Course Descriptions

MATH-I 555 - Introduction to Biomathematics

Credits:

3

Description:

The class will explore how mathematical methods can be applied to study problems in life-sciences. No prior knowledge of life-sciences is required. Wide areas of mathematical biology will be covered at an introductory level. Several selected topics, such as dynamical systems and partial differential equations in neuroscience and physiology, and mathematical modeling of biological flows and tissues, will be explored in depth.

Prerequisite:

MATH-I 266 or MATH-I 366, MATH-I 351 or MATH-I 511, MATH-I 426, or consent of instructor.

MATH-I 559 - Applied Computational Methods I

Credits:

3

Description:

Computer arithmetic, interpolation methods, methods for nonlinear equations, methods for solving linear systems, special methods for special matrices, linear least square methods, methods for computing eigenvalues, iterative methods for linear systems; methods for systems of nonlinear equations.

Prerequisite:

MATH-I 266 or MATH-I 366, and MATH-I 351 or MATH-I 511.

MATH-I 561 - Projective Geometry

Credits:

3

Description:

Projective invariants, Desargues' theorem, cross-ratio, axiomatic foundation, duality, consistency, independence, coordinates, and conics.

Prerequisite:

MATH-I 351.

MATH-I 562 - Introduction to Differential Geometry and Topology

Credits:

3

Description:

Smooth manifolds, tangent vectors, inverse and implicit function theorems, submanifolds, vector fields, integral curves, differential forms, the exterior derivative, DeRham cohomology groups, surfaces in E3, Gaussian curvature, two-dimensional Riemannian geometry, and Gauss-Bonnet and Poincare theorems on vector fields.

Prerequisite:

MATH-I 351 and MATH-I 445.

MATH-I 563 - Advanced Geometry

Credits:

3

Description:

Topics in Euclidean and non-Euclidean geometry.

Prerequisite:

MATH-I 300 or consent of instructor.

MATH-I 567 - Dynamical Systems I

Credits:

3

Description:

Covers the basic notions and theorems of the theory of dynamical systems and their connections with other branches of mathematics. Topics covered include fundamental concepts and examples, one-dimensional systems, symbolic dynamics, topological entropy, hyperbolicity, structural stability, bifurcations, invariant measures, and ergodicity.

Prerequisite:

MATH-I 545 and MATH-I 571.

MATH-I 571 - Elementary Topology

Credits:

3

Description:

Topological spaces, metric spaces, continuity, compactness, connectedness, separation axioms, nets, and function spaces.

Prerequisite:

MATH-I 444.

MATH-I 572 - Introduction to Algebraic Topology

Credits:

3

Description:

Singular homology theory, Ellenberg-Steenrod axioms, simplicial and cell complexes, elementary homotopy theory, and Lefschetz fixed point theorem.

Prerequisite:

MATH-I 571.

MATH-I 574 - Mathematical Physics I

Credits:

1-3

Description:

Covers the basic concepts and theorems of mathematical theories that have direct applications to physics. Topics to be covered include special functions ODEs and PDEs of mathematical physics, groups and manifolds, mathematical foundations of statistical physics.

Prerequisite:

MATH-I 530 and MATH-I 545.

MATH-I 578 - Mathematical Modeling of Physical Systems I

Credits:

3

Description:

Linear systems modeling, mass-spring-damper systems, free and forced vibrations, applications to automobile suspension, accelerometer, seismograph, etc., RLC circuits, passive and active filters, applications to crossover networks and equalizers, nonlinear systems, stability and bifurcation, dynamics of a nonlinear pendulum, van der Pol oscillator, chemical reactor, etc., introduction to chaotic dynamics, identifying chaos, chaos suppression and control, computer simulations, and laboratory experiments.

Prerequisite:

MATH-I 266 or MATH-I 366, PHYS-I 152, PHYS-I 251, and consent of instructor.

MATH-I 581 - Introduction to Logic for Teachers

Credits:

3

Description:

Logical connectives, rules of sentential inference, quantifiers, bound and free variables, rules of inference, interpretations and validity, theorems in group theory, and introduction to set theory.

Prerequisite:

MATH-I 351.

MATH-I 583 - History of Elementary Mathematics

Credits:

3

Description:

P: 26100. A survey and treatment of the content of major developments of mathematics through the eighteenth century, with selected topics from more recent mathematics, including non-Euclidean geometry and the axiomatic method.

MATH-I 585 - Mathematical Logic I

Credits:

3

Description:

Formal theories for propositional and predicate calculus with study of models, completeness, and compactness. Formalization of elementary number theory; Turing machines, halting problem, and the undecidability of arithmetic.

Prerequisite:

MATH-I 351 or an undergraduate proof course; MATH-I 587 recommended.

MATH-I 587 - General Set Theory

Credits:

3

Description:

An introduction to set theory, including both so-called "naive" and formal approaches, leading to a careful development using the Zermelo-Fraenkel axioms for set theory and an in-depth discussion of cardinal and ordinal numbers, the Axiom of Choice, and the Continuum Hypothesis.

Prerequisite:

MATH-I 351 or equivalent proof course in Linear Algebra.

MATH-I 588 - Mathematical Modeling of Physical Systems II

Credits:

3

Description:

Depending on the interests of the students, the content may vary from year to year. Emphasis will be on mathematical modeling of a variety of physical systems. Topics will be chosen from the volumes by Avner Friedman. Researchers from local industries will be invited to present real-world applications. Each student will undertake a project in consultation with one of the instructors or an industrial researcher.

Prerequisite:

MATH-I 578.

MATH-I 598 - Topics in Mathematics

Credits:

1-6

Description:

By arrangement. Directed study and reports for students who wish to undertake individual reading and study on approved topics.

MATH-I 611 - Methods of Applied Mathematics I

Credits:

3

Description:

Introduction to Banach and Hilbert spaces, linear integral equations with Hilbert-Schmidt kernels, eigenfunction expansions, and Fourier transforms.

Prerequisite:

Consent of instructor.

MATH-I 612 - Methods of Applied Mathematics II

Credits:

3

Description:

Continuation of theory of linear integral equations; Sturm-Liouville and Weyl theory for second-order differential operators, distributions in n dimensions, and Fourier transforms.

Prerequisite:

MATH-I 611.

MATH-I 626 - Mathematical Formulation of Physical Problems I

Credits:

3

Description:

Nature of applied mathematics, deterministic systems and ordinary differential equations, random processes and partial differential equations, Fourier analysis, dimensional analysis and scaling.

Prerequisite:

Advanced calculus or vector calculus, partial differential equations, linear algebra.

MATH-I 627 - Mathematical Formulation of Physical Problems II

Credits:

3

Description:

Theories of continuous fields, continuous medium, field equations of continuum mechanics, inviscid fluid flow, viscous flow, turbulence. Additional topics to be discussed include application of the theory of dynamical systems, methods for analysis of nonlinear ordinary and partial differential equations, and others. This course is an advancement of topics covered in MATH-I 626.

Prerequisite:

MATH-I 626.