- Explanation of Course Numbers
- New Mathematics Books Utilizing Wolfram Technology
- Catalog Record: Calculus using Mathematica | HathiTrust Digital Library
- Available Projects

Sequences and series; Cauchy sequences and convergence. Required for all mathematics majors.

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Point-set topology in metric spaces with attention to n-dimensional space; completeness, compactness, connectedness, and continuity of functions. Topics in sequences, series of functions, uniform convergence, Fourier series and polynomial approximation. Theoretical development of differentiation and Riemann integration. Properties, singularities, and representations of analytic functions, complex integration.

### Explanation of Course Numbers

Riemann surfaces. Relevance to the theory of physical problems. The theme of this course will be the interplay between geometry and complex analysis, algebra and other fields of mathematics. An effort will be made to highlight significant, unexpected connections between major fields, illustrating the unity of mathematics. The choice of text s and syllabus itself will be flexible, to be adapted to the range of interests and backgrounds of pre-enrolled students.

Possible topics include: the Mobius group and its subgroups, hyperbolic geometry, elliptic functions, Riemann surfaces, applications of conformal mapping, and potential theory in classical physical models. Introduction to the theory of convex sets and functions and to the extremes in problems in areas of mathematics where convexity plays a role.

Among the topics discussed are basic properties of convex sets extreme points, facial structure of polytopes , separation theorems, duality and polars, properties of convex functions, minima and maxima of convex functions over convex set, various optimization problems. Prereq: MATH or consent. An introductory survey to Scientific Computing, from principles to applications.

Topics include accuracy and efficiency, conditioning and stability, numerical solution of linear and nonlinear systems, optimization, interpolation, quadrature rules, numerical solutions of ODEs and PDEs. Coreq: MATH This course is intended for upper undergraduate students in Mathematics, Cognitive Science, Biomedical Engineering, Biology or Neuroscience who have an interest in quantitative investigation of the brain and its functions.

Students will be introduced to a variety of mathematical techniques needed to model and simulate different brain functions, and to analyze the results of the simulations and of available measured data. The mathematical exposition will be followed — when appropriate — by the corresponding implementation in Matlab. The course will cover some basic topics in the mathematical aspects of differential equations, electromagnetism, Inverse problems and Imaging related to brain functions.

Validation and falsification of the mathematical models in the light of available experimental data will be addressed.

## New Mathematics Books Utilizing Wolfram Technology

This course will be a first step towards organizing the different brain investigative modalities within a unified mathematical framework. A final presentation and written report are part of the course requirements. Nonlinear discrete dynamical systems in one and two dimensions.

Chaotic dynamics, elementary bifurcation theory, hyperbolicity, symbolic dynamics, structural stability, stable manifold theory. The purpose of this seminar is to introduce students to some of the research being done at Case that explores questions at the intersection of mathematics and biology.

Students will explore roughly five research collaborations, spending two weeks with each research group. In the first three classes of each two-week block, students will read and discuss relevant papers, guided by members of that research group, and the two-week period will culminate in a talk in which a member of the research group will present a potential undergraduate project in that area. Together, they will write up this project as a research proposal, introducing the problem, explaining how it connects to broader scientific questions, and outlining the proposed work.

It is expected that students will use the associated research group as a resource, but the proposal should be their own work. Students will submit a first draft, receive feedback, and then submit a revised draft.

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## Catalog Record: Calculus using Mathematica | HathiTrust Digital Library

Introduction to mathematical logic, different classes of automata and their correspondence to different classes of formal languages, recursive functions and computability, assertions and program verification, denotational semantics. Cross-listed as EECS A two-semester course 2 credits per semester in the joint B. Study of the techniques utilized in a specific research area and of recent literature associated with the project.

Work leading to meaningful results which are to be presented as a term paper and an oral report at the end of the second semester. Supervising faculty will review progress with the student on a regular basis, including detailed progress reports made twice each semester, to ensure successful completion of the work. Mathematics Senior Project.

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Students pursue a project based on experimental, theoretical or teaching research under the supervision of a mathematics faculty member, a faculty member from another Case department or a research scientist or engineer from another institution. A departmental Senior Project Coordinator must approve all project proposals and this same person will receive regular oral and written progress reports. Final results are presented at the end of the second semester as a paper in a style suitable for publication in a professional journal as well as an oral report in a public Mathematics Capstone symposium.

A three credit course on mathematical modeling as it applies to the origins sciences.

Students gain practical experience in a wide range of techniques for modeling research questions in cosmology and astrophysics, integrative evolutionary biology including physical anthropology, ecology, paleontology, and evolutionary cognitive science , and planetary science and astrobiology. An introduction to the various two-dimensional geometries, including Euclidean, spherical, hyperbolic, projective, and affine. The course will examine the axiomatic basis of geometry, with an emphasis on transformations.

Topics include the parallel postulate and its alternatives, isometrics and transformation groups, tilings, the hyperbolic plane and its models, spherical geometry, affine and projective transformations, and other topics. We will examine the role of complex and hypercomplex numbers in the algebraic representation of transformations. The course is self-contained. An introduction to the mathematical theory of knots and links, with emphasis on the modern combinatorial methods.

Reidemeister moves on link projections, ambient and regular isotopies, linking number tricolorability, rational tangles, braids, torus knots, seifert surfaces and genus, the knot polynomials bracket, X, Jones, Alexander, HOMFLY , crossing numbers of alternating knots and amphicheirality. Connections to theoretical physics, molecular biology, and other scientific applications will be pursued in term projects, as appropriate to the background and interests of the students. This is the first introduction to algebraic geometry — the study of solutions of polynomial equations — for advanced undergraduate students.

Recent application of this large and important area include number theory, combinatorics, theoretical physics, coding theory, and robotics. In this course, we will learn the basic objects and notions of algebraic geometry. Topics that are planned to be covered are affine and projective varieties, the Zariski topology, the correspondence between ideals and varieties, the sheaf of regular functions, regular and rational maps, dimensions and tang spaces.

Examples such as Grassmannians, curves, and blow-ups will be discussed. Depending on time constraints, we may also touch upon the modern language of schemes, line bundles and the Riemann Roch formula, and algorithmic techniques such as Groebner bases. Building on the material in Biology , this course focuses on the mathematical tools used to construct and analyze biological models, with examples drawn largely from ecology but also from epidemiology, developmental biology, and other areas.

## Available Projects

By the end of the course, students should be able to recognize basic building blocks in biological models, be able to perform simple analysis, and be more fluent in translating between verbal and mathematical descriptions. Computer simulations and mathematical analysis of neurons and neural circuits, and the computational properties of nervous systems.

Students are taught a range of models for neurons and neural circuits, and are asked to implement and explore the computational and dynamic properties of these models. The course introduces students to dynamical systems theory for the analysis of neurons and neural circuits, as well as a cable theory, passive and active compartmental modeling, numerical integration methods, models of plasticity and learning, models of brain systems, and their relationship to artificial and neural networks.

Term project required. Students enrolled in MATH will make arrangements with the instructor to attend additional lectures and complete additional assignments addressing mathematical topics related to the course. Combinatorial analysis. Permutations and combinations. Axioms of probability. Sample space and events. Equally likely outcomes.

Conditional probability. Independent events and trials. Discrete random variables, probability mass functions. Expected value, variance. Bernoulli, binomial, Poisson, geometric, negative binomial random variables. Continuous random variables, density functions. Expected value and variance.