Applied Algebra and Geometry Third meeting
21 September 2017
L3 Mathematical Institute,
Andrew Wiles Building,
University of Oxford,
Radcliffe Observary Quarter,
Oxford OX2 6GG.
- 10:00-11:00 Welcome and coffee/tea/juice
- 11:00-12:00 Gavin Brown
- 12:00-13:15 Lunch (at the Pi Cafe)
- 13:15-14:15 Hamid Ahmadinezhad: Linkages with screw motion
- 14:15-15:15 Kevin Houston: Applications of sparsity methods within geometry
- 15:15-15:45 Coffee/tea/juice
- 15:45-16:45 Elizabeth Baldwin
Gavin Brown: Flops from Calabi-Yau potentials
I will describe joint work with Michael Wemyss that applies recent advances in the theory of noncommutative deformations to the construction of apparently simple but elusive structures in 3-dimensional complex geometry, namely neighbourhoods of smooth contractible rational curves on smooth varieties. I will explain a bit of the history of the latter, and our place in that development, as well as the role of the modern theory.
Hamid Ahmadinezhad: Linkages with screw motion
The mobility of a linkage, a closed chain of robot arms connected via joints, can be decided using (complex) algebraic manipulation if the joints only rotate. For joints with screw motion the relations become transcendental. I will talk about all these and explain how to study the mobility problem for the screw case using known results in number theory.
Kevin Houston: Applications of Sparsity Methods within Geometry
Sparsity assumptions are useful in real world applications. For example, in picture compression systems such as JPEG and in compressive sensing one can make a 1 pixel camera. An important problem in applied geometry is to produce bijective maps between scanned objects as this allows us to compare them. We would like these bijections to be in some discrete sense continuous or differentiable. In this talk, I will look at the search for a sparse basis similar to an eigenfunction basis that can be used in this situation and many others.
Elisabeth Baldwin: Geometry in Consumer Preferences: Equilibrium and Auction Design
(with Paul Klemperer and Paul Goldberg) This talk covers several papers in which we introduce techniques from tropical geometry into economics. Specifically, we gain new insights into the structure of consumer preferences when goods are indivisible, allowing a new system of classification that is both mathematically straightforward and economically meaningful. These insights in turn lead to new theorems on competitive equilibrium. And they allow us to extend the design of auctions in which multiple differentiated products are sold simultaneously -- auctions not only interesting in theory but also have already been put to real-world use.