Course Goals and Objectives:

• Revisit fundamental calculus concepts, including the real numbers, limits, continuity, and differentiability, with greater rigor and precision than in the calculus sequence.
• Develop students’ ability to read, write, and construct proofs in the setting of real analysis.
• Explore the theoretical foundations underlying the major results of single-variable calculus.
• Strengthen students’ capacity to reason carefully about mathematical definitions and to construct counterexamples.
• Prepare students for further study in analysis and other proof-intensive areas of mathematics.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to state and apply precise definitions of limits, continuity, uniform continuity, and differentiability for functions of one real variable.
• Students who successfully complete this course will be able to construct rigorous proofs of key results in single-variable calculus, including properties of continuous and differentiable functions.
• Students who successfully complete this course will be able to identify errors in mathematical arguments and construct counterexamples to false statements in analysis.
• Students who successfully complete this course will be able to communicate mathematical reasoning clearly and correctly in written proofs using complete sentences.

Course Goals and Objectives:

• Develop the rigorous theory of integration, sequences, and infinite series for functions of one real variable.
• Deepen students’ understanding of convergence concepts, including pointwise and uniform convergence of sequences and series of functions.
• Explore the construction and properties of key functions in analysis, including trigonometric, exponential, and logarithmic functions.
• Further develop students’ skills in reading, writing, and constructing proofs in analysis.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to state and apply the definition of the Riemann integral and prove results about integrability and properties of the integral, including the Fundamental Theorem of Calculus.
• Students who successfully complete this course will be able to determine the convergence or divergence of numerical series and sequences using appropriate tests and rigorous arguments.
• Students who successfully complete this course will be able to distinguish between pointwise and uniform convergence of sequences and series of functions and apply these concepts in proofs.
• Students who successfully complete this course will be able to work with power series, including determining radii of convergence and justifying term-by-term operations.
• Students who successfully complete this course will be able to communicate mathematical reasoning clearly and correctly in written proofs.

Course Goals and Objectives:

• Develop a rigorous, axiomatic treatment of single-variable calculus, building from the completeness of the real numbers through sequences, limits, continuity, differentiation, and Riemann integration.
• Introduce fundamental ideas from the topology of the real line, including open and closed sets, compactness, and connectedness.
• Develop students’ ability to read, write, and discover proofs and to construct counterexamples in analysis.
• Explore convergence of sequences and series of functions, including the distinction between pointwise and uniform convergence.
• Prepare students for graduate-level mathematics and further study in analysis.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to construct rigorous proofs from the axioms of the real numbers, including arguments involving completeness, convergence of sequences and series, and properties of continuous and differentiable functions.
• Students who successfully complete this course will be able to state and apply precise definitions of limits, continuity, uniform continuity, differentiability, and Riemann integrability.
• Students who successfully complete this course will be able to prove and apply results about the topology of the real line, including properties of open sets, closed sets, and compact sets.
• Students who successfully complete this course will be able to analyze convergence of sequences and series of functions, distinguishing between pointwise and uniform convergence.
• Students who successfully complete this course will be able to communicate mathematical reasoning clearly and correctly in written proofs.

Course Goals and Objectives:

• Extend the rigorous, axiomatic treatment of analysis from the real line to the more general setting of metric spaces and multivariable calculus.
• Develop students’ facility with abstract concepts such as compactness, completeness, and connectedness in metric spaces.
• Explore convergence of sequences and series of functions, including uniform convergence and power series, in the metric space setting.
• Introduce the theory of differentiation in several variables, including the inverse and implicit function theorems.
• Further develop students’ ability to read, write, and discover proofs in analysis at a level that prepares them for graduate study.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to work with the basic theory of metric spaces, including proving results about open and closed sets, compactness, completeness, and continuity in this setting.
• Students who successfully complete this course will be able to analyze convergence of sequences and series of functions and apply uniform convergence to justify limit operations.
• Students who successfully complete this course will be able to state and apply the definition of the derivative for functions between Euclidean spaces and prove results about differentiable mappings.
• Students who successfully complete this course will be able to apply the contraction mapping theorem, inverse function theorem, and implicit function theorem in appropriate contexts.
• Students who successfully complete this course will be able to communicate mathematical reasoning clearly and correctly in written proofs.

Course Goals and Objectives:

• Develop a rigorous treatment of the foundations of real analysis, including the real and complex number systems, metric spaces, sequences and series, continuity, differentiation, and Riemann integration.
• Introduce the basic topology of metric spaces, with emphasis on compactness, connectedness, and their role in analysis.
• Deepen students’ ability to read, write, and construct proofs in analysis at a level that bridges undergraduate and graduate mathematics.
• Strengthen critical thinking and mathematical writing skills through sustained engagement with rigorous arguments.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to state and apply the axiomatic properties of the real number system, including completeness, and prove foundational results about sequences, series, and convergence in metric spaces.
• Students who successfully complete this course will be able to prove and apply results about the topology of metric spaces, including properties of open sets, closed sets, and compact sets.
• Students who successfully complete this course will be able to state and apply precise definitions of continuity and differentiability for functions on metric spaces and on the real line, and prove key theorems such as the mean value theorem and Taylor’s theorem.
• Students who successfully complete this course will be able to develop and present rigorous proofs of results about the Riemann integral, including the fundamental theorem of calculus.
• Students who successfully complete this course will be able to communicate mathematical reasoning clearly and correctly in written proofs.

Course Goals and Objectives:

• Develop the theory of sequences and series of functions, with emphasis on uniform convergence and its consequences for limit operations.
• Introduce the Lebesgue integral, including measurable sets, measurable functions, and the convergence theorems that distinguish Lebesgue integration from Riemann integration.
• Explore topics in approximation theory and equicontinuity, including the Weierstrass approximation theorem and the Arzelà–Ascoli theorem.
• Further develop students’ ability to read, write, and construct proofs in analysis at a level that bridges undergraduate and graduate mathematics.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to determine the pointwise and uniform convergence of sequences and series of functions and apply uniform convergence to justify interchange of limit operations.
• Students who successfully complete this course will be able to state and apply key results in approximation theory, including the Weierstrass approximation theorem and the Arzelà–Ascoli theorem.
• Students who successfully complete this course will be able to work with the construction of Lebesgue measure and the Lebesgue integral, including proving properties of measurable sets and measurable functions.
• Students who successfully complete this course will be able to state and apply the major convergence theorems of Lebesgue integration, including the monotone convergence theorem and the dominated convergence theorem.
• Students who successfully complete this course will be able to communicate mathematical reasoning clearly and correctly in written proofs.

Course Goals and Objectives:

• Introduce the theory of functions of a complex variable, including analytic functions, contour integration, and series representations.
• Develop students’ ability to work with complex numbers, their geometry, and the algebraic and topological structure of the complex plane.
• Explore the central results of complex analysis, including the Cauchy-Goursat theorem, Cauchy integral formula, and the calculus of residues.
• Develop students’ ability to write clear, complete, and logical solutions to problems involving complex-valued functions.
• Illustrate connections between complex analysis and other areas of mathematics and its applications.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to verify analyticity of complex functions using the Cauchy-Riemann equations and work with elementary complex functions such as the exponential, logarithmic, and trigonometric functions.
• Students who successfully complete this course will be able to evaluate contour integrals using the Cauchy-Goursat theorem and the Cauchy integral formula.
• Students who successfully complete this course will be able to compute Taylor and Laurent series expansions of analytic functions and determine their regions of convergence.
• Students who successfully complete this course will be able to classify isolated singularities, compute residues, and apply the residue theorem to evaluate integrals.
• Students who successfully complete this course will be able to communicate solutions to problems in complex analysis clearly and logically in written form.

Course Goals and Objectives:

• Introduce fundamental concepts and techniques in combinatorics, including enumeration, generating functions, and graph theory.
• Develop students’ ability to solve counting problems using a variety of methods, including permutations, combinations, partitions, and inclusion-exclusion.
• Strengthen students’ capacity for precise and creative reasoning about discrete mathematical structures.
• Build conceptual and computational competency in a core area of discrete mathematics.
• Prepare students for further study in combinatorics, discrete mathematics, and related fields requiring creative problem-solving skills.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to solve enumeration problems using appropriate counting techniques, including permutations, combinations, and the inclusion-exclusion principle.
• Students who successfully complete this course will be able to set up and manipulate generating functions to solve recurrence relations and counting problems.
• Students who successfully complete this course will be able to apply basic concepts from graph theory, including properties of graphs, trees, and graph coloring, to solve combinatorial problems.
• Students who successfully complete this course will be able to communicate solutions to combinatorial problems clearly and logically in written form.

Course Goals and Objectives:

• Continue the development of combinatorial techniques begun in MAD 4203, covering advanced topics such as extremal combinatorics, Ramsey theory, matching theory, and probabilistic methods.
• Deepen students’ ability to analyze and solve problems involving discrete mathematical structures.
• Strengthen students’ capacity for rigorous reasoning and clear argumentation in combinatorics.
• Build conceptual and computational competency in advanced areas of discrete mathematics.
• Prepare students for further study or research in combinatorics and related fields.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to apply advanced combinatorial methods, including extremal and probabilistic techniques, to solve counting and optimization problems.
• Students who successfully complete this course will be able to analyze problems in graph theory involving matchings, colorings, planarity, and related structural properties.
• Students who successfully complete this course will be able to apply results from Ramsey theory and extremal combinatorics in appropriate contexts.
• Students who successfully complete this course will be able to communicate solutions to combinatorial problems clearly and logically in written form.

Course Goals and Objectives:

• Introduce the theory and applications of graphs, including fundamental concepts, trees, connectivity, planarity, coloring, matching, and network flow.
• Develop students’ ability to reason rigorously about discrete structures and to construct proofs involving graphs and digraphs.
• Explore algorithmic and structural aspects of graph theory, emphasizing both theoretical results and their applications.
• Strengthen students’ capacity for precise mathematical reasoning and clear written communication in a proof-based setting.
• Prepare students for further study in combinatorics, discrete mathematics, and fields that rely on graph-theoretic methods.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to apply fundamental graph-theoretic concepts, including connectivity, trees, and matchings, to solve problems and prove results about graphs.
• Students who successfully complete this course will be able to analyze planarity and coloring properties of graphs using appropriate theorems and techniques.
• Students who successfully complete this course will be able to construct rigorous proofs of results in graph theory using appropriate definitions and logical reasoning.
• Students who successfully complete this course will be able to communicate mathematical arguments about graphs clearly and correctly in written solutions and proofs.

Course Goals and Objectives:

• Introduce the theory and application of fundamental numerical methods, including techniques for solving nonlinear equations, linear systems, interpolation, numerical differentiation and integration, and ordinary differential equations.
• Develop students’ ability to analyze the accuracy, stability, and convergence properties of numerical algorithms.
• Build hands-on experience with the implementation of numerical methods using scientific programming tools such as MATLAB, Octave, or Python.
• Develop and analyze mathematical models of scientific problems through the lens of numerical computation.
• Strengthen students’ ability to interpret numerical results and communicate computational findings clearly.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to apply standard numerical methods to solve nonlinear equations, linear systems, and ordinary differential equations accurately and efficiently.
• Students who successfully complete this course will be able to construct polynomial and other approximations to functions and data using interpolation and least-squares techniques.
• Students who successfully complete this course will be able to analyze the error, convergence, and stability of numerical algorithms in appropriate contexts.
• Students who successfully complete this course will be able to implement numerical methods in a scientific programming language and interpret the computational results.
• Students who successfully complete this course will be able to communicate the setup, methodology, and conclusions of numerical computations clearly in written solutions.

Course Goals and Objectives:

• Introduce the mathematical foundations of probability theory and stochastic processes, including Markov chains, Poisson processes, and Brownian motion.
• Develop students’ ability to model real-world phenomena involving randomness using a rigorous probabilistic framework.
• Build conceptual and computational competency in core probabilistic methods used across mathematics, statistics, and the applied sciences.
• Strengthen students’ capacity to reason precisely about random processes and their long-term behavior.
• Prepare students for graduate study in probability, statistics, or related disciplines, and for careers requiring strong quantitative and analytical skills.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to apply the language and rules of probability, including conditional probability, independence, and Bayes’ formula, to solve problems.
• Students who successfully complete this course will be able to analyze Markov chains, including classifying states, computing first passage times, and determining invariant measures.
• Students who successfully complete this course will be able to describe and apply key stochastic processes, including the Poisson process and Brownian motion, and interpret their properties.
• Students who successfully complete this course will be able to formulate and solve problems from pure and applied settings using probabilistic arguments.
• Students who successfully complete this course will be able to communicate solutions to problems in probability clearly and logically in written form.

Course Goals and Objectives:

• Extend the study of ordinary differential equations beyond the introductory course, developing techniques for solving systems of ODEs using matrix methods and linear algebra.
• Introduce series solutions of differential equations and their role in solving equations arising in applications.
• Explore qualitative and geometric methods for analyzing nonlinear systems, including phase-plane analysis and stability.
• Familiarize students with boundary value problems, eigenvalue problems, and introductory partial differential equations, including connections to Fourier series.
• Develop and analyze mathematical models from the physical and biological sciences using differential equations.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to solve homogeneous and non-homogeneous linear systems of ODEs with constant coefficients using matrix methods, including eigenvalue and eigenvector techniques.
• Students who successfully complete this course will be able to construct power series solutions to ordinary differential equations and determine their convergence.
• Students who successfully complete this course will be able to analyze the stability and qualitative behavior of equilibria for nonlinear systems using phase-plane methods and linearization.
• Students who successfully complete this course will be able to solve boundary value problems and apply separation of variables and Fourier series to elementary partial differential equations.
• Students who successfully complete this course will be able to communicate the setup, solution, and interpretation of differential equations problems clearly in written form.

Course Goals and Objectives:

• Introduce the qualitative theory of nonlinear ordinary differential equations, including phase portraits, stability, and bifurcations.
• Develop students’ ability to analyze the long-term behavior of dynamical systems using both analytical and geometric methods.
• Explore chaotic dynamics, including strange attractors, one-dimensional maps, and routes to chaos.
• Analyze mathematical models arising in physics, biology, and engineering using the language of dynamical systems.
• Build conceptual and computational competency in an area of applied mathematics that connects differential equations, linear algebra, and physical systems.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to classify fixed points of one- and two-dimensional systems, determine their stability, and sketch phase portraits.
• Students who successfully complete this course will be able to identify and analyze bifurcations, including saddle-node, transcritical, pitchfork, and Hopf bifurcations.
• Students who successfully complete this course will be able to apply linearization and qualitative techniques such as the Poincaré–Bendixson theorem to analyze nonlinear systems.
• Students who successfully complete this course will be able to analyze discrete maps and identify hallmarks of chaotic behavior, including sensitivity to initial conditions and period-doubling cascades.
• Students who successfully complete this course will be able to communicate solutions and analyses of dynamical systems clearly in written form using proper notation.

Course Goals and Objectives:

• Introduce the basic theory and solution techniques for first- and second-order partial differential equations, with emphasis on the Fourier method for initial and boundary value problems.
• Develop students’ ability to classify second-order PDEs as hyperbolic, parabolic, or elliptic and to select appropriate solution strategies accordingly.
• Explore separation of variables in rectangular, polar, cylindrical, and spherical coordinate systems, including connections to special functions and orthogonal expansions.
• Develop and analyze mathematical models arising in the physical sciences, including the wave, heat, and Laplace equations.
• Build conceptual and computational competency in an area of mathematics that bridges differential equations, analysis, and applications.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to classify second-order linear PDEs and solve initial and boundary value problems for the wave, heat, and Laplace equations using separation of variables and Fourier series.
• Students who successfully complete this course will be able to apply the method of characteristics to solve first-order PDEs.
• Students who successfully complete this course will be able to formulate and solve Sturm–Liouville eigenvalue problems and use the resulting eigenfunctions in Fourier expansions.
• Students who successfully complete this course will be able to set up and solve PDE problems in polar, cylindrical, and spherical coordinates using appropriate special functions.
• Students who successfully complete this course will be able to communicate the formulation, solution, and interpretation of PDE problems clearly in written form.

Course Goals and Objectives:

• Introduce the basic theory of Fourier series, including convergence properties and the representation of functions by trigonometric series.
• Develop the theory of the Fourier transform on the real line and in higher dimensions.
• Explore connections between Fourier analysis and applications in the physical and medical sciences.
• Build conceptual and computational competency in a core area of applied analysis that bridges pure mathematics and its applications.
• Strengthen students’ ability to work with rigorous analytic arguments involving convergence, approximation, and integral transforms.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to compute Fourier series representations of periodic functions and analyze their convergence properties.
• Students who successfully complete this course will be able to compute and apply the Fourier transform and its inverse to appropriate classes of functions.
• Students who successfully complete this course will be able to apply Fourier analytic techniques to solve problems arising in differential equations and the physical sciences.
• Students who successfully complete this course will be able to work with key theoretical results in Fourier analysis, including convolution, Parseval’s identity, and summability methods.
• Students who successfully complete this course will be able to communicate solutions to problems in Fourier analysis clearly and correctly in written form.

Course Goals and Objectives:

• Introduce the principles and methods of mathematical modeling as applied to biological systems, such as population dynamics, infectious disease, and biochemical kinetics.
• Develop students’ ability to formulate, analyze, and simulate mathematical models using tools such as differential equations, difference equations, and stochastic processes.
• Explore qualitative and quantitative analysis of biological models, such as equilibrium analysis, stability, and parameter interpretation.
• Build hands-on experience with computational tools for simulating and visualizing model behavior.
• Develop and analyze mathematical models of scientific problems, emphasizing the interplay between mathematical reasoning and biological interpretation.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to formulate mathematical models of biological systems using mathematical frameworks appropriate to the application.
• Students who successfully complete this course will be able to analyze the qualitative behavior of biological models, such as determining equilibria, stability, and long-term dynamics.
• Students who successfully complete this course will be able to implement and simulate mathematical models using scientific computing software and interpret the computational results in biological context.
• Students who successfully complete this course will be able to communicate the formulation, analysis, and conclusions of a mathematical model clearly in written solutions.

Course Goals and Objectives:

• Develop the theory of linear algebra rigorously, covering vector spaces, linear transformations, matrices, determinants, eigenvalues, and inner product spaces.
• Further develop students’ ability to read, write, and construct proofs in an abstract algebraic setting.
• Build both conceptual understanding and computational fluency in a subject fundamental to nearly all branches of pure and applied mathematics.
• Foster rigorous and precise reasoning about abstract mathematical structures and their properties.
• Prepare students for advanced coursework in algebra, analysis, and applied fields that rely on linear algebra.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to work with vector spaces and subspaces, including verifying subspace properties, constructing bases, and determining dimension.
• Students who successfully complete this course will be able to represent linear transformations as matrices and use properties of matrices, rank, and nullity to analyze them.
• Students who successfully complete this course will be able to compute and apply determinants, eigenvalues, and eigenvectors to solve problems involving diagonalization and related topics.
• Students who successfully complete this course will be able to construct rigorous proofs of results in linear algebra using appropriate definitions, notation, and logical reasoning.
• Students who successfully complete this course will be able to communicate mathematical reasoning clearly and correctly in written solutions and proofs.

Course Goals and Objectives:

• Deepen students’ knowledge of linear algebra by focusing on topics most essential for data science, including matrix decompositions, orthogonality, and high-dimensional geometry.
• Introduce the theory and numerical methods required for linear problems associated with large data sets and machine learning.
• Develop students’ ability to connect abstract linear algebra concepts to practical data analysis problems.
• Build computational competency through hands-on programming in Python, using standard numerical and data science libraries.
• Prepare students for further study or careers in data science, machine learning, and related quantitative fields.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to apply linear algebra concepts such as orthogonality, matrix decompositions, and dimensionality reduction to analyze real-world data sets.
• Students who successfully complete this course will be able to implement and evaluate data science methods, including clustering, classification, and regression, using Python.
• Students who successfully complete this course will be able to construct and train basic neural network models and explain the role of linear algebra in their operation.
• Students who successfully complete this course will be able to select appropriate linear algebraic methods for a given data analysis task and critically assess the results.
• Students who successfully complete this course will be able to communicate mathematical reasoning and computational findings clearly in written analyses and code documentation.

Course Goals and Objectives:

• Introduce the fundamental concepts and results of elementary number theory, including divisibility, congruences, arithmetic functions, and quadratic reciprocity.
• Develop students’ ability to write rigorous proofs in the setting of number theory.
• Explore the structure of the integers through topics such as the fundamental theorem of arithmetic, the Chinese remainder theorem, and primitive roots.
• Strengthen students’ ability to communicate mathematical ideas effectively in written form.
• Prepare students for further study in algebra, number theory, and related areas of mathematics.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to apply the Euclidean algorithm and the fundamental theorem of arithmetic to solve problems involving divisibility and prime factorization.
• Students who successfully complete this course will be able to solve systems of linear congruences using the Chinese remainder theorem and apply Fermat’s little theorem and Euler’s theorem.
• Students who successfully complete this course will be able to evaluate and manipulate arithmetic functions, including Euler’s totient function and the Möbius function, and apply Möbius inversion.
• Students who successfully complete this course will be able to determine quadratic residues and apply the law of quadratic reciprocity.
• Students who successfully complete this course will be able to construct rigorous proofs of number-theoretic results and communicate mathematical reasoning clearly in written form.

Course Goals and Objectives:

• Introduce the fundamental structures of abstract algebra, focusing on groups, subgroups, homomorphisms, and isomorphisms.
• Develop students’ ability to read, write, and construct proofs in the setting of abstract algebra.
• Explore the role of symmetry and structure through a wide range of examples, including permutation groups, cyclic groups, and factor groups.
• Strengthen students’ capacity to reason rigorously and precisely within abstract algebraic systems.
• Prepare students for further study in algebra and other proof-intensive areas of mathematics.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to state and apply definitions and fundamental results about groups, subgroups, normal subgroups, and factor groups.
• Students who successfully complete this course will be able to construct and verify group homomorphisms and isomorphisms and apply results such as Lagrange’s theorem and the fundamental theorem of finite abelian groups.
• Students who successfully complete this course will be able to work with permutation groups and compute in symmetric and dihedral groups.
• Students who successfully complete this course will be able to construct rigorous proofs of algebraic results using appropriate definitions, notation, and logical reasoning.

Course Goals and Objectives:

• Develop the theory of rings, ideals, and fields, building on the group theory covered in Abstract Algebra I.
• Introduce the fundamental results of Galois theory, including the correspondence between intermediate fields and subgroups of the Galois group.
• Explore classical applications of field theory and Galois theory, including the impossibility of certain ruler-and-compass constructions and the insolvability of the general quintic by radicals.
• Further develop students’ ability to read, write, and construct proofs in abstract algebra.
• Prepare students for graduate study in algebra and related areas of mathematics.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to state and apply fundamental results about rings, ideals, polynomial rings, and field extensions.
• Students who successfully complete this course will be able to compute splitting fields and determine the degree of field extensions, including extensions obtained by adjoining roots of polynomials.
• Students who successfully complete this course will be able to determine Galois groups of field extensions and apply the fundamental theorem of Galois theory to relate subgroups to intermediate fields.
• Students who successfully complete this course will be able to apply Galois-theoretic arguments to prove the impossibility of classical geometric constructions and the insolvability of the general quintic by radicals.
• Students who successfully complete this course will be able to construct rigorous proofs of algebraic results using appropriate definitions, notation, and logical reasoning.

Course Goals and Objectives:

• Introduce the fundamentals of logic, set theory, and proof techniques as a foundation for upper-division mathematics courses.
• Develop students’ ability to think rigorously and critically about mathematical statements and arguments.
• Familiarize students with standard proof strategies, including direct proof, proof by contradiction, proof by contraposition, and mathematical induction.
• Strengthen clear and precise written communication of mathematical reasoning.
• Prepare students for advanced mathematics courses in which constructing and verifying proofs is essential.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to construct direct proofs, indirect proofs, and proofs by mathematical induction.
• Students who successfully complete this course will be able to analyze the logical structure of mathematical statements using propositional and predicate logic.
• Students who successfully complete this course will be able to identify errors in mathematical reasoning and construct counterexamples to disprove false statements.
• Students who successfully complete this course will be able to apply basic concepts involving sets, functions, and relations in proof-based contexts.
• Students who successfully complete this course will be able to communicate mathematical arguments clearly and rigorously in written form.

Course Goals and Objectives:

• Introduce the axioms of Zermelo–Fraenkel set theory and develop set theory as a foundation for mathematics.
• Explore key constructions and results in set theory, including ordinal numbers, cardinal numbers, and the axiom of choice.
• Develop students’ ability to read, write, and present rigorous proofs in an abstract mathematical setting.
• Strengthen students’ capacity to reason precisely within an axiomatic system and to work with transfinite objects.
• Prepare students for further study in logic, foundations of mathematics, and other proof-intensive areas.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to state and apply the axioms of ZFC set theory to construct and verify basic set-theoretic arguments.
• Students who successfully complete this course will be able to work with ordinal and cardinal numbers, including performing transfinite arithmetic and comparing cardinalities using the Schröder–Bernstein theorem.
• Students who successfully complete this course will be able to apply the axiom of choice and equivalent formulations such as Zorn’s lemma in proof-based contexts.
• Students who successfully complete this course will be able to construct rigorous proofs of theorems in set theory using appropriate definitions, notation, and logical reasoning.
• Students who successfully complete this course will be able to communicate mathematical arguments clearly and correctly in written proofs and presentations.

Course Goals and Objectives:

• Introduce the basics of formal logic, including propositional and predicate logic, and their role in the foundations of mathematics.
• Explore models of mathematical theories and the relationship between formal proofs and mathematical truth.
• Develop an understanding of computability, including algorithms, Turing machines, and undecidability.
• Familiarize students with fundamental results in mathematical logic, including Goedel’s incompleteness theorems and their implications.
• Strengthen students’ ability to read, write, and present rigorous mathematical arguments.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to work with formal systems of propositional and predicate logic, including constructing proofs and evaluating the validity of logical arguments.
• Students who successfully complete this course will be able to analyze models of formal theories and determine whether given sentences are true in a given model.
• Students who successfully complete this course will be able to describe the basic theory of computability, including Turing machines, decidable and undecidable problems, and their significance.
• Students who successfully complete this course will be able to explain the statements and significance of Goedel’s completeness and incompleteness theorems.
• Students who successfully complete this course will be able to communicate mathematical reasoning clearly in written proofs and oral presentations.

Course Goals and Objectives:

• Introduce the fundamental concepts of point-set topology, including topological spaces, continuous functions, connectedness, compactness, and separation axioms.
• Develop students’ facility with abstract mathematical structures by studying how topological properties generalize familiar ideas from analysis and metric spaces.
• Explore key constructions in topology, including product spaces, quotient spaces, and metric spaces, and their role in modern mathematics.
• Develop students’ ability to read, write, and construct rigorous proofs and counterexamples in an axiomatic setting.
• Prepare students for graduate study in topology, analysis, and other areas of mathematics that rely on topological foundations.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to state and apply the definitions of topological spaces, bases, continuous functions, and homeomorphisms, and work with product, subspace, and quotient topologies.
• Students who successfully complete this course will be able to determine whether a topological space is connected or compact and apply these properties in proofs.
• Students who successfully complete this course will be able to work with countability and separation axioms and distinguish between key classes of topological spaces using examples and counterexamples.
• Students who successfully complete this course will be able to apply properties of complete metric spaces, including the contraction mapping theorem, in appropriate contexts.
• Students who successfully complete this course will be able to construct rigorous proofs of topological results using appropriate definitions, notation, and logical reasoning.

Course Goals and Objectives:

• Develop advanced topics in general topology, including the Tietze extension theorem, Tychonoff theorem, and Stone–Čech compactification.
• Introduce the central ideas of algebraic topology, including the fundamental group, covering spaces, and the classification of surfaces.
• Explore how algebraic invariants, particularly the fundamental group, can be used to distinguish topological spaces and answer geometric questions.
• Further develop students’ ability to read, write, and construct rigorous proofs and counterexamples in topology.
• Prepare students for graduate study in topology, geometry, and related areas of mathematics.

Expectations and Student Learning Outcomes:

• Students who successfully complete this course will be able to state and apply key results in general topology, including the Tietze extension theorem and the Tychonoff theorem.
• Students who successfully complete this course will be able to compute the fundamental group of topological spaces using techniques such as covering spaces and the Seifert–van Kampen theorem.
• Students who successfully complete this course will be able to work with covering spaces, including lifting properties and the correspondence between covering spaces and subgroups of the fundamental group.
• Students who successfully complete this course will be able to apply the classification theorem for compact surfaces to identify and distinguish surfaces.
• Students who successfully complete this course will be able to construct rigorous proofs of topological and algebraic-topological results using appropriate definitions, notation, and logical reasoning.