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Series: Other Talks

After a few remarks to tie up some loose ends from last week's talk on locally
ringed spaces, I will discuss exact sequences of sheaves and give some natural
examples coming from real, complex, and algebraic geometry. In the context of these
examples, we'll see that a surjective map of sheaves (meaning a morphism of sheaves
which is surjective on the level of stalks) need not be surjective on global
sections. This observation will be used to motivate the need for "sheaf cohomology"
(which will be discussed in detail in subsequent talks).

Series: Other Talks

I will discuss how various geometric categories (e.g. smooth manifolds, complex manifolds) can be be described in terms of locally ringed spaces. (A locally ringed space is a topological spaces endowed with a sheaf of rings whose stalks are local rings.) As an application of the notion of locally ringed space, I'll define what a scheme is.

Series: Other Talks

We discuss the convergence properties of first-order methods for two problems that
arise in computational geometry and statistics: the minimum-volume enclosing ellipsoid problem
and the minimum-area enclosing ellipsoidal cylinder problem for a set of m points in R^n.
The algorithms are old but the analysis is new, and the methods are remarkably effective
at solving large-scale problems to high accuracy.

Series: Other Talks

In these talks we will introduced the basic definitions and examples of presheaves, sheaves and sheaf spaces. We will also explore various constructions and properties of these objects.

Series: Other Talks

In this talk Professor Tapia identifies elementary mathematical frameworks for the study of popular drag racing beliefs. In this manner some myths are validated while others are destroyed. Tapia will explain why dragster acceleration is greater than the acceleration due to gravity, an age old inconsistency. His "Fundamental Theorem of Drag Racing" will be presented. The first part of the talk will be a historical account of the development of drag racing and will include several lively videos.

Series: Other Talks

Series: Other Talks

Series: Other Talks

This will be an informal seminar with a discussion on some mathematical problems in relativistic astrophysics, and discuss plans for future joint seminars between the Schools of Mathematics and Physics.

Series: Other Talks

An old conjecture of Erdos and Szemeredi states that if A is a finite set of integers then the sum-set or the product-set should be large. The sum-set of A is A + A={a+b | a,b \in A\}, and the product set is defined in a similar way, A*A={ab | a,b \in A}. Erdos and Szemeredi conjectured that the sum-set or the product set is almost quadratic in |A|, i.e. max(|A+A|,|A*A|)> c|A|^{2-\epsilon}. In this talk we review some recent developments and problems related to the conjecture.

Series: Other Talks

Twistor theory is now over 45 years old. In December 1963, I proposed the initial ideas of this scheme, based on complex-number geometry, which presents an alternative perspective to that of standard 4-dimensional space-time, for the basic arena in which (quantum) physics takes place. Over the succeeding years, there were numerous intriguing developments. But many of these were primarily mathematical, and there was little interest expressed by the physics community. Things changed rather dramatically, in December 2003, when E. Witten produced a 99-page article initiating the subject of “twistor-string theory” this providing a novel approach to high-energy scattering processes. In this talk, I shall provide an account of the original geometrical and physical ideas, and also outline various recent developments, some of which may help our understandings of the seeming paradoxes of quantum mechanics.