Elliptic Growth and Variation of the Green Function

Abstract: An elliptic growth process is one wherein a domain in Euclidean space grows quasi-statically, pushed outward by its own Green function. The Green function can be that of the Laplacian (in which case the process is better known as Laplacian growth) or that of a more general elliptic operator. A natural inverse question arises: given a movie of a domain growing, can we identify an elliptic operator so that the process isgoverned by elliptic growth? Taking inspiration from Hadamard's variational formula for the Green function (which we'll briefly discuss) we can derive a variational formula for the Green function with respect to the underlying operator and begin to study the local inverse problem.

Tight Closure and its Applications: Introduction and First Concepts #5

Z-stability and Strict Comparison (continued)

Nourdin-Peccati Analysis on Wiener and Wiener-Poisson Space for General Distributions

Abstract: Given a reference random variable, we study the solution of its Stein's equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet-Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis for convergence to Normal random variables. We do these in both Wiener space and the more general Wiener-Poisson space. In the former, we study conditions for convergence under several particular cases and give a characterization onwhen two random variables have the same distribution. As an example, we applied this tool to bilinear functionals of Gaussian subordinated fields where the underlying process is a fractional Brownian motion with Hurst parameter bigger than 1/2. In the latter we give sufficient conditions for a sequence of multiple (Wiener-Poisson) integrals to converge to a Normal random variable.

Part 2: Cloning Genes Using Next Generation Sequencing

With the advent of next generation sequencing, data costs are a fraction of what they were ten years ago. Whole genomes are sequenced in days; all of the genes expressed in a species can be analyzed to answer biological questions. We can now ask questions in a different way. We should, therefore, perform experiments in a different way. EMS (Ethyl Methane Sulfonate) treatment causes G to A and C to T mutations in DNA. Populations of plants treated with EMS can be screened for individuals disrupted in valuable or otherwise interesting biological processes. The genomes of these individuals can then be sequenced to identify the SNPs (Single Nucleotide Polymorphisms) that arise from EMS mutagenesis and are responsible for the change in phenotype. Noise in the sequence data, the quality of the reference genome, and errors in the alignment of sequencing data to the reference genome can produce artifactual results and obscure the true causative mutations. By changing the way we use sequence data to maximize the signal to noise ratio, we have found that we can easily identify causative mutations. We screened SNPs for only those due to the EMS treatment and incorporated knowledge about the possible biological processes affected by the mutations. These "filtering" steps eliminated many of the errors in the data and increased the likelihood of identifying gene(s) controlling the phenotype. To demonstrate that this procedure will work in the complex genomes of crops, and not just in the favorite model organisms of geneticists, we have carried out gene cloning by next generation sequencing in Sorghum bicolor. The first example was a sorghum EMS treated line that showed altered accumulation of the cyanogenic glucoside dhurrin. Genes suspected to act in the dhurrin biosynthetic pathway were scanned for mutations and a single disruption was identified, and later confirmed to be responsible for this phenotype. The second project utilized a sorghum TILLING (Targeting Induced Local Lesion IN Genomes) line. Dwarf and non-dwarf pools of plants were sequenced using Illumina technology and a pipeline is currently under construction to align reads to the reference genome for SNP variant calling and mutant identification. Progress on the identification the dwarf gene, improvements to our pipeline, and prospects gene cloning in species with sequenced and un-sequenced genomes will be discussed. This talk is Part 2 of a two part series. Brian Dilkes delivered Part 1 ("Part 1: Bioinformatics for Next Generation Genetics") on February 7, 2012.

Associated Reading

SHOREmap: Simultaneous Mapping and Mutation Identification by Deep Sequencing Nature Methods 6, 550 - 551 (2009).

Click here for a full schedule of BIOINFORMATICS SEMINARS, past and present.

Proof of Theorem A

Profinite Completions of Mapping Class Groups

Abstract: The mapping class group of a genus g curve with n marked points has a few different profinite completions. I will discuss the so-called congruence subgroup problem in this setting which loosely equates two of these completions. The so-called congruence subgroups arise from mapping class actions on spaces of representations. These actions are of broad interest to many areas of mathematics, though I doubt I will have time to talk much on this.

Curvature Dimension Inequality and Volume Growth on Riemannian Manifolds

Abstract: The purpose of this talk is to prove Li-Yau type inequalities, a scale invariant Harnack inequality, upper and lower Gaussian bounds and a global volume doubling estimate when the manifold satisfies the Curvature Dimension Inequality. This inequality can be viewed as the analytic counterpart of the Ricci tensor being bounded from below. This talk is based on some recent joint works of N. Garofalo and F. Baudoin and will have an expository character.

Stochastic volatility with long-memory in discrete and continuous time — Part 2

It is commonly accepted that certain financial data exhibit long-range dependence. A continuous time stochastic volatility model is considered in which the stock price is geometric Brownian motion with volatility described by a fractional Ornstein-Uhlenbeck process. Two discrete time models are also studied: a discretization of the continuous model via an Euler scheme and a discrete model in which the returns are a zero mean iid sequence where the volatility is a fractional ARIMA process. A particle filtering algorithm is implemented to estimate the empirical distribution of the unobserved volatility, which we then use in the construction of a multinomial recombining tree for option pricing. We also discuss appropriate parameter estimation techniques for each model. For the long memory parameter, we compute an implied value by calibrating the model with real data. We compare the performance of the three models using simulated data and we price options on the S&P 500 index.

The Subelliptic Heat Kernel on the CR Sphere

Traces in Monoidal Categories

New Proofs for the Abhyankar-Gurjar Inversion Formula and the Equivalence of the Jacobian Conjecture and the Vanishing Conjecture

Abstract: Let Δ be the Laplace operator in n-variables, i.e., Δ=∑_i=1ⁿ ∂²/∂ z_i². The Vanishing Conjecture claims that for any homogeneous polynomial f (of degree 4) with complex coefficients such that Δ^m (f^m)=0 for all m ≥ 1, we have Δ^m (f^m+1)=0 when m is large enough. In this talk we first give a new proof for the Abhyankar-Gurjar inversion formula, and then use it to give a new proof for the equivalence of the Jacobian Conjecture and the Vanishing Conjecture.

Main reference: arXiv:0907.3991v2[math.AG]

Sampling Frequency and Model Selection for Diffusion Processes with an Application to Spot Interest Rate Models

We show the crucial role of sampling frequencies in model selection testing for diffusion models with high frequency data. The model selection is based on the observed likelihood functions of two candidate diffusion models. Our new framework emphasizes the effect of sampling frequencies and sampling spans on the rankings of the likelihoods. When a model has a superior diffusion function specification, it dominates the other model under high frequency sampling regardless of its drift function specification. The relative importance of the drift and diffusion function specifications depends on the ratio of the sampling frequency and the sampling span, and the drift functions become relevant only if the sampling span is long enough relative to the sampling frequency. We prove that sampling frequency must be higher than the square root of the sampling span asymptotically to ensure the effects of the drift specifications do not influence the asymptotic distribution of our test statistic under the null hypothesis that the models have an equal Kullback-Leibler information divergence from the true process. When the sampling frequency is lower, or the sampling span is longer, than this asymptotic bound, the drift specifications cannot be ignored in determining the asymptotic distribution of the test statistic. However, when two models have the same diffusion function specification, the rankings of the models are determined by the drift functions only, and the sampling frequency-span ratio does not matter. We show that how the strategic choice of sampling frequencies can help distinguish a superior model and interpret results in spot interest rate model selection testing.

An Introducton to KK-Theory: The KK Groups

Diophantine Equations With Products of Consecutive Values of a Quadratic Polynomial

Abstract: Let us consider the Diophantine equation

∏k=1ⁿ(ak²+bk+c)=d y^l, \gcd(a, b, c)=1,\; l≤ 2,where a, b, c, d are given nonnegative integers with a, d ≤ 1 and ax2+bx+c is an irreducible quadratic polynomial. In this talk, we will show that one can obtain a sharp computable bound for n. Using this bound, we entirely prove some conjectures of Amdeberhan-Medina-Moll. Moreover, we will obtain the solutions of other related equations.

Simulation-based maximum likelihood inference for partially observed Markov process models

Estimation of static (or time constant) parameters in a general class of nonlinear, non-Gaussian, partially observed Markov process models is an active area of research. In recent years, simulation-based techniques have made estimation and inference feasible for these models and have offered great flexibility to the modeler. An advantageous feature of many of these techniques is that there is no requirement to evaluate the state transition density of the model, which is often high-dimensional and unavailable in closed-form. Instead, inference can proceed as long as one is able to simulate from the state transition density — often a much simpler problem. In this talk, we introduce a simulation based maximum likelihood inference technique known as iterated filtering that uses an underlying sequential Monte Carlo (SMC) filter. We discuss some key theoretical properties of iterated filtering. In particular, we prove the convergence of the method and establish connections between iterated filtering and well-known stochastic approximation methods. We then use the iterated filtering technique to estimate parameters in a nonlinear, non-Gaussian mechanistic model of malaria transmission and answer scientific questions regarding the effect of climate factors on malaria epidemics in Northwest India. Motivated by the challenges encountered in modeling the malaria data, we conclude by proposing an improvement technique for SMC filters used in an off-line, iterative setting.

A Characterization of Noncommutative Brownian Motion

Abstract: We prove an equivalence between a certain class of noncommutative Brownian motions, a class of functors between the category of real Hilbert spaces with contractions and the category of finite von Neumann algebras with unital completely positive maps, and certain real-valued positive definitefunctions on the infinite symmetric group. This generalizes the construction of q-Brownian motion of Bożejko and Speicher and is related to the approach of Guta and Maassen. In certain cases, we are able to establish the weak* CBAP for these algebras. This is joint work withMarius Junge.

Sharp Estimates for Levy-Fourier Multiplers

Abstract: Using the machinery of D.L. Burkholder which he invented in the 80's to derive sharp martingale inequalities, one can prove sharp bounds on norms of a large class of Fourier multiplier operators that arise from transformation of Lèvy processes via the Lèvy-Khintchine formula. This class of multipliers contains many of the classical Fourier (singular integral) operators in harmonic analysis. The original motivation for these results comes from efforts to bring to bear stochastic analysis techniques on a well-known 1982 conjecture of Tadeusz Iwaniec on the norm of the Beurling-Ahlfors operator, a Calderon-Zygmund singular integral of considerable importance to applications in PDE's. Our Lèvy multiplier results have applications to Schauder estimates of solutions to "elliptic" non-local (integral) operators.

Students in both analysis (broadly interpreted) and probability (also broadly interpreted) are very welcome to attend this talk which will be made as non-technical as possible.

The talk is based on joint work with Adam Osekowski of Warsaw University, Poland.

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Mathieu Subspaces

Abstract: Motivated by the study of the Jacobian Conjecture, the notion of Mathieu subspaces, as a generalization of the notion of ideals, has been introduced recently. In this talk we will discuss some examples, recent developments, and open problems of Mathieu subspaces, especially the connections of Mathieu subspaces with the Jacobian conjecture.

Joint with UIUC
Using the criterion-predictor factor model to compute the probability of detecting prediction bias with ordinary least squares regression

The study of prediction bias is important and the last five decades includes research studies that examined whether test scores differentially predict academic or employment performance. Previous studies used ordinary least squares (OLS) to assess whether groups differ in intercepts and slopes. This study shows that OLS yields inaccurate inferences for prediction bias hypotheses. This paper builds upon the criterion-predictor factor model by demonstrating the effect of selection, measurement error, and measurement bias on prediction bias studies that use OLS. The range restricted, criterion-predictor factor model is used to compute type I error and power rates associated with using regression to assess prediction bias hypotheses. In short, OLS is not capable of testing hypotheses about group differences in latent intercepts and slopes. Additionally, a theorem is presented which shows that researchers should not employ hierarchical regression to assess intercept differences with selected samples.

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On Square Values of Quadratics

Inference for Size Demography from Point Pattern Data using Integral Projection Models

Population dynamics with regard to evolution of traits has typically been studied using matrix projection models (MPMs). Recently, to work with continuous traits, integral projection models (IPMs) have been proposed. Imitating the path with MPMs, IPMs are handled first with a fitting stage, then with a projection stage. Fitting these models has so far been done only with individual-level transition data. This data is used to estimate the demographic functions (survival, growth, fecundity) that comprise the kernel of the IPM specification. Then, the estimated kernel is iterated from an initial trait distribution to project steady state population behavior under this kernel. When trait distributions are observed over time, such an approach does not align projected distributions with these observed temporal benchmarks.

The contribution here, focusing on size distributions, is to address this issue. Our concern is that the above approach introduces an inherent mismatch in scales. The redistribution kernel in the IPM proposes a mechanistic description of population level redistribution. A kernel of the same functional form, fitted to data at the individual level, would provide a mechanistic model for individual-level processes. Resulting parameter estimates and the associated estimated kernel are at the wrong scale and do not allow population-level interpretation.

Our approach views the observed size distribution at a given time as a point pattern over a bounded interval. We build a three-stage hierarchical model to infer about the dynamic intensities used to explain the observed point patterns. This model is driven by a latent deterministic IPM and we introduce uncertainty by having the operating IPM vary around this deterministic specification. Further uncertainty arises in the realization of the point pattern given the operating IPM. Fitted within a Bayesian framework, such modeling enables full inference about all features of the model. Such dynamic modeling, optimized by fitting data observed over time, is better suited to projection.

Exact Bayesian model fitting is very computationally challenging; we offer approximate strategies to facilitate computation. We illustrate with simulated data examples as well as well as a set of annual tree growth data from Duke Forest in North Carolina. A further example shows the benefit of our approach, in terms of projection, compared with the foregoing individual level fitting.

Joint with Alan E. Gelfand and James S. Clark.

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