Topics vary from year to year. Fall Term: Numerical properties and vanish theorems for ample, nef, and big line bundles and vector bundles; multiplier ideals and their applications

This course provides an introduction to algebraic number theory. Topics covered include dedekind domains, unique factorization of prime ideals, number fields, splitting of primes, class group, lattice methods, finiteness of the class number, Dirichlet's units theorem, local fields, ramification, discriminants.

Content varies from year to year. Introduces new and significant developments in algebraic topology with the focus on homotopy theory and related areas. Spring 2003: An introduction to higher algebraic K-theory.

In this graduate-level course, we will be covering advanced topics in combinatorial optimization. We will start with non-bipartite matchings and cover many results extending the fundamental results of matchings, flows and matroids. The emphasis is on the derivation of purely combinatorial results, including min-max relations, and not so much on the corresponding algorithmic questions of how to find such objects. The intended audience consists of Ph.D. students interested in optimization, combinatorics, or combinatorial algorithms.

This is an introductory (i.e. first year graduate students are welcome and expected) course in generalized geometry, with a special emphasis on Dirac geometry, as developed by Courant, Weinstein, and Severa, as well as generalized complex geometry, as introduced by Hitchin. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry. For this reason, the latter is intimately related to the ideas of mirror symmetry.

This course will focus on various aspects of mirror symmetry. It is aimed at students who already have some basic knowledge in symplectic and complex geometry (18.966, or equivalent). The geometric concepts needed to formulate various mathematical versions of mirror symmetry will be introduced along the way, in variable levels of detail and rigor.

This course will give a detailed introduction to the theory of tensor categories and review some of its connections to other subjects (with a focus on representation-theoretic applications). In particular, we will discuss categorifications of such notions from ring theory as: module, morphism of modules, Morita equivalence of rings, commutative ring, the center of a ring, the centralizer of a subring, the double centralizer property, graded ring, etc.

Geometry of pseudoconvex domains, the Monge-Ampere equation, Hodge theory on Kaehler manifolds, the theory of toric varieties and (time permitting) some applications to combinatorics.

This graduate-level course focuses on one-dimensional nonparametric statistics developed mainly from around 1945 and deals with order statistics and ranks, allowing very general distributions. For multidimensional nonparametric statistics, an early approach was to choose a fixed coordinate system and work with order statistics and ranks in each coordinate. A more modern method, to be followed in this course, is to look for rotationally or affine invariant procedures. These can be based on empirical processes as in computer learning theory. Robustness, which developed mainly from around 1964, provides methods that are resistant to errors or outliers in the data, which can be arbitrarily large. Nonparametric methods tend to be robust.

The main goal of this course is to study the generalization ability of a number of popular machine learning algorithms such as boosting, support vector machines and neural networks. Topics include Vapnik-Chervonenkis theory, concentration inequalities in product spaces, and other elements of empirical process theory.

This lesson introduces students to the “Tragedy of the Commons,” an extended metaphor for problems of shared environmental or man-made resources that are overused and eventually depleted. In this metaphor, shared resources are compared to a common grazing pasture, or “commons,” on which any dairy farmer can graze as many cows as he/she wishes. If too many cows are added to the commons, they will overeat the grass in the pasture and the shared resource will become depleted – a disadvantage to everyone. In this lesson, students will be inspired to think about possible solutions to this problem. To get there, they will use basic math to frame the problem and will discover how useful this can be in considering consequences of various actions. Most importantly, they will become comfortable with the concept of problems of shared resources – and will learn to recognize, and seek out, examples all around them. An exposure to algebra 1 and basic functions is the only math prerequisite necessary. The lesson will take around 50 minutes to complete and the required materials for this lesson are paper and pens or pencils, as well as some sort of prize to provide the winning team with in the final activity. For all five activities, students are asked to work in groups of 4, but groups of 3 or 5 would also be okay. Students will work with their groups to discuss the logic behind the tragedy of the commons, to consider some options for preventing this tragedy and to examine examples of problems of shared resources that are relevant to them. They will also come up with functions that fit behavior described in the video, and be asked to think about the behavior of functions provided in the video and accompanying materials.

Examines a number of famous trials in European and American history. Considers the salient issues (political, social, cultural) of several trials, the ways in which each trial was constructed and covered in public discussion at the time, the ways in which legal reasoning and storytelling interacted in each trial and in later retellings of the trial, and the ways in which trials serve as both spectacle and a forum for moral and political reasoning. Students have an opportunity to study one trial in depth and present their findings to the class.

This course provides an introduction to probability and statistics, with emphasis on engineering applications. Course topics include events and their probability, the Total Probability and Bayes' Theorems, discrete and continuous random variables and vectors, uncertainty propagation and conditional analysis. Second-moment representation of uncertainty, random sampling, estimation of distribution parameters (method of moments, maximum likelihood, Bayesian estimation), and simple and multiple linear regression. Concepts illustrated with examples from various areas of engineering and everyday life.

it would be ideal if students already have learned that DNA is the genetic material, and that DNA is made up of As, Ts, Gs, and Cs. It also would help if students already know that each human has two versions of every piece of DNA in their genome, one from mom and one from dad. The lesson will take about one class period, with roughly 30 minutes of footage and 30 minutes of activities.

This video is meant to be a fun, hands-on session that gets students to think hard about how machines work. It teaches them the connection between the geometry that they study and the kinematics that engineers use -- explaining that kinematics is simply geometry in motion. In this lesson, geometry will be used in a way that students are not used to. Materials necessary for the hands-on activities include two options: pegboard, nails/screws and a small saw; or colored construction paper, thumbtacks and scissors. Some in-class activities for the breaks between the video segments include: exploring the role of geometry in a slider-crank mechanism; determining at which point to locate a joint or bearing in a mechanism; recognizing useful mechanisms in the students' communities that employ the same guided motion they have been studying.

The major purpose of this lesson is to promote the learning of eye function by associating eye problems and diseases to parts of the eye that are affected. Included in this module are discussions and activities that teach about eye components and their functions. The main activity is dissecting a cow eye, which in many high schools is part of the anatomy curriculum. This lesson extends the curriculum by discussing eye diseases that students might be familiar with. An added fun part of the lesson is discussion of what various animals see.

This video lesson will introduce students to algorithmic thinking through the use of a popular field in graph theory—social networking. Specifically, by acting as nodes in a graph (i.e. people in a social network), the students will experientially gain an understanding of graph theory terminology and distance in a graph (i.e. number of introductions required to meet a target person). Once the idea of distance in a graph has been built, the students will discover Dijkstra's Algorithm. The lesson should take approximately 90 minutes and can be comfortably partitioned across two class sessions if necessary (see the note in the accompanying Teacher Guide). There are no special supplies needed for this class and all necessary hand-outs can be downloaded from this website.

Beavers are generally known as the engineers of the animal world. In fact the beaver is MIT's mascot! But honeybees might be better engineers than beavers! And in this lesson involving geometry in interesting ways, you'll see why! Honeybees, over time, have optimized the design of their beehives. Mathematicians can do no better. In this lesson, students will learn how to find the areas of shapes (triangles, squares, hexagons) in terms of the radius of a circle drawn inside of these shapes. They will also learn to compare those shapes to see which one is the most efficient for beehives. This lesson also discusses the three-dimensional shape of the honeycomb and shows how bees have optimized that in multiple dimensions. During classroom breaks, students will do active learning around the mathematics involved in this engineering expertise of honeybees. Students should be conversant in geometry, and a little calculus and differential equations would help, but not mandatory.

The aim of this lesson is to introduce the concept of Neutralization and its application in our daily lives. Students are encouraged to construct their knowledge of Neutralization through brainstorming sessions, experiments, and mind mapping. This video lesson presents a series of stories relating to Neutralization—beginning with a story of a girl being cured from a stomach ache with the help of Neutralization. Prerequisites for this lesson are knowledge of the basic concept of Neutralization, chemical equations and the pH indicator scale. The lesson will take about 50 minutes to complete, but you may want to divide into two classes if the activities require more time.

This lesson is an introduction to Multiple Regression Analysis or MRA, a statistical process used widely in many professions to estimate the relationship among variables. The aim of this video is to make it easier for students to understand the introduction to the concept of MRA based upon a property valuation setting. In order to facilitate students’ understanding of this, a scaffolding method is used whereby students are first exposed to basic equations. Then they will be introduced to the concept of variables, teaching them to calculate property value based on only 2 variables. Their understanding is further enhanced by exposing them to multiple variables related to property valuation. Finally, they are asked to calculate property value based on multiple variables. It is shown in this video that finding the value of two variables is possible using the paired comparison method, but that the same method cannot be applied if we have more than 2 equations; that is when Multiple Regression Analysis is needed. MRA can solve problems related to more than 2 equations. A prerequisite for students is an understanding of basic statistics such as total, average, mod, mean and median.