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Department of Mathematics and Computer Science

Professors Emeritus:ÌýJosé Barría, Leonard Klosinski, Edward F. Schaefer

Professors:ÌýFrank A. Farris, Tamsen McGinley (Department Chair), Daniel Ostrov, Dennis C. Smolarski, S.J., Venkatesh Srinivasan, Nicholas Q. Tran

Associate Professors:ÌýGlenn D. Appleby, Robert A. Bekes, Evan Gawlik, Michael Hartglass, Shiva Houshmand, Sara Krehbiel, ÌýNicolette Meshkat, Byron L. Walden

Assistant Professors:ÌýJavier Gonzalez Anaya, Shamil Asgarli, Tiantian Chen, Smita Ghosh,ÌýChi-Yun Hsu, Ray Li

Teaching Professors:ÌýLinda Burks, Natalie Linnell, Mona Musa,ÌýLaurie Poe

Associate Teaching Professors:ÌýCorey Irving, Mary Long, Norman Paris, Luvreet Sangha

Assistant Teaching Professors: Mehdi Ahmadi, ÌýKatelyn Byington, Will Dana, George Schaeffer, Vaishavi Sharma

Halmos Professor: Bill Dunham

The Department of Mathematics and Computer Science offers major programs leading to the bachelor of science in mathematics or the bachelor of science in computer science, as well as required and elective courses for students majoring in other fields. Either major may be pursued with any of three principal goals: preparation for graduate studies leading to advanced degrees in pure mathematics, applied mathematics, computer science, statistics, operations research, or other fields; preparation for secondary school teaching of mathematics or computer science; or preparation for a research career in business, industry, or government. The major in mathematics may be taken with an emphasis in applied mathematics, data science, financial mathematics, mathematical economics, or mathematics education. The emphasis in mathematics education is designed to prepare majors to earn single subject teaching credentials in mathematics. The major in computer science offers emphasesÌýspecializing in algorithms and complexity, data science, security, or software. Minors in mathematics andÌýcomputer science are also available.

The Department of Mathematics and Computer Science maintains a program for the discovery, encouragement, and development of talent in mathematics andÌýcomputer science among undergraduates. This program includes special sections, seminars, individual conferences, and directed study guided by faculty members. Students are also encouraged to participate in research projects directed by faculty.

Requirements for the Major

In addition to fulfilling undergraduate Core Curriculum requirements for the bachelor of science degree, students majoring in mathematics orÌýcomputer science must complete the following departmental requirements for the respective degree:

Major in Mathematics

• CSCI 10 (or demonstrated equivalent proficiency in computer programming)
• MATH 11, 12, 13, 14, 23, 51, 52, and 53
• PHYS 31 and 32. ÌýÌýSubstitutions for PHYS 32 can be made at the discretion of the Department chair. ÌýStudents planning to teach in secondary schools may substitute, with approval of the department chair, PHYS 11 and 12 for PHYS 31 and 32.
• Seven approved 5-unit upper-division courses in mathematics (CSCI 162ÌýalsoÌýpermitted), which must include at least one course in analysis (MATH 102, 105, or 153), at least one course in algebra (MATH 103 or 111), and at least one course selected from geometry (MATH 101, 113, or 174), or from discrete mathematics (MATH 176 or 177), or from applied mathematics (MATH 122, 125, 141, 144, 146, 155, or 166). ÌýMATH 100, and courses 190 and above do not count toward the seven courses.

Students are Ìýnot required to select an emphasis for the major in mathematics. Students planning to undertake graduate studies in pure mathematics should take MATH 105, 111, 112, 113, 153, and 154. Students planning to undertake graduate studies in applied mathematics should complete the emphasis in applied mathematics and take MATH 105, 144, 153, 154, and 155.

Emphasis in Applied Mathematics

Complete the requirements for a bachelor of science in mathematics with the following specifications and additions:

• MATH 102, 122, and 123
• Two courses from MATH 125, 141, 144, 146, 147,Ìý155, 166, 178, or an approved alternative 5-unit upper-division mathematics course
• One course from MATH 166, 147

Emphasis in Data Science

Complete the requirements for a bachelor of science degree in mathematics with the following specifications and additions:

• MATH 122, 123
• CSCI 10, 60, 61, 62, 183
• CSCI 184 or CSEN/COENÌý178
• Two courses from CSCI 127, 163; MATH 146, 147, 166; CSEN/COENÌý166, 169; ECON 174

Emphasis in Financial Mathematics

Complete the requirements for a bachelor of science degree in mathematics with the following specifications and additions:

• MATH 102, 122, 123, 125
• Either MATH 146 and 147 or MATH 144 and 166
• BUSN 70
• ACTG 11, 12
• FNCE 121, 124

Emphasis in Mathematical Economics

Complete the requirements for a bachelor of science degree in mathematics with the following specifications and additions:

• MATH 102, 122, 123, 146, 147
• ECON 113
• TwoÌýcourses from MATH 125, MATHÌý166, ECON 170–174

Emphasis in Mathematics Education

Complete the requirements for a bachelor of science degree in mathematics with the following specifications and additions:

MATH 101, 102, 122, 123 (or 8), 170, 175 (or 178)Students are strongly advisedÌýto complete the College’sÌýUrban Education minor.

Major in Computer Science

• MATH 11, 12, 13, 14, 51, 53
• CSCI 10, 60, 61, 62
• One of PHYS 31, CHEM 11, ENVS 21, or ENVS 23
• CSEN/COENÌý20 and 20L, ÌýECEN 21 and 21L
• MATH 122, CSCI 161,ÌýandÌý163, and CSEN/COENÌý177 and 177L
• Five additional 4- or 5-unit upper-division courses in one of the following emphases:

Algorithms and Complexity emphasis:

• CSCI 162, CSCI 164Ìý
• Two more courses from CSCI 146, 147, 165,Ìý181, MATH 101, 175, 176, 177, 178 Ìý
• One more course from the list above or any other additional 4-5 unit upper-division CSCI course below 190 or CSEN/COEN course below 188

Data Science emphasis:

• CSCI 183, 184, 185
• Two moreÌýcourses from CSCI 127, 146, 147, 164, 166,ÌýMATH 123,ÌýCSEN/COENÌý166, or any other additional 4-5 unit upper-division CSCI course below 190 or CSEN/COENÌýcourse below 188 excluding CSEN 166

Security emphasis:

• MATH 178, CSCI 180, CSCI 181
• Two more courses from MATH 175, CSEN/COEN 152 and 152L, CSEN/COENÌý161 and 161L, CSEN/COENÌý146 and 146L, or any other additional 4-5 unit upper-division CSCI course below 190 or CSEN/COENÌýcourse below 188

Software emphasis:

• CSCI 169, CSCI 187,ÌýCSEN/COENÌý146 and 146L
• OneÌýmore course from CSCI 183, 180, 168, or any otherÌýadditional 4-5 unit upper-division CSCI course below 190
• One other course from CSCI 183, 180, 168, CSEN/COEN 161, 178 or any other additional 4-5 unit upper-division CSCI course below 190 or CSEN/COEN course below 188

It is highly recommended that students (especially students in the Software emphasis) take additional upper-division courses beyond the minimum required for the degree, such asÌýCSEN/COENÌý178.

For the major in either mathematics or computer science, at least four of the required upper-division courses in the major must be taken at Santa Clara. A single upper-division course in the Department of Mathematics and Computer Science can only be used to satisfy one major or one minor requirement in the department.ÌýÌý(Exceptions may be approved by the Chair.)

Students should decide their emphasis by the end of junior year. Only one emphasis is allowed. Data Science may not be used as an emphasis for both majors.

Requirements for the Minors

Minor in Mathematics

Students must fulfill the following requirements for a minor in mathematics:

• MATH 11, 12, 13, 14, and either 52 or 53
• Three approved 5-unit upper-division mathematics courses.ÌýMATH 100, 192, and 195 do not count toward the minor. (Substitutions may be approved by the Chair.)

Minor in Computer Science

Students must fulfill the following requirements for a minor in computer science:

• CSCI 10, 60, 61, and 62
• MATH 51
• CSEN/COENÌý20 and 20L
• A total of three 4 or 5-unit upper-division courses, as follows: Two upper-division CSCI courses and one upper-division CSCI or CSEN/COENÌýcourse. CSCI 192 does not count toward the minor.

Preparation in Mathematics for Admission to Teacher Training Credential Programs

ÌýIn order to teach mathematics or computer science in California secondary schools, the State of California requires a secondary teaching credential, which can be completed as a fifth year of study or an intern program. Students who are contemplating secondary school teaching in mathematics or computer science should consult with the coordinator in the Department of Mathematics and Computer Science as early as possible.

Lower-Division Courses: Mathematics

4. The Nature of Mathematics

For students majoring in arts and humanities. Topics chosen from set theory, logic, counting techniques, number systems, graph theory, financial management, voting methods, and other suitable areas. Material will generally be presented in a setting that allows students to participate in the discovery and development of important mathematical ideas. Emphasis on problem solving and doing mathematics. (4 units)

6. Finite Mathematics for Social Science

Introduction to finite mathematics with applications to the social sciences. Sets and set operations, Venn diagrams, trees, permutations, combinations, probability (including conditional probability and Bernoulli processes), discrete random variables, probability distributions, and expected value. (4 units)

8. Introduction to Statistics

Elementary topics in statistics, including descriptive statistics, regression, probability, random variables and distributions, the central limit theorem, confidence intervals and hypothesis testing for one population and for two populations, goodness of fit, and contingency tables. (4 units)

9. Precalculus

College algebra and trigonometry, at an accelerated pace, for students intending to take calculus. Does not fulfill the undergraduate Core Curriculum requirement in mathematics. Students with credit for calculus are not admitted to Math 9 without permission. Ìý (Recommended for students with CRE scores 55 - 75). (4 Units)

MATH 9S Precalculus with support

College algebra and trigonometry, at an accelerated pace with the support of a weekly student-centered discussion and problem-solving section that focuses on building overall strength in precalculus skills and non-routine problem solving. For students intending to take calculus. Does not fulfill the undergraduate Core Curriculum requirement in mathematics. Students with credit for calculus are not admitted to Math 9 without permission. Ìý(Recommended for students with CRE scores 40 - 54). (4 units)

MATH 9A and MATH 9B Prelude to Calculus

A two-quarter sequence that builds foundational knowledge in college algebra and trigonometry while helping students review the mathematical skills needed to succeed in Calculus. Does not fulfill the undergraduate Core Curriculum requirement in mathematics. Students with credit for calculus are not admitted to Math 9 without permission. (Recommended for students with CRE scores below 40) (4 units)

11. Calculus and Analytic Geometry I

Limits and differentiation. Methods and applications of differentiation. Ordinarily, only one of MATH 11, 30, or 35 may be taken for credit. Note: MATH 11 is not a suitable prerequisite for MATH 31 or 36Ìýwithout additional preparation. Prerequisite: MATH 9 or a passing grade on the Calculus Readiness Exam. If MATH 9 is taken, a grade of C-Ìýor higher is strongly recommended before taking MATH 11.Ìý(4 units)

12. Calculus and Analytic Geometry II

Further applications of differentiation. Integration and the fundamental theorem of calculus. Methods and applications of integration. Only one of MATH 12, 31, or 36 may be taken for credit. Note: MATH 30 and 35 areÌýnot suitable prerequisites for MATH 12 without additional preparation. Prerequisite: MATH 11 or equivalent. A grade of C-Ìýor higher in MATH 11 is strongly recommended before taking MATH 12. (4 units)

13. Calculus and Analytic Geometry III

Taylor series, vectors, quadric surfaces, and partial derivatives, including optimization of functions with multiple variables. Prerequisite: MATH 12 or equivalent. Students who have taken MATHÌý31, MATHÌý36, or an equivalent course may take MATHÌý13 after consultation with an instructor.ÌýA grade of C-Ìýor higher in MATH 12 is strongly recommended before taking MATH 13. (4 units)

14. Calculus and Analytic Geometry IV

Vector functions, line integrals, multiple integrals, flux, divergence theorem, and ³§³Ù´Ç°ì±ð²õ’Ìýtheorem. Prerequisite: MATH 13 or equivalent. A grade of C-Ìýor higher in MATH 13 is strongly recommended before taking MATH 14. (4 units)

23. Series and Differential Equations

Sequences, series, and analytic functions. Use of explicit, numerical, and series methods to solve ordinary differential equations. Complex numbers. Only one of MATH 22, 23, or AMTH 106 may be taken for credit. Prerequisite: MATH 13. (4 units)

30. Calculus for Business I

Differentiation and its applications to business, including marginal cost and profit, maximization of revenue, profit, utility, and cost minimization. Natural logarithms and exponential functions and their applications, including compound interest and elasticity of demand. Study of the theory of the derivative normally included in MATH 11, except trigonometric functions not included here. Ordinarily, only one of MATH 11,Ìý30, or 35Ìýmay be taken for credit. Note: MATH 30 is not a suitable prerequisite for MATH 12 or 36 without additional preparation. Prerequisite: MATH 9 or a passing grade on the Calculus Readiness Exam.ÌýIf MATH 9 is taken, a grade of C-Ìýor higher is strongly recommended before taking MATH 30. (4 units)

31. Calculus for Business II

Integration and its applications to business, including consumer surplus and present value of future income. Functions of several variables and their derivatives. Emphasis throughout the sequence on mathematical modeling, the formulation of practical problems in mathematical terms. Only one of MATH 12, 31, or 36 may be taken for credit. Note: MATH 11 and 35 are not suitable prerequisites for MATH 31 without additional preparation. Prerequisite: MATH 30 or equivalent. A grade of C−Ìýor higher in MATH 30 is strongly recommended before taking MATH 31. (4 units)

35. Calculus for Life Sciences I

Modeling with functions, limits, and derivatives. Derivative rules and tools. Applications to the life sciences. Ordinarily, only one of MATH 11, 30, or 35 may be taken for credit. Note: MATH 35 is not a suitable prerequisite for MATH 12 or 31 without additional preparation. Prerequisite: MATH 9 or a passing grade on the Calculus Readiness Exam. If MATH 9 is taken, a grade of C- or higher is strongly recommended before taking MATH 35. (4 units)

36. Calculus for Life Sciences II

Integration, differential equations, and probability. Applications to the life sciences. Only one of MATH 12, 31, or 36 may be taken for credit. Note: MATH 11 and 31 are not suitable prerequisites for MATH 36 without additional preparation. Prerequisite: MATH 35 or equivalent. A grade of C- or higher in MATH 35 is strongly recommended before taking MATH 36. (4 units)

51. Discrete Mathematics

Predicate logic, methods of proof, sets, functions, sequences, modular arithmetic, cardinality, induction, elementary combinatorial analysis, recursion, and relations. Also listed as CSENÌý19. (4 units)

52. Introduction to Abstract Algebra

Groups, homomorphisms, isomorphisms, quotient groups, fields, integral domains; applications to number theory. Prerequisite: MATH 51 or permission of the instructor. (4 units)

53. Linear Algebra

Vector spaces, linear transformations, algebra of matrices, eigenvalues and eigenvectors, and inner products. Prerequisite: MATH 13. (4 units)

90. Lower-Division Seminars

Basic techniques of problem solving. Topics in algebra, geometry, and analysis. (1–4 units)

Upper-Division Courses: Mathematics

Note: Although CSCI 10 is not explicitly listed as a formal prerequisite, some upper-division courses suggested for computer science majors may presuppose the ability to write computer programs in some language. A number of upper-division courses do not have specific prerequisites. Students planning to enroll should be aware, however, that all upper-division courses in mathematics require some level of maturity in mathematics. Those without a reasonable background in lower-division courses are advised to check with instructors before enrolling.

100. Writing in the Mathematical Sciences

An introduction to writing and research in mathematics. Techniques in formulating research problems, standard proof methods, and proof writing. Practice in mathematical exposition for a variety of audiences. Strongly recommended for mathematics and computer science majors beginning their upper-division coursework. MATH 100 may not be taken to fulfill any mathematics or computer science upper-division requirements for students majoring or minoring in mathematics or computer science. Offered only on demand. Prerequisites: CTW 1, CTW 2.Ìý(5 units)

101. A Survey of Geometry

Topics from advanced Euclidean, projective, and non-Euclidean geometries. Symmetry. Offered in alternate years. Prerequisite: MATH 13. (5 units)

102. Advanced Calculus

Topics to be chosen from the following:Ìý Open and closed subsets of , the definition of limits and continuity for functions on , the least upper bound property on R, the intermediate and extreme value theorems for functions on , the derivative of a function on Ìýin terms of a matrix, the matrix interpretation of the chain rule, Taylor's theorem in multiple variables with applications to critical points, the inverse and implicit function theorems, multiple integrals,Ìýline and surface integrals, Green’s theorem,Ìý ³§³Ù´Ç°ì±ð²õ’ theorem, the divergence theorem, and differential forms. Prerequisites: MATH 14, 51, and 53. (5 units)

103. Advanced Linear Algebra

Abstract vector spaces, dimensionality, linear transformations, isomorphisms, matrix algebra, eigenspaces and diagonalization, Cayley-Hamilton Theorem, canonical forms, unitary and Hermitian operators, applications. Prerequisite: MATH 53. (5 units)

105. Theory of Functions of a Complex Variable

Analytic functions. Cauchy integral theorems, power series, conformal mapping. Riemann surfaces. Offered in alternate years. Prerequisite: MATH 14 (MATH 23 and 51 recommended). (5 units)

111. Abstract Algebra I

Topics from the theory of groups. Offered in alternate years. Prerequisites: MATH 52 and 53. (5 units)

112. Abstract Algebra II

Rings and ideals, algebraic extensions of fields, and the Galois theory. Offered in alternate years. Prerequisite: MATH 111. (5 units)

113. Topology

Topological spaces and continuous functions. Separability and compactness. Introduction to covering spaces or combinatorial topology. Offered in alternate years. Prerequisites: MATH 14 and 51 (102 recommended). (5 units)

122. Probability and Statistics I

Sample spaces; conditional probability; independence; random variables; discrete and continuous probability distributions; expectation; moment-generating functions; weak law of large numbers; central limit theorem. Pre- or corequisite: MATH 14. (5 units)

123. Probability and Statistics II

Confidence intervals and hypothesis testing. Maximum likelihood estimation. Analysis of variance (ANOVA) and analysis of categorical data. Simple and multiple linear regression. Optional topics may include sufficiency, the Rao-Blackwell theorem, logistic regression, and nonparametric statistics. Applications. Pre- or corequisite: MATH 53 or permission of instructor, and MATH 122. (5 units)

125. Mathematical Finance

Introduction to Ito calculus and stochastic differential equations. Discrete lattice models. Models for the movement of stock and bond prices using Brownian motion and Poisson processes. Pricing models for equity and bond options via Black-Scholes and its variants. Optimal portfolio allocation. Solution techniques will include Monte Carlo and finite difference methods. Also listed as FNCE 116, FNCE 3489, and AMTH 367. Prerequisites: MATH 53 or permission of instructor and MATH 122 or AMTH 108. (5 units)

133. Logic and Foundations

Deductive theories. Theories and models. Consistency, completeness, decidability. Theory of models. Cardinality of models. Some related topics of metamathematics and foundations. Open to upper-division science and mathematics students and to philosophy majors having sufficient logical background. Offered on demand. (5 units)

134. Set Theory

Naive set theory. Cardinal and ordinal arithmetic. Axiom of choice and continuum hypothesis. Axiomatic set theory. Offered on demand. (5 units)

141. Mathematical Models

Matching the real world with mathematical structure using such things as differential equations, difference equations, and stochastic processes, at the instructor’s discretion. ÌýTopics related to continuous or discrete models may be selected from topics, including the following: qualitative analysis, simulation, parameter estimation, steady-state analysis, shock waves, random walks, Markov chains, agent-based models, and cellular automata. Offered in alternate years. Prerequisites: MATH 53 and one of MATH 22, 23, and AMTH 106. (5 units)

144. Partial Differential Equations

Linear partial differential equations with applications in physics and engineering, including wave (hyperbolic), heat (parabolic), and Laplace (elliptic) equations. Solutions on bounded and unbounded domains using Fourier series and Fourier transforms. Introduction to nonlinear partial differential equations. Offered in alternate years. Prerequisite: MATH 14. Recommended: MATH 22 or 23 or AMTH 106. (5 units)

146. Optimization I

Methods for finding local maxima and minima of functions of multiple variables in either unconstrained or constrained domains: the Hessian matrix, Newton’s method, Lagrangians, Karush-Kuhn-Tucker conditions. Convex sets, convex functions, and convex programming. Methods for determining functions that optimize an objective like maximizing profit or minimizing task completion time: calculus of variations, optimal control, and both deterministic and stochastic dynamic programming. Also listed as CSCI 146. Prerequisites: A grade of C− or better in CSCI 10 (or prior programming experience), a grade of C− or better in MATH 14 and MATH 53. (5 units)

147. Optimization II

Algorithms for computing local and global optima. Numerical methods for finding local optima, including gradient descent and Nelder-Mead. The simplex method for linear programming and the duality of zero-sum games. Metaheuristics for approximating global optima. Also listed as CSCI 147. Prerequisites: A grade of C− or better in MATH 122 or AMTH 108, and a grade of C− or better in MATH/CSCI 146. (5 units)

153. Intermediate Analysis I

Rigorous investigation of the real number system. Concepts of limit, continuity, differentiability of functions of one real variable, uniform convergence, and theorems of differential and integral calculus. Offered in alternate years. Prerequisite: MATH 51 and either 102 or 105, or permission of the instructor. (5 units)

154. Intermediate Analysis II

Continuation of MATH 153. Offered in alternate years. Prerequisite: MATH 153. (5 units)

155. Ordinary Differential Equations

Systems of linear differential equations with constant coefficients. Numerical methods and applications. Autonomous systems, critical points, and stability of linear and nonlinear systems. Elementary Liapunov theory. Existence and uniqueness of solutions. Offered in alternate years. Prerequisite: MATH 53 or permission of instructor. Recommended: ÌýCSCI 10 and either MATH 22 or 23 or AMTH 106. (5 units)

166. Numerical Analysis

Numerical algorithms and techniques for solving mathematical problems. Linear systems, integration, approximation of functions, solution of nonlinear equations. Analysis of errors involved in the various methods. Direct methods and iterative methods. Also listed as CSCI 166. Prerequisites: A grade of C− or better in CSCI 10 or equivalent, and a grade of C− or better in MATH 53, or permission of the instructor. (5 units)

170. Development of Mathematics

A selection of mathematical concepts with their historical context. Offered in alternate years. Prerequisite: Upper-division standing in a science major. (5 units)

172. Problem Solving

Use of induction, analogy, and other techniques in solving mathematical problems. Offered in alternate years. (5 units)

174. Differential Geometry

Introduction to curves and surfaces. Frenet-Serret formulas, Gauss’ Theorema Egregium, Gauss-Bonnet theorem (as time permits). Offered in alternate years. Prerequisite: MATH 53. (5 units)

175. Theory of Numbers

Fundamental theorems on divisibility, primes, congruences. Number theoretic functions. Diophantine equations. Quadratic residues. Offered in alternate years. Prerequisite: MATH 52. (5 units)

176. Combinatorics

Permutations and combinations, generating functions, recursion relations, inclusion-exclusion, Pólya counting theorem, and a selection of topics from combinatorial geometry, graph enumeration, and algebr