Past Colloquia

ABSTRACT: I will talk on the pheromone response signaling pathway in yeast and the goal of the Alpha Project research which is to make quantitative predictions about yeast response to pheromone and defined experimental perturbations. The pheromone pathway is well-studied by biologists. It is qualitatively understood, i.e. scientists agree on the components (proteins, genes, etc.) of the pathway and how they interact. However, quantitative information about the rates of reactions and numbers of molecules is lacking due to experimental and measurement limitations. The model I develop for the pheromone pathway is a system of differential equations. I introduce some ideas on how to use methods from dynamical systems theory to analyze this complex biological system including bifurcation and stability analysis.

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ABSTRACT: The Italian mathematician Mario Pieri (1860-1913) played a central role in the research groups surrounding Corrado Segre and Giuseppe Peano, and thus a major role in the development of algebraic geometry and foundations of mathematics in the years surrounding the turn of the nineteenth century. Pieri's accomplishments were duly recognized in his own time both within and outside of Italy. In today's world they are largely forgotten, despite the fact that their impact can be seen in the work of contemporary mathematicians. Only one of Pieri's fifty-eight papers his been translated into English. His results are often credited to others or used with no citation of the papers in which they originated. The publication and analysis of Pieri's research are long overdue. James T. Smith and I have been studying Pieri's life and work with the intention of producing three volumes that explore them from a viewpoint ninety years after Pieri's premature death at the age of 52. In this talk I plan to provide an overview of our research and share some of the more important contributions Pieri has made which remain in the shadow of those of some of his more famous contemporaries.

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ABSTRACT: The talk will present Sharkovsky's remarkable theorem about periodic points of continous maps of the real line. The theorem is particularly interesting because its proof is surprisingly elementary. It requires nothing more than the intermediate value property of continuous functions. Sharkovsky's intial argument was long and complicated, but modern proofs are accessible to undergraduates and even high school students.

The talk will outline what I believe is the clearest version of the proof so far developed. This is joint work with Boris Hasselblatt.

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ABSTRACT: Fighting is endemic to our world. Animals fight primarily for food, mates, or territory. Humans, tribes, gangs, and nation-states fight for endless reasons over anything ranging from the possession of a valued asset to the setting of social norms. But why would rational actors fight when they could negotiate a settlement in peace? Bargaining under the threat or the actual imposition of violence is a central topic of game theory that assumes rational decision making. I will review some basic concepts of bargaining and game theory. I will then discuss some recent advances that offer new insights into why people and institutions prefer to fight rather than settle peacefully.

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ABSTRACT: Frames are known for their flexibility and great benefits so infused in many applications. There have been observations where more freedom than that of a frame is beneficial if not necessary. An extension of frames in the notion of pseudoframes for subspaces (PFFS) is thereby considered. PFFS functions in a manner just like a frame for a subspace in a Hilbert space. Yet sequences of frame elements and their duals are not necessarily contained in the subspace. This gives rise to properties exactly centering around the flexibility. Characterizations, constructions and properties of PFFS will be briefly discussed. More focus of this talk will be on examples of PFFS applications. We show how to construct compactly supported (pseudoframe) bi-orthogonal duals of a B-spline Riesz sequence. We demonstrate the existence of tight pseudo-duals of frames of translates. We explain how PFFS can be applied in the construction of bi-orthogonal wavelets so as to obtain results that are more favorable. We exhibit examples where pseudo-duals or alternative duals can be used to reduce quantization noise in a redundant frame or pseudoframe representation. We shall also discuss how PFFS is sufficient and even necessary in a quite general optimal noise suppression problem. Using the dual frame formula, we can also construct tight frames iteratively.

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ABSTRACT: I will give a summary on what an orbifold is and how to calculate its topological invariants, such as the fundamental group and the cohomology. Then I will explain the McKay correspondence and how its weak version is proven using motivic integration. I will finalize by stating the open questions regarding orbifolds.

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ABSTRACT: Mathematical models of biological systems have helped to advance our basic understanding of the pathogenesis of some diseases. They have provided estimates of important biological parameters and have evaluated pathogenic mechanisms that are difficult to test experimentally. These models, especially when used in conjunction with data, can suggest potential (or unforeseen) biological mechanisms and lead to new treatment strategies. I will present examples of models that use systems of differential equations to answer specific research questions. One model investigates host-pathogen interactions and suggests a mechanism of wild type HIV-1 (Human Immunodeficiency Virus type 1) resurgence following antiretroviral therapy. A second model predicts the effect of vaccines on the HSV-2 (Herpes Simplex Virus type 2) epidemic, using rigorous uncertainty and sensitivity analyses. Predictions such as these can be used to guide future therapeutic interventions.

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ABSTRACT: Marginal structural models (MSM) provide a powerful tool for estimating the causal effect of a treatment or risk variable on the distribution of a disease outcome in a population. These models, as originally introduced by Robins (Robins, 1998; Robins, 2000), model the marginal distributions of treatment-specific counterfactual outcomes, possibly conditional on a subset of the baseline covariates. Marginal structural models are particularly useful in the context of longitudinal data structures, in which each subject's treatment and covariate history are measured over time, and an outcome is recorded at a final time point. In addition to the simpler, weighted regression approaches (inverse probability of treatment weighted estimators), more general (and robust) estimators have been developed and studied in detail for parameters of the distribution treatment-specific counterfactuals (Robins, 2000; Neugebauer and van der Laan, 2004; Yu and van der Laan, 2003). This talk concerns applications where one is interested in modeling the difference between a treatment-specific counterfactual population distribution of a disease outcome and the actual current population distribution of the same outcome. Relevant parameters describe the effect of a hypothetical intervention on a population, and therefore we refer to these models as population intervention models. Estimators have been developed for the effect on an intervention in terms of a difference of means, ratio in means (e.g., relative risk if the outcome is binary), a so called switch relative risk for binary outcomes, and difference in entire distributions as measured by the quantile-quantile function. In this talk, before introducing this new estimator, we discuss a philosophy of estimation of causal effects in observational studies.

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ABSTRACT: Diffusion Tensor Imaging (DTI) is a new form of Magnetic Resonance Imaging (MRI) that is revolutionizing brain research as it allows insight into the anatomical structure of the white matter. As opposed to standard MRI, DTI measurements at each volume element are not scalars but three-dimensional ellipsoids, represented by 3x3 positive definite matrices (also called diffusion tensors). This is a new form of multivariate data for which standard statistical methods do not apply. In this work, I propose a new log-normal probability model for random positive definite matrices based on normal random matrix theory. Based on this model, I derive estimation and testing tools for the tensors eigenvalues and eigenvectors. These methods are illustrated in the context of a DTI study of reading ability in children.

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ABSTRACT: We analyze the EEG-sleep signal of neonates. A rigorous methodology to analyze EEG-sleep pattern are needed to assess functional brain maturation of infants. Changes in ultradian rhythm of the sleep (change between sleep stages) is one of the important characteristics related the brain maturation.
We consider the EEG signal as a piecewise stationary time series. Several EEG spectral and non-linear dynamics characteristics are being estimated. We find the characteristics which are significantly different for the different sleep stages. The non-parametric change-point detection algorithm and cluster analysis is applied to these characteristics to obtain the sleep stage separation.
Amplitude distributions of EEG-signal of infants is studied. We find that Smoothly Truncated L\'evy (STL) distributions provide a reasonable fit for the amplitude distribution of the EEG-signal observed during quasi-stationary segments. The properties of the STL distributions are studied and Numerical Maximum Likelihood Estimators for the their parameters are constructed.

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ABSTRACT: Rotating spiral wave patterns are a signature of oscillating chemical reactions (the Belusov-Zhabotinsky reaction, and AMP pulses in slime mold), and are believed to be involved in heart fibrillation and neural seizures. This talk will discuss a mathematical characterization of these spiral wave patterns and their time evolution, in terms of phase maps and the homotopy of phase maps. A quantization condition that is necessary and sufficient for the (mathematical) existence of a rotating spiral wave pattern will be derived.

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ABSTRACT: Survival analysis deals with data representing time durations before the occurrence of a certain event of interest, such as death from a particular disease, committing another crime, making an insurance claim, etc. Such data are typically subject to censoring and/or truncation, so that special techniques are required to account for theses situations. A long-term survivor is an individual who will never experience the event of interest, such as a cancer patient who has been completely cured of the cancer, a HIV carrier who will never develop AIDS symptoms, and an insurance policyholder who never needs to make a claim, etc. The presence of long-term survivors could have a significant impact on the analysis of survival data. In this talk, we will introduce appropriate statistical models for survival data with long-term survivors. Nonparametric and parametric approaches are used to estimate the distributions and/or parameters under such models. Test for the presence of long-term survivors will also be discussed.

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ABSTRACT: We have known since the late 1960s that mod p modular forms give rise to mod p representations of the Galois group of Q. In the early 1970s, Serre and Tate asked themselves whether the converse might be true. By the mid 1980s, Serre became convinced that the converse was indeed true; he made a precise conjecture to this effect that predicted a specific finite-dimensional space in which one would find the modular form that give rise to a given representation. This conjecture is explained in detail in my "Lectures on Serre's conjectures," written with William Stein; see http://modular.fas.harvard.edu/papers/serre/ribet-stein.pdf for this article.

Over the last year or so, there has been tremendous progress on this conjecture, due mainly to the work of Chandrashekhar Khare, some of which is joint with Jean-Pierre Wintenberger. For example, they prove Serre's conjecture for those Galois representations that correspond to modular forms on the full modular group SL(2,Z). (In general, one gets forms on congruence subgroups of SL(2,Z).) In my talk, I will summarize the strategy of their proof and talk about the ingredients that go into their proof. These ingredients include modularity lifting theorems (as in the proof of Fermat's Last Theorem), new results in deformation theory, and some theorems about "potential modularity" that were proved by Richard Taylor about 5 years ago.

For background reading, see http://math.berkeley.edu/~ribet/cms.pdf, which are the slides for a talk that I gave last June, and the first couple of papers on the page http://www.math.utah.edu/~shekhar/papers.html.

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ABSTRACT: The Drosophila Genome Project is a coordinated research effort whose aim is to sequence, compare and contrast 12 Drosophila genomes with the goal of significantly advancing comparative genomics methods. We will provide an overview of our recent work on annotation and alignment of the genomes, which focuses on the related problems of transposable element identification, gene finding and multiple sequence alignment. In particular, we emphasize the importance of robust alignment methods, and their relevance for identifying functional elements in genomes. A key concept is the alignment polytope, which will be explained and illustrated.

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ABSTRACT: Hyperbolic systems, and among them Anosov flows, are one of the central and most studied examples of structurally stable chaotic dynamical systems. Despite their long history, many fundamental questions concerning their existence and classification still remain open. Prompted by the lack of other types of examples, Alberto Verjovsky conjectured in the early 1970's that every codimension one Anosov flow in dimensions greater than three is a suspension of a linear Anosov diffeomorphism on a torus. As a corollary, we obtain a complete dynamical classification of such flows. I will talk about the background and history of Verjovsky's conjecture and outline the main ideas from a recent proof for volume preserving flows.

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