Catalogue description: Church's thesis and models of computation. Elementary computability theory: enumeration and recursion theorems, the halting problem, relative computability, Turing degrees, and basic priority constructions. Post's problem. Truth vs. provability, Godel's incompleteness theorem. Decidable and undecidable problems in number theory and other areas of mathematics.
Steven is studying finite automata presentable groups. In particular, he is working towards a structural characterisation of the Thurston automatic groups (finitely generated groups whose Cayley graphs are recognisable by finite automata).
Anna learned about finite automata presentable sets of strings and sets of trees. She provided connections between the definitions of labelled trees and finite automata on strings. Throughout, cardinality questions came up and Anna became more familiar with arguments involving countable and uncountable sets.
Students present and discuss the subject matter taken from current journals or books. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.
This seminar will explore the important notions of Algorithmic Randomness and Kolmogorov Complexity. Algorithmic Randomness describes what it means for a string of bits to be random using notions from computability theory, information theory, and probability theory. The Kolmogorov Complexity of a string is its intrinsic information and is defined in terms of incompressibility. This seminar will explore these important notions and their applications. Possible topics for student projects include coding theory, learning theory, alternative notions of randomness, and entropy in physics.
Six-week review of one-variable calculus, emphasizing material not on the high-school AB syllabus: integration techniques and applications, polar coordinates, improper integrals, infinite series. Prerequisites: one year of high-school calculus or the equivalent, with a score of 4 or 5 on the AB Calculus test (or the AB portion of the BC test, or an equivalent score on a standard international exam), or equivalent college transfer credit, or a passing grade on the first half of the 18.01 advanced standing exam.
First half of multivariable calculus taught during the last six weeks of the Fall semester. Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals.
Topics include functions and graphs, limits and continuity, differentiation and integration of algebraic, trigonometric, inverse trig, logarithmic, and exponential functions; applications of differentiation, including graphing, max-min problems, tangent line approximation, implicit differentiation, and applications to the sciences; the mean value theorem; and antiderivatives, definite and indefinite integrals, the fundamental theorem of calculus, substitution in integration, the area under a curve.
Differential and integral calculus of functions in several variables, line and surface integrals as well as the theorems of Green, Stokes and Gauss.
First course in mathematical logic providing precise definitions of the language of mathematics and the notion of proof (propositional and predicate logic). The completeness theorem says that we have all the rules of proof we could ever have. The Godel incompleteness theorem says that they are not enough to decide all statements even about arithmetic. The compactness theorem exploits the finiteness of proofs to show that theories have unintended (nonstandard) models. Possible additional topics: the mathematical definition of an algorithm and the existence of noncomputable functions; the basics of set theory to cardinality and the uncountability of the real numbers.
Functions, limits, derivatives, continuity; applications to optimization, rate problems and curve sketching; mean value theorem; anti-derivatives, fundamental theorem; Riemann integral; applications to areas, volumes, arc length, work, pressure and force; inverse functions; inverse trigonometric functions; exponential, logarithmic and hyperbolic functions.
More integration techniques; numerical integration, improper integrals. Curves, speed, velocity. Functions of several variables, partial derivatives, differentials, error estimates, gradient, maxima and minima. Sequences, series, power series; Taylor polynomial approximations. Double and triple integrals, polar and cylindrical coordinates; applications to mass, center of mass, moment, etc.
During my first semester at MIT, my class was videotaped and I reviewed the tape with a senior faculty member. This provided an opportunity for reflecting on my growth as a teacher since I had been taped once before as part of the TA training program at Cornell.
This seminar provides an opportunity for all members of the Cornell math department to discuss teaching strategies, philosophy, and content-specific questions. I led seminar meetings on topics such as groupwork in tutorials/ recitations and helping students and teachers find the resources best suited to them.
I led a mentoring program for first-time TAs in the math department. We met regularly to help the new TAs overcome standard challenges (grading, time management) and to provide support.
I helped plan and run the three day workshop for incoming graduate students TAs. This included sessions on topics such as "Professionalism" and "How to run a recitation" and microteaching, where I observed and gave feedback on sample recitations given by the new TAs.
As part of this committee, I participated in decisions regarding the courses in the mathematics & engineering program.