model was actually introduced decades ago to model protein synthesis1. In this talk, I will describe recent efforts to develop a comprehensive theory for protein synthesis, building on asymmetric exclusion processes with extended objects, modeling ribosomes covering multiple codons. We discuss the effects of local hopping rates and ribosome size on density profiles and particle currents. The latter translate directly into synthesis rates for the corresponding protein. Some intriguing results for real genes will be presented. 1C.T. MacDonald, J.H. Gibbs and A.C. Pipkin, Kinetics of biopolymerization on nucleic acid templates, Biopolymers,6 1 (1968); C.T. MacDonald and J.H. Gibbs, Concerning the kinetics of polypeptide synthesis on polyribosomes, Biopolymers, 7, 707, (1969).
Ken Jackson : Numerical Methods for the Valuation of Synthetic Collateralized Debt Obligations (CDOs)
- Geometry and Topology ( 156 Views )Our numerical computation group has studied several problems in computational finance over the past decade. One that we've looked at recently is the pricing of "collateralized debt obligations" (CDOs). The market for CDOs has grown rapidly to over US$1 trillion annually in 2006, since the appearance of JP Morgan's Bistro deal, the first synthetic CDO, in December 1997. Much of the turmoil in the financial markets recently has been due to such credit derivatives. As this suggests, there are still many open problems associated with the pricing and hedging of these complex financial instruments. I will talk briefly about some work that we have done recently in this area.
Fernando Schwartz : On the topology of black holes
- Graduate/Faculty Seminar ( 146 Views )3+1 dimensional black holes have spherical topology, but in higher dimensions this is no longer true. In this talk I will explain the preceding statement and show a construction, in terms of Riemannian geometry, of outermost apparent horizons with nonspherical topology.
Mary Lou Zeeman : Modeling the Menstrual Cycle:How does estradiol initiate the LH surge?
- Applied Math and Analysis ( 142 Views )In vertebrates, ovulation is triggered by a surge of luteinizing hormone (LH) from the pituitary. The precise mechanism by which rising estradiol (E2) from the ovaries initiates the LH surge in the human menstrual cycle remains a mystery. The mystery is due in part to the bimodal nature of estradiol feedback action on LH secretion, and in part to disagreement over the site of the feedback action.
We will describe a differential equations model in which the mysterious bimodality of estradiol action arises from the electrical connectivity of a network of folliculo-stellate cells in the pituitary. The mathematical model is based as closely as possible on current experimental data, and is being used to design and conduct new experiments. No biological background will be assumed.
Matthew Hedden : On Floer homology and knots admitting lens space surgeries
- Geometry and Topology ( 170 Views )J. Berge discovered a simple condition on a knot, K, in the three-sphere which ensures that Dehn surgery on K yields a lens space. It is an open conjecture, known as the Berge conjecture, that any knot on which one can perform surgery and obtain a lens space satisfies his condition. I will discuss a strategy, developed jointly with Ken Baker and Eli Grigsby by which the knot Floer homology invariants of Ozsvath, Szabo, and Rasmussen could be used to prove this conjecture.
Paul Aspinwall : The Ubiquity of the ADE Classification
- Graduate/Faculty Seminar ( 247 Views )Many classes of mathematical objects turn out to be classified in the same way --- two infinite series and 3 "exceptional" objects. These include symmetries of 3-dimensional solids, rigid singularities, certain types of Lie algebras, positive definite even intersection forms, etc. Discovering why such classes should have the same classification has led to many beautiful ideas and observations. I will give a review of some of the basic ideas (assuming very little in the way of prerequisites) and I may have time to say why string theory has been important in this context.
Paolo Aluffi : Chern class identities from string theory
- Applied Math and Analysis ( 143 Views )(joint with Mboyo Esole) String theory considerations lead to a non-trivial identity relating the Euler characteristics of an elliptically fibered Calabi-Yau fourfold and of certain related surfaces. After giving a very sketchy idea of the physics arguments leading to this identity, I will present a Chern class identity which confirms it, generalizing it to arbitrary dimension and to varieties that are not necessarily Calabi-Yaus. The relevant loci are singular, and this plays a key role in the identity.
Anda Degeratu : Analysis on crepant resolutions of Calabi-Yau orbifolds
- Geometry and Topology ( 144 Views )A Calabi-Yau orbifold is locally modeled on C^n/G with G a finite subgroup of SU(n). If the singularity is isolated, then the crepant resolution (if it exists) is an ALE manifold, for which index-type results are well known. However, most of the time the singularity is not isolated, and for the corresponding crepant resolution there is no index theorem so far. In this talk, I present the first step towards obtaining such a result: I will introduce the class of iterated cone-edge singular manifolds and the corresponding quasi-asymptotically conical spaces (of which orbifolds and their resolutions of singularities are examples), and build-up the general set-up for studying Fredholm properties of geometrical elliptic operators on these spaces. This is joint work with Rafe Mazzeo.
Svetlana Tlupova : Numerical Solutions of Coupled Stokes and Darcy Flows Based on Boundary Integrals
- Applied Math and Analysis ( 151 Views )Coupling between free fluid flow and flow through porous media is important in many industrial applications, such as filtration, underground water flow in hydrology, oil recovery in petroleum engineering, fluid flow through body tissues in biology, to name a few.
Stokes flows appear in many applications where the fluid viscosity is high and/or the velocity and length scales are small. The flow through a porous medium can be described by Darcy's law. A region that contains both requires a careful coupling of these different systems at the interface through appropriate boundary conditions.
Our objective is to develop a method based on the boundary integral formulation for computing the fluid/porous medium problem with higher accuracy using fundamental solutions of Stokes and Darcy's equations. We regularize the kernels to remove the singularity for stability of numerical calculations and eliminate the largest error for higher accuracy.
Santosh Vempala : Logconcave Random Graphs
- Probability ( 145 Views )We propose the following model of a random graph on $n$ vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair ij with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by picking a random point X from F and defining a graph on n vertices whose edges are pairs ij for which X_{ij} \le p. The standard Erd\H{o}s-R\'{e}nyi model is the special case when F is uniform on the 0-1 unit cube. We determine basic properties such as the connectivity threshold for quite general distributions. We also consider cases where the X_{ij} are the edge weights in some random instance of a combinatorial optimization problem. By choosing suitable distributions, we can capture random graphs with interesting properties such as triangle-free random graphs and weighted random graphs with bounded total weight. This is joint work with Alan Frieze (CMU) and Juan Vera (Waterloo). The talk will be self-contained and no prior knowledge of random graphs is assumed.
Richard Schoen : Positive scalar curvature and connections with relativity
- Gergen Lectures ( 257 Views )In this series of three lectures we will describe positivity conditions on Riemannian metrics including the classical conditions of positive sectional, Ricci, and scalar curvature. We will discuss open problems and recent progress including our recent proof of the differentiable sphere theorem (joint with Simon Brendle). That proof employs the Ricci flow, so we will spend some time explaining that technique. Finally we will discuss problems related to positive scalar curvature including some high dimensional issues which occur in that theory. If time allows we will describe recent progress on black hole topologies. These lectures, especially the first two, are intended for a general audience.
Richard Schoen : Ricci flow and 1/4-pinching
- Gergen Lectures ( 239 Views )In this series of three lectures we will describe positivity conditions on Riemannian metrics including the classical conditions of positive sectional, Ricci, and scalar curvature. We will discuss open problems and recent progress including our recent proof of the differentiable sphere theorem (joint with Simon Brendle). That proof employs the Ricci flow, so we will spend some time explaining that technique. Finally we will discuss problems related to positive scalar curvature including some high dimensional issues which occur in that theory. If time allows we will describe recent progress on black hole topologies. These lectures, especially the first two, are intended for a general audience.
Anne Catlla : Mean, Lean ODE-fighting Machine
- Graduate/Faculty Seminar ( 145 Views )Our brains are composed of networks of cells, including neurons and glial cells. While the significance of neurons has been established by biologists, the role of glial cells is less understood. One hypothesis is that glial cells facilitate neural communication in nearby neurons, while suppressing communication among more distant neurons via a reaction-diffusion process. I consider this proposed mechanism using partial and ordinary differential equation models. By analyzing the ordinary differential equation model, I can determine conditions for this hypothesis to hold. I then compare the results of this analysis with simulations of the partial differential equation model and discuss the biological implications.
Janet Best : Parkinsons: two mathematical views of a neurological disease
- Applied Math and Analysis ( 135 Views )Parkinson's Disease (PD) is the most common movement disorder in the United States, with symptoms due to progressive loss of neurons within the basal ganglia. In the first part of the talk, we present and analyze a minimal model for the lack of cross-correlations in neuronal activity in the healthy basal ganglia.
The second part of the talk focuses on experimentally-observed changes in neuronal firing patterns that accompany PD and that may result in the motor symptoms. We have constructed a neuronal network model for the increases in correlated activity within the basal ganglia following the onset of PD. We then apply dynamical systems methods to understand transitions between irregular and rhythmic, correlated firing in the model. Geometric singular perturbation theory and one-dimensional maps are used to understand how an excitatory-inhibitory neuronal network with fixed architecture can generate both activity patterns for possibly different values of the intrinsic and synaptic parameters. We discuss hypotheses arising from the model as well as ongoing experiments to test these predictions.
Aaron Hoffman : Existence and Orbital Stability for Counterpropagating Waves in the FPU model
- Applied Math and Analysis ( 146 Views )The Fermi-Pasta-Ulam (FPU) model of coupled anharmonic oscillators has long been of interest in nonlinear science. It is only recently (Friesecke and Wattis 1994, Frieseck and Pego 1999-2003, and Mizumachi (submitted)) that the existence and stability of solitary waves in FPU has been completely understood. In light of the fact that the Korteweg-deVries (KdV) equation may recovered as a long wave limit of FPU and that the theory of soliton interaction is both beautiful and completely understood in KdV, it is of interest to describe the interaction of two colliding solitary waves in the FPU model. We show that the FPU model contains an open set of solutions which remain close to the linear sum of two long wave low amplitude solitions as time goes to either positive or negative infinity.
Jimmy Dillies : On some K3 automorphisms
- Algebraic Geometry ( 171 Views )In order to construct a viable model of string theory, one seeks to build Calabi Yau threefolds with prescribed conditions. Borcea and Voisin were able to built a family of Calabi-Yau threefolds using elliptic curves and K3 surfaces admitting non symplectic involutions. We will display how the construction can be generalized by studying higher order non symplectic automorphisms on K3 surfaces
Fédéric Rochon : On the uniqueness of certain families of holomorphic disks
- Geometry and Topology ( 133 Views )A Zoll metric is a metric whose geodesics are all circles of equal length. In this talk, we will first review the definition of the twistor correspondence of LeBrun and Mason for Zoll metrics on the sphere $S^{2}$. It associates to a Zoll metric on $S^{2}$ a family of holomorphic disks in $CP_{2}$ with boundary in a totally real submanifold $P\subset CP_{2}$. For a fixed $P\subset CP_{2}$, we will indicate how one can show that such a family is unique whenever it exists, implying that the twistor correspondence of LeBrun and Mason is in some sense injective. One of the key ingredients in the proof will be the blow-up and blow-down constructions in the sense of Melrose.
Anja Sturm : Coexistence and convergence for voter model with selection
- String Theory ( 220 Views )We consider variations of the usual voter model, which favor types that are locally less common. Such voter models with selection are dual to systems of branching annihilating random walks that are parity preserving. We consider coexistence of types in the voter models which is related to the survival of particles in the branching annihilating random walk. We find conditions for the uniqueness of a homogeneous coexisting invariant law as well as for convergence to this law from homogeneous and coexisting initial laws. For a particular one dimensional model we also show a complete convergence result for any initial condition. This is based on comparison with oriented percolation of the associated branching annihilating random walk.
Vestislav Apostolov : Old and new trends in Bihermitian geometry
- Geometry and Topology ( 158 Views )A bihermitian structure is a Riemannian metric compatible with two distinct orthogonal complex structures. In the mathematical literature this notion appeared in 90's in the study of the curvature of conformal 4-manifolds. However, bihermitian metrics were already studied in the physics literature in the 80's, as a building bloc of what Gates, Hull and Rocek call `the target space for a (2,2) super-symmetric sigma model'. There has been a great deal of interest in bihermitian geometry more recently, motivated by its link with the notion of generalized Kaehler geometry, introduced by Gualtieri and Hitchin. In this talk I will explain some main features of 4-dimensional bihermitian manifolds, as developed in the 90's, and report on recent classification results that I obtained with M. Gualtieri and G. Dloussky.
Katia Koelle : Exploration, innovation, and selective sweeps in the ecology
- Nonlinear and Complex Systems ( 138 Views )For many biological systems, the timescale at which ecological interactions occur is much shorter than the timescale at which evolutionary changes occur. For rapidly evolving pathogens such as influenza, however, this is not the case; influenza researchers therefore need to understand both the ecological interactions between the host and the pathogen and the virus?s evolutionary changes in order to ultimately control the disease in humans. Recently, a study looking at the evolutionary patterns of influenza showed that, while the virus?s genetic evolution occurred gradually, its antigenic evolution occurred in a punctuated manner. (Genetic evolution refers to how the virus?s nucleotides change over time; antigenic evolution refers to how the virus changes over time with respect to how our immune system recognizes it.) Previous research from our group hypothesized that these differences in evolutionary patterns could be explained by the presence of /neutral networks/ in the virus?s genotype space: networks of sequences that differ genetically from one another but fold into the same protein conformation and thereby share antigenic properties. Here, I will present a simple epidemiological model that implicitly incorporates these neutral networks. I show that this model can reproduce (1) the seasonal and interannual outbreak patterns of influenza, (2) the quantitative patterns of influenza?s antigenic evolution, and (3) the patterns of the virus?s genetic evolution, including its characteristic phylogenetic tree. I end with how this model may be useful in understanding patterns of viral diversity in other host species (e.g., avian and equine hosts).
Matthew Surles : Approximating Layer Potentials on and near curve segments, the long and the short of it.
- Graduate/Faculty Seminar ( 139 Views )In many problems in fluids and electromagnetics, we may formulate solutions to the Dirichlet and Neumann problems in terms of double and single layer potentials. Such boundary integral representations can result in computational difficulties at points on and near the boundary due to singularities and near singularities. The case of a smooth closed boundary has been well-studied, but I will focus on computational issues that arise from a boundary that is only piecewise smooth, consisting of connected curve segments. I will give an overview of my research in approximating singular and nearly singular integrals, as well as discuss an approach for the computation of double layer potentials at points on and near a curve segment.