## Description

Supported by the NSF, this mini series of lectures will be held in the Mathematics Department at Tulane University, New Orleans, from January 13 to January 15, 2012. The principal speaker will be Professor Frederick Cohen from the University of Rochester, who will deliver three lectures centered on the topics of the title. The lectures will be aimed at the graduate students audience. In addition there will be lectures given by

• Ryan Budney (University of Victoria),
• Daniel Cohen (Louisiana State University),
• Elizabeth Denne (Smith College),
• Daniel Koditschek (University of Pennsylvania),
• Laurence Taylor (University of Notre Dame),
• Ismar Volic (Wellesley College).

Abstract: Algebraic properties of configuration spaces frequently reflect properties of various geometric problems which are ubiquitous in nature. These types of structures are directly connected to several subjects such as knots, links, homotopy groups as well as the structure of function spaces. There is also a deep interest in these objects in the engineering sciences, especially in problems such as motion planning for robots. Two of the lectures in the series will present these fascinating connections. Throughout the series, open problems, conjectures, and new directions will be discussed.

## Outline

The main subject of these lectures are the structure and applications of configurations spaces of ordered $$q$$-tuples of distinct points in a space $$M$$ denoted $$Conf(M,q) = \{(m_1, \ldots, m_q)\ |\ m_i \neq m_j \ \hbox {if }\ i \neq j\}.$$ These spaces have been the subject of intense applications in many areas which trace back to Ptolemy as well as Gauss. These lectures will give definitions, together with elementary as well as more sophisticated properties for classical configuration spaces

For example, a picture of a braid is a loop in a configuration space of points in the plane. Thus the space of braids can be thought of as the space of (pointed) loops for the configuration space. On the other-hand, spaces of loops measure collisions and linking phenomena in several ways which will be described as well as where and how these subjects fit.
• The initial setting of classical applications to collisions/ non-collisions, Borsuk-Ulam Theorem, and coincidence points will be an introduction. Further versions of collisions, non-collisions and coincidences such as Borromean braids and links are given as well as how these structures provide a disguise for classical homotopy groups.
• Geometric and algebraic invariants given by cohomology rings, homotopy groups as well as Pontrjagin rings of pointed loop spaces will be developed. These were first encountered by Ptolemy in construction of epicycles while trying to describe motions of planetary objects. These epicycles in turn give rise to 'tautologous cohomology classes' present in the cohomology of configuration spaces for 'most' manifolds.
• Basic geometric properties such as fibration sequences, cofibration sequences, Thom spaces, and the cohomology of configuration spaces will be developed. Cohomology algebras will be given in many cases with some examples given by $$M = N \times \mathbb{R}$$ . Some of the natural connections to classical representation theory and early work of Hall, and Witt will be described. Further connections to early work of Milnor as well as work of Kohno on Vassiliev invariants will be given.
• Stable decompositions of these and similar spaces together with some natural applications to other spaces such as spaces of packings will be developed. Some examples with configuration spaces having simple singularities will be given.
• Properties of pure braid groups. We discuss a number of properties of the Artin pure braid group, with particular attention paid to the following question attributed to Kervaire: How do you know the pure braid group is not a direct product of free groups?
• Topological complexity of configuration spaces. The topological complexity of a space is a homotopy type invariant motivated by the motion planning problem from robotics. We discuss this invariant in the context of various configuration spaces, including configuration spaces of ordered points on orientable surfaces and toric complexes.
• There is also a deep interest in these objects in the engineering sciences, especially in problems such as motion planning for robots. Two of the lectures in the series will present these fascinating connections.

## Program

All lectures will take place in the conference room 103 at Dinwiddie Hall, see the situation map or Bldg #3 on the campus map.

 January 13 (Friday) January 14 (Saturday) January 15 (Sunday) 9:00-9:50 Name tags and coffee 8:30-9:00  Name tags and coffee 8:30-9:00  Name tags and coffee 9:50-10:00  Opening address 9:00-10:00  Dan Cohen 9:00-10:00 Dan Koditschek 10:00-11:00  Fred Cohen 30 min break 30 min break 30 min break 10:30-11:30  Fred Cohen 10:30-11:30 Dan Koditschek 11:30-12:30  Larry Taylor Lunch Lunch Lunch 1:00-2:00  Larry Taylor 1:00-2:00  Fred Cohen 2:00-3:00  Dan Cohen 30 min break Farewell coffee 30 min break 2:30-3:30  Ismar Volic departure 3:30-4:30 Elizabeth Denne 30 min break 4:00-5:00  Ryan Budney 5:30-8:00 dinner social at the faculty club*

*Faculty club is located on the 2nd floor of Lavin-Bernick Center for University Life (Bldg #29 on the campus map) also see the situation map

## Registration and financial support

Application for the finantial support is now closed. However anybody is welcome to attend, there is no registration fee, you will be able to register and obtain a name tag at the conference.

## Accomodations

Participants whose lodging will be arranged by the conference will stay in

Other recommended hotels are

## Transportation

The Louis Armstrong New Orleans International Airport - MSY offers a shuttle service to hotels including Parkview Hotel , the shuttle stop is located outside the baggage claim area. Taxi fare from/to the airport should not exceed \$30. The Mathematics Department of Tulane is located at 6823 St. Charles Ave.