Courses tagged with "Mathematics" (283)
This course begins a series of classes illustrating the power of computing in modern biology. Please join us on the frontier of bioinformatics to look for hidden messages in DNA without ever needing to put on a lab coat. After warming up our algorithmic muscles, we will learn how to apply popular bioinformatics software tools to real experimental datasets.
In this class, we will compare DNA from an individual against a reference human genome to find potentially disease-causing mutations. We will also learn how to identify the function of a protein even if it has been bombarded by so many mutations compared to similar proteins with known functions that it has become barely recognizable.
This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted.
This course offers biweekly problem sets with solutions, two term tests and a final exam, all with solutions.
Biologists still cannot read the nucleotides of an entire genome as you would read a book from beginning to end. However, they can read short pieces of DNA. In this course, we will see how graph theory can be used to assemble genomes from these short pieces. We will further learn about brute force algorithms and apply them to sequencing mini-proteins called antibiotics. Finally, you will learn how to apply popular bioinformatics software tools to sequence the genome of a deadly Staphylococcus bacterium.
How do we infer which genes orchestrate various processes in the cell? How did humans migrate out of Africa and spread around the world? In this class, we will see that these two seemingly different questions can be addressed using similar algorithmic and machine learning techniques arising from the general problem of dividing data points into distinct clusters.
Geometry and Quantum Field Theory, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals. It covers the basics of classical field theory, free quantum theories and Feynman diagrams. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and String Theory.
This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations.
This is a literature seminar with a focus on classic papers in Algebraic Topology. It is named after the late MIT professor Daniel Kan. Each student gives one or two talks on each of three papers, chosen in consultation with the instructor, reads all the papers presented by other students, and writes reactions to the papers. This course is useful not only to students pursuing algebraic topology as a field of study, but also to those interested in symplectic geometry, representation theory, and combinatorics.
Une fonction discontinue peut-elle être solution d'une équation différentielle? Comment définir rigoureusement la masse de Dirac (une "fonction" d'intégrale un, nulle partout sauf en un point) et ses dérivées? Peut-on définir une notion de "dérivée d'ordre fractionnaire"? Cette initiation aux distributions répond à ces questions - et à bien d'autres.
This course emphasizes concepts and techniques for solving integral equations from an applied mathematics perspective. Material is selected from the following topics: Volterra and Fredholm equations, Fredholm theory, the Hilbert-Schmidt theorem; Wiener-Hopf Method; Wiener-Hopf Method and partial differential equations; the Hilbert Problem and singular integral equations of Cauchy type; inverse scattering transform; and group theory. Examples are taken from fluid and solid mechanics, acoustics, quantum mechanics, and other applications.
This graduate-level course covers fluid systems dominated by the influence of interfacial tension. The roles of curvature pressure and Marangoni stress are elucidated in a variety of fluid systems. Particular attention is given to drops and bubbles, soap films and minimal surfaces, wetting phenomena, water-repellency, surfactants, Marangoni flows, capillary origami and contact line dynamics.