Global Arc

A study of the emergence of a distinctive Western European civilization out of Christian, Greco-Roman, and Germanic institutions and ideas from the decline of the Roman Empire to about A.D. 1050. Two lectures, one preceptorial.

An analysis of typical institutions, social and economic structures, and forms of thought and expression from about 1050 to about 1350. Emphasis is placed on the elements of medieval civilization that have influenced the subsequent history of European peoples. Two lectures, one preceptorial.

The Crusades were a central phenomenon of the Middle Ages. This course examines the origins and development of the Crusades and the Crusader States in the Islamic East. It explores dramatic events, such as the great Siege of Jerusalem, and introduces vivid personalities, including Richard the Lionheart and Saladin. We will consider aspects of institutional, economic, social and cultural history and compare medieval Christian (Western and Byzantine), Muslim and Jewish perceptions of the crusading movement. Finally, we will critically examine the resonance the movement continues to have in current political and ideological debates

Introduction to geometry of surfaces. Surfaces in Euclidean space, second fundamental form, minimal surfaces, geodesics, Gauss curvature, Gauss-Gonnet formula, uniformization of surfaces, elementary notions of contact geometry. Prerequisite: MAT218 or MAT300, or MAT203 or equivalent.

Introduction to point-set topology, the fundamental group, covering spaces, methods of calculation and applications. Prerequisite: MAT202 or 204 or 218 or equivalent.

The fundamental theorems and algorithms of graph theory. Topics include: connectivity, matchings, graph coloring, planarity, the four-color theorem, extremal problems, network flows, and related algorithms. Prerequisite: MAT202 or 204 or 217 or equivalent.

Combinatorics is the study of enumeration and structure of discrete objects. These structures are widespread throughout mathematics, including geometry, topology and algebra, as well as computer science, physics and optimization. This course will give an introduction to modern techniques in the field, and how they relate to objects such as polytopes, permutations and hyperplane arrangements.

Games in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations, solution concepts; related topics and applications. Prerequisite: MAT202 or 204 or 217 or equivalent. MAT215 or equivalent is recommended.

Sequence of independent trials, applications to number theory and analysis, Monte Carlo method. Markov chains, ergodic theorem for Markov chains. Entropy and McMillan theorem. Random walks, recurrence and non-recurrence; connection with the linear difference equations. Strong laws of large numbers, random series and products. Weak convergence of probability measures, weak Helly theorems, Fourier transforms of distributions. Limit theorems of probability theory. Prerequisite: MAT203 or 218 or equivalent.

Linear programs, duality, Dantzig's simplex method; theory of dual linear systems; matrix games, von Neumann's minimax theorem, simplex solution; algorithms for assignment, transport, flow; brief introduction to nonlinear programming.