This task presents an incomplete problem and asks students to choose numbers …
This task presents an incomplete problem and asks students to choose numbers to subtract (subtrahends) so that the resulting problem requires different types of regrouping. This way students have to recognize the pattern and not just follow a memorized algorithm--in other words, they have to think about what happens in the subtraction process when we regroup. This task is appropriate to use after students have learned the standard US algorithm.
The purpose of this task is to engage students in geometric modeling, …
The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints. The modelling process is a challenging one, and will likely elicit a variety of attempts from the students.
The course consists of a sampling of topics from algebraic combinatorics. The …
The course consists of a sampling of topics from algebraic combinatorics. The topics include the matrix-tree theorem and other applications of linear algebra, applications of commutative and exterior algebra to counting faces of simplicial complexes, and applications of algebra to tilings.
The main aims of this seminar will be to go over the …
The main aims of this seminar will be to go over the classification of surfaces (Enriques-Castelnuovo for characteristic zero, Bombieri-Mumford for characteristic p), while working out plenty of examples, and treating their geometry and arithmetic as far as possible.
Topics vary from year to year. Fall Term: Numerical properties and vanish …
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 …
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 …
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 …
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 …
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 …
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 …
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, …
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 …
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 …
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.
Students overlay USGS topographic maps into Google Earth’s satellite imagery. By analyzing …
Students overlay USGS topographic maps into Google Earth’s satellite imagery. By analyzing Denali, a mountain in Alaska, they discover how to use map scales as ratios to navigate maps, and use rates to make sense of contour lines and elevation changes in an integrated GIS software program. Students also problem solve to find potential pathways up a mountain by calculating gradients.
Student teams investigate the migration of small-particle plastic pollution by exposing invertebrates …
Student teams investigate the migration of small-particle plastic pollution by exposing invertebrates found in water samples from a local lake or river to fluorescent bead fragments in a controlled environment of their own designs. Students begin by reviewing the composition of food webs and considering the ethics of studies on live organisms. In their model microcosms, they set up a food web so as to trace the microbead migration from one invertebrate species to another. Students use blacklights and microscopes to observe and quantify their experimental results. They develop diagrams that explain their investigations—modeling the ecological impacts of microplastics.
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