The focus of the lecture and homework components of this year's edition of the course will be to give students hands-on experience with a full autonomous robotic vehicle navigation stack. Student projects can cover a range of topics including theoretical, computational, hardware or even user studies, using simulation and/or real robots. Note unusual timing: Tues 12:30 - 2 & Fri 10:30 - noon.

This course covers a few topics in algorithm analysis, including:
- Rounding of linear programs in the design of approximation algorithms
- Randomized rounding of semidefinite programs
- Metric embedding
- Optimal stopping time problems

This course will give a thorough introduction to the special class of discrete optimization known as submodular optimization. We are given a ground set V and a set function f which assigns a real value f(S) to each subset of V. Such a function is submodular if it has the diminishing returns property: f(A+i)-f(A) \geq f(B+i)-f(B) whenever A is a subset of B.

We study models and algorithms of the form

min/max f(S): S \in \mathcal{F}

where \mathcal{F} denotes a structured family of "feasible" sets
(such as shortest paths, matchings etc.).

In the past two decades scientists have begun to formulate and build a new type of computer called a quantum computer. Immense gains in computational power have been shown to be possible with these kinds of computers, and the first commercial quantum computers are starting to appear. This class will teach students the fundamental aspects and applications of quantum computers.



We will begin by understanding the postulates of quantum mechanics and the matrix framework of quantum information science. This will expose some of the most bizarre concepts in quantum theory that are actually crucial to the operation of quantum computers: the uncertainty principle, quantum measurement, entanglement, and spooky action at a distance. We will apply this framework to study quantum cryptography, quantum algorithms, and quantum error correction, and we will make use of development kits from IBM’s quantum experience program. Finally, we learn about the implementation of quantum computers, including aspects such as decoherence, control, and quantum logic.

This is a course for graduate-level students in the Sciences, Engineering or Mathematics interested in fluid dynamics relevant to the large-scale low-frequency motions in strongly rotating fluid systems like planetary atmospheres, the ocean and liquid planetary core. By the end of the course, students should be able to 1. write down the 'standard equations' of geophysical fluid dynamics (GFD), identify the different terms, evaluate their relative importance based on scaling arguments, and explain how different dynamical features depend on these terms; 2. define standard terms and concepts used in GFD (the "language'' of GFD), and identify them when they arise in the context of dynamical interpretations; 3. use standard mathematical techniques to simplify complex equation sets relevant to GFD; and 4. use the appropriate approximations and mathematical techniques to simplify and solve particular ``canonical'' GFD problems. A background in fluid dynamics, geophysics, atmospheric sciences, and/or oceanography is not required, however this course is mathematical and assumes a working knowledge of vector calculus (e.g. div, grad, curl), partial differential equations (i.e. you can solve at least some of them), and some exposure to complex analysis (e.g. you know that if $z=x+iy$, then $e^z = e^x\cos(y)+ie^x\sin(y)$).

In this course we formulate and analyze continuum-based
mathematical models of phenomena in a wide range of areas of
application. A list of the topics for the course is given below. The
mathematical modeling typically leads to PDE/ODE models, free and
moving boundary value problems, etc. For each topic covered I plan to
give some insight into the formulation of appropriate mathematical
models from the underlying physics. Wherever possible, asymptotic
analysis is then used to systematically simplify the underlying model
into a more tractable form. Asymptotic and numerical methods, based
on MATLAB, will then be used to provide insight into the solution
behavior. Finally, for each topic considered, we will examine a recent
journal article (either in an Applied Math or Engineering journal)
that is closely related to the topic.

Topics to be covered:
1) Materials Science Modeling: MEMS modeling, theroelastic contact
2) Lubrication theory and slow viscous flow phenomena;
singularity formation
3) Bifurcation problems in combustion theory and nonlinear heat transfer
4) Nonlinear Oscillators: Forced Oscillators, Entrainment,
Stick-Slip Oscillations
5) Dynamical Hysteresis for ODE's; Slow passage problems
6) Floquet Theory: stability of periodic solutions
7) Delay-Differential Equations; Machine-Tool Vibrations, Car-Following
Models

This graduate course will run in term II, taught by Leah Edelstein-Keshet. While the main area of application will be cell biology, the mathematical methods will include dimensional analysis, ODEs, PDES, reaction -diffusion systems, non-local (integro-PDE) equations, stability analysis, and pattern formation - applicable to a wide variety of scientific problems.
Days, times, and website TBA

New measurement technologies like single-cell RNA sequencing are bringing ‘big data’ to biology. This course introduces a mathematical framework for thinking about questions like: How does a stem cell transform into a muscle cell, a skin cell, or a neuron? How do cell types destabilize in diseases like cancer? Can we reprogram a skin cell into a stem cell? We will learn how to model developing organisms as stochastic processes in gene expression space. We will cover random matrices • stochastic processes • entropy • optimal transport • convex optimization • duality • gradient flows • geodesic interpolation • and developmental genetics.

Advanced course on analytical dynamics describing the kinematics and kinetics of isolated and connected rigid bodies.

Elementary introduction to the theory of groups and Lie algebras, representation theory and applications to problems in quantum mechanics.

Introduction to quantum field theory up to Feynman diagrams, elementary renormalization theory and computing fundamental processes in electrodynamics.