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ABSTRACT: Recent advances in computational modeling and imaging technology are making it possible that computational models can be built based on patient-specific image data and accurate predictions can be made for realistic clinical applications. In this talk, patient-specific MRI-based computational models for blood flow in arteries and blood flow in human heart (a right ventricle/left ventricle (RV/LV) model) will be presented. We will demonstrate how computational models can be used to a) understand fundamental biological and disease processes; b) assess atherosclerotic plaque rupture risk (plaque rupture often leads to stroke and heart attack); and c) conduct virtual surgery to optimize right ventricle volume reduction surgical procedures. It is well-accepted that atherosclerosis initiation and progression correlate positively with low and oscillating flow wall shear stresses. However, this ``low and oscillating shear stress hypothesis" cannot explain why intermediate and advanced plaques continue to grow under elevated high flow shear stress conditions. It is also natural that people think that plaque rupture may be related to maximum stress conditions. We will challenge those popular views and present evidence which support new hypotheses for plaque progression and rupture conditions. Patient-specific multi-year serial MRI were acquired to provide plaque morphology and progression data. A 3D multi-component model with fluid-structure interactions (FSI) was introduced to obtain the flow and stress/strain distributions in the plaque to better understand mechanisms governing plaque progression and rupture process. Our results indicate that plaque thickness and plaque progression correlate positively with low structure wall stress for intermediate and advanced plaques which supports a possible new hypothesis: Low structure stress in the plaque has positive correlation with plaque growth, and may create favorable mechanical conditions for further plaque progression. For plaque vulnerability assessment, our results also indicate that maximum stress conditions are often found at healthy site of the vessel and are not good indicators of rupture risk. A computational plaque vulnerability index (CPVI) based on local stress conditions at critical sites was proposed. Plaque assessments (34 plaque MRI samples) using CPVI method had 90% agreement rate with histopathological analysis. With more patient study validations, our research may serve as the starting points for further plaque progression and rupture investigations. Results from a human RV/LV patch model for surgery optimization will also be presented. The work has been supported by the National Sciences Foundation (DMS and BIO), National Institutes of Health (NIBIB and NIGMS), and the Whitaker Foundation.

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ABSTRACT: Let $B$ denote the open unit ball in the $n$-dimensional complex Euclidean space $C^n$. The subspace of holomorphic functions in $L^p(B,dv)$, where $dv$ is Lebesgue volume measure, is called a Bergman space and is denoted by $A^p$. I will give a brief history of the theory of Bergman spaces and discuss several recent characterizations of such spaces.

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ABSTRACT: We consider links between various combinatorial properties of words on a finite alphabet in Theoretical Computer Science, and transcendence theory and diophantine approximation in Number Theory. The talk will be divided in two parts: From Words to Numbers and from Numbers to Words.
In the first part, we consider different ways in which a word determines a real number: For instance, an infinite word $W$ on the alphabet $\{0,1,...,b-1\}$ determines the real number whose base $b$-expansion is $W$. Alternatively, an infinite word $W$ on the alphabet $\{1,2,...,k\}$ determines the real number whose sequence of partial quotients (in its continued fraction expansion) is equal to $W.$ In the first case, eventually periodic words correspond to rational numbers, while in the second case, they correspond to quadratic irrationals. We consider families of non periodic words possessing rich combinatorial structures. For example, words generated by finite substitution rules. We then illustrate how certain basic word combinatorial properties translate into deep number theoretic properties.
In Numbers to Words we consider the reverse problem of associating words to real numbers: We consider a symbolic dynamical system parametrized by a vector of real numbers $(\alpha _1,\alpha _2,..., \alpha_k),$ and take as words the natural coding of the orbits of points. The dynamical system in question is an interval exchange transformation in which the unit interval is sub-divided into $k$ non-overlapping sub-intervals of lengths $\alpha _1,\alpha _2,..., \alpha_k$ (labeled $1$ through $k),$ which are then re-arranged according to a fixed permutation of $\{1,2,...,k\}.$ We define a new induction rule which allows us to code the system by an infinite path in a finite graph $\Gamma _k;$ each vertex $V$ of $\Gamma _k$ is a tree of relations which in turn defines a total circular order on the vertices of $V.$ Curiously enough, this relation between tree structure and circular order is analogous to the so-called "secondary structure" of ribonucleic acid (RNA). In the base case $k=2,$ this coding reduces to the usual continued fraction algorithm. In general, our induction defines a new multi-dimensional continued fraction algorithm whose number theoretic properties are still largely unexplored.

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ABSTRACT: During the last 30 years, the study of hyperplane arrangements has been a very active area of research in Combinatorics. Rather than attempting the impossible task of surveying this vast discipline, in this talk I will present some selected results and techniques that I believe convey the depth and beauty of the subject.

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As old as Euclid, prime numbers have recently started to yield their secrets. Mathematicians from California to India and elsewhere have shown us that primes regularly fall into strict patterns, they display unusual ``clumping," and they are computationally easy to detect. While many mysteries remain, it does seem that this first decade of the new millennium is indeed a prime time for primes.

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A famous old problem of Paul Erdös is whether for each number B the set of integers may be covered with a finite collection of congurence classes with distinct moduli each at least B. In fact, Erdös wrote of this as his ``favorite problem". In recent work with Filaseta, Ford, Konyagin, and Yu, we have found some new results in this area; for example, if such finite collections should exist, the largest modulus cannot be O(B). These new results settle some conjectures of Erdös, Graham, and Selfridge.

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ABSTRACT: This talk consists of three parts. First, it points out the presence of Finsler geometry in everyday life and gives a case study of geodesics. Second, it presents a pedestrian definition of flag curvatures and illustrates with a practical example. Third, it explains what Ricci curvatures are, states a question due to Chern, and proposes a notion of Ricci flow for answering this question.

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ABSTRACT: Statistical models exhibit interesting mathematical structures that can be alternatively classified as differential geometry, conxex analysis, commutative algebra. Statistical models are usually presented as parameterized families of probability densitities. A fully non parametric presentation is possible, leading to an interesting approach even in the case of a finite state space model. Some work is in progress to generalize this theory to extended exponential models.
(Slides available here.)

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ABSTRACT: The Matching Game is a game designed for teaching 4th graders in California about probability. It is a two-player game, each player draws a ball from a bag of balls with 2 different colors. The first player wins if the balls drawn are of the same color, otherwise the second player wins. We will examine some generalizations of the Matching Game and find the "fair games" in those situations.

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ABSTRACT: This talk is about the axiomatic method, the choice of primitive notions for axiomatic Euclidean geometry, a logical framework for geometry, definitions in that framework, and some delicate undefinability results. It emphasizes advances by Mario PIERI and Alfred TARSKI in the era 1900?1935, but provides background starting with ARISTOTLE around 350 B.C.E. and a taste of advanced results as recent as 1991. The talk will be understandable by undergraduates: its only requirements are facility with high-school and analytic geometry and the notation commonly used in upper-division mathematics courses, and comfort with critical thinking about basic issues.

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