Course Descriptions

Biomathematics

Below are representative sample course descriptions. The actual content of courses varies from year to year.

**Computational Methods in Biology (Biomathematics II)**, MAP 5486

Description:
Applications of mathematics to biology will be discussed. Knowledge of a computer programming language. This course introduces biological topics where mathematical and computational methods are applicable, including discrete and continuous models of biological systems, nonlinear differential equations, and stochastic methods.

Prerequisites:
MAP 5165, or equivalent knowledge of dynamical systems.

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**Spatial and Temporal Models in Biology**, MAP 5932

Description:
Biological models described using partial differential equations. Topics may include chemotaxis, Turing pattern, biofluids, biofilms, cancer, and calcium signaling.

Prerequisites:
MAP5345 (Elementary PDEs I).

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**Graph Theory and Networks**, MAD 5306

Description:
Mathematical methods for studying networks. Topics may include description of real networks, directed and undirected networks, diffusion on a network, network centrality, and random networks.

Prerequisites:
MAS3105, or equivalent knowledge of linear algebra.

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**Elementary Partial Differential Equations I**, MAP 5345

Description:
Separation of variables; Fourier series; Sturm-Liouville problems; multidimensional initial boundary value problems; nonhomogeneous problems; Bessel functions and Legendre polynomials.

Prerequisites:
MAC 2313; MAP 2302 or 3305.

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**Elementary Partial Differential Equations II**, MAP 5346

Description:
Solution of first order quasi-linear partial differential equations; classification and reduction to normal form of linear second order equations; Greens function; infinite domain problems; the wave equation; radiation condition; spherical harmonics.

Prerequisites:
MAP 4341 or 5345 (Elementary PDEs I).

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**Theory of Functions of a Complex Variable I**, MAA 5406

Description:
Algebra and geometry of complex numbers; elementary functions and their mappings. Analytic functions; integration in the complex plane; Cauchy's integral theorem and related theorems. Representation theorems including the Taylor and Laurent expansions. Calculus of residues. Entire and meromorphic functions.

Prerequisites:
MAA 4227 or 5307; alternatively MAA 4226 and 4402.

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**Theory of Functions of a Complex Variable II**, MAA 5407

Description:
Continuation of MAA 5406.

Prerequisites:
MAA 5406 (Complex Analysis I).

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**Foundations of Computational Mathematics I**, MAD 5403

Description:
Analysis and implementation of numerical algorithms. Conditioning, numerical errors,
interpolation, quadrature, approximation theory, numerical methods for ordinary differential equations.

Prerequisites:
Linear algebra, competence in a programming language suitable for numeric computation.

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**Foundations of Computational Mathematics II**, MAD 5404

Description:
Direct and iterative solution of linear systems and least squares problems, root finding,
systems of nonlinear equations, numerical optimization.

Prerequisites:
MAD 5403 (Foundations of Computational Mathematics I).

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**Methods of Applied Mathematics I**, MAP 5165

Description:
Continuous and discrete models from physics, chemistry, biology, and engineering are analyzed using perturbation methods, analytical and geometrical tools and dynamical systems theory.

Prerequisites:
MAP 2302, MAC 2313, and MAS 3105.

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**Methods of Applied Mathematics II**, MAP 5423

Description:
Ordinary differential equations in the complex plane, special functions, asymptotic methods, integral transforms.

Prerequisites:
MAP 4341 or MAP 5345.

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**Biomedical Mathematics Projects**, MAP 6437

Description:
This courses give students an opportunity to apply and supplement knowledge gained from coursework to real problems in biology or medicine. Students will give class presentations and will present a written report at the end of the semester.

Prerequisites:
This is the projects course for the Master's degree. Students should have three semesters of coursework in biomathematics.

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**Distribution Theory**, STA 5326

Description:
Axioms and basic properties of probability, Combinatorial probability, Conditional probability and independence, Applications of the Law of Total Probability and Bayes Theorem, Random variables, Cumulative distribution, density, and mass functions, Distributions of functions of a random variable, Expected values, Computations using indicator random variables, Moments and moment generating functions, Common families of distributions, Location and scale families. Exponential families, Joint and conditional distributions, Bivariate transformations, Covariance and correlation, Hierarchical Models, Variance and Conditional variance. Introduction to Brownian motion.

Prerequisites:
Three semesters of calculus and an undergraduate course in probability (or some exposure to probability plus a sufficiently strong math background).

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**Statistical Inference**, STA 5327

Description:
Methods of estimation, Bayesian models, Fisher information, large sample theory, introduction to hypothesis testing.

Prerequisites:
STA 5326 (Distribution Theory).

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**Epidemiology for Statisticians**, STA 5198

Description:
Identification of risk factors for disease, including exposure-disease association, design of cohort,
matched and randomized studies, cross-sectional and longitudinal studies, statistical analysis of
data arising from such studies, confounding, adjustment and causality, and evaluation of diagnostic and
screening tests.

Prerequisites:
STA 2171

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**Molecular Biology**, PCB 5525

Description:
Introduction to molecular biology and molecular genetics. The emphasis will be on the activities of DNA, RNA, regulation of gene expression, gene cloning, bioinformatics, and biotechnology.

Prerequisites:
PCB 3063, or the equivalent, or permission of the instructor.

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**Cell and Molecular Neuroscience**, PCB 5845

Description:
Students are introduced to basic principles of neurophysiology, including intracellular signaling, membrane potentials, synaptic communication, sensory and motor systems and neural development and plasticity.

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**Membrane Biophysics**, BSC 5936

Description:
The primary objective of this course is to train the graduate student with the necessary mathematical, physiological, and molecular background that he or she will need to be able to design competitive research in the field of membrane biophysics.

This course is an integrated approach to modern biophysics with an emphasis on neural applications. Modern biophysics requires a strong working knowledge of physical laws, molecular approaches, physiological responses, structural proteins, and the mechanics of the equipment used to measure the physical properties of biological membranes.

It is a tandem objective of this course that the student will be able to apply this working knowledge to a deep comprehension of the primary literature. Towards this end, the class will collectively build a literature resource that can be drawn upon for a firm foundation for comprehensive research directives in two fields: (1) Ion Channels, and (2) Biophysical Methods.

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**Advanced Cell Biology**, PCB 5137

Description:
Principles of cell organization; membrane structure and transport; cyto skeleton; signaling; organelle structure and function; energy metabolism; cellular aspects of cancer and immunity.

Prerequisites:
A course in Molecular Biology.

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**Statistical Modeling with Application to Biology**, STA 5176

Description:
This is an interdisciplinary course, focusing on application of statistical and computational methods to biological problems.

Methods covered are Expectation Maximization (EM), Hidden Markov Model (HMM), Bayesian Network (BN), Monte Carlo (MC) methods and Markov Chain Monte Carlo (MCMC), maximum likelihood estimation (MLE), regression, logistic regression, bootstrapping, machine learning methods such as clustering, classification, and variable selection (feature selection).

The biological problems used to illustrate the methods include DNA sequence analysis/alignment, microarray and genomic data analysis, protein sequence alignment, protein structure prediction, and gene regulations.

Prerequisites:
An undergraduate course in probability.

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**Mathematical Statistics**, STA 5325

Description:
Sufficiency, point estimation, confidence intervals, hypothesis testing, regression, linear models, Bayesian models.

Prerequisites:
An undergraduate course in probability.

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**Statistics in Epidemiology**, STA 5172

Description:
This course introduces the statistical methods developed for and used in epidemiology. Statistical design issues in epidemiological studies, measures of disease occurrence, measures of association, and adjusting for confounding without and with multivariate models.

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**Applied Survival Analysis**, STA 5179

Description:
This course provides a focused introduction to methods for describing, analyzing, and modeling survival data in medical studies.

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