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2006 IMA PI Summer Program for Graduate Students
Topology and its Applications
July 10 - 28, 2006 (Travel Days: 7/9 and 7/29)


IMA Participating Institutions Graduate Students Summer Programs
Letter to Graduate Students:    pdf    word doc

Mississippi State University, Starkville

Lead Organizer:

Kevin P. Knudson
Department of Mathematics and Statistics
Mississippi State University
http://www2.msstate.edu/~kk116/

Speakers:

Gunnar Carlsson
Department of Mathematics
Stanford University
http://math.stanford.edu/~gunnar/

John Harer
Department of Mathematics
Duke University
http://fds.duke.edu/db/aas/math/faculty/harer/

Konstantin Mischaikow
School of Mathematics
Georgia Institute of Technology
http://www.math.gatech.edu/~mischaik/

During July 10-28, 2006 , Mississippi State University, Starkville will be the host of the Institute for Mathematics and its Applications (IMA) summer graduate program in mathematics. The course will concentrate on Topology and its Applications.

This program is open to graduate students from IMA Participating Institutions. Students are nominated by their department head. Participating institution department heads nominate graduate students from their institution by an e-mail to with the students' names and e-mail addresses.

Those students may then register by filling out the application form. Places are guaranteed for two graduate students from each participating institution, with additional students accommodated as space allows.

Course Description:

In a number of diverse areas, topological issues have begun to surface. In molecular biology, for example, the geometric features of the surface of a molecule have been shown to influence certain protein docking processes. Knot theory is becoming increasingly important in the study of DNA. Computer scientists encounter topological problems in attempts to reconstruct surfaces from sampled data. Topology in phase space can help overcome the inherent sensitivity in longtime simulations of dynamical systems. During the three-week meeting in the period July 9-29, 2006, there will be three week-long courses in the following areas:

  • Week 1: Applications to Dynamical Systems
  • Week 2: Topological Approximation and Surface Reconstruction
  • Week 3: Applications to Molecular Biology
Below are the Lecturers and the description for each course.


Konstantin Mischaikow (School of Mathematics, Georgia Institute of Technology)

Applications to Dynamical Systems (Week 1)

Konstantin Mischaikow We will discuss computational homology and its use in the study of nonlinear dynamical systems. The lectures will survey five topics:
(1) Algorithms for computing the homology of spaces.
(2) The use of homology in investigating and classifying nonlinear systems based on the patterns observed in experiments or numerical simulations.
(3) Algorithms for computing the homology of maps.
(4) Conley index theory and associated algorithms.
(5) Computer assisted proofs in dynamics.
The associated projects will involve the application of these tools to specific problems.

Talk Materials


Gunnar Carlsson (Department of Mathematics, Stanford University)

Topological Approximation and Surface (Week 2) Reconstruction

Gunnar
Carlsson High dimensional data is now being generated at a very rapid rate in many different disciplines. Further, the data is frequently noisy, and is not equipped with any theoretical model. Rather, the data analysis needs to be used to discover the model. Since there is frequently no model, and therefore no preferred coordinate system, it is important to study those properties which don't change under continuous coordinate changes, which are called topological. The goal of this course is to provide an introduction to a recently developed computational version of algebraic topology, called persistent homology, which allows one to infer topological properties of geometric objects from "point clouds" sampled from them. We will introduce algebraic topology itself, the theory of persistent homology, software which permits its computation, and demonstrate how it is used in several real world examples.

Talk Materials


John Harer (Department of Mathematics, Duke University)

Applications to Molecular Biology (Week 3)

John Harer

  1. Protein surface and docking geometry
    - Intro to protein structure
    - Constructing the interface surface
    - Flattening
    - Docked structure characterization
    - Elevation
    - Extended persistence
    - Docking predition

  2. Topology for Point Clouds
    - /alpha - /beta witness complexes
    - Local homology and manifold recognition
    - Stratified spaces and intersection homology
    - Persistence for intersection homology

  3. Segmentation of Medical Image Data - Watershed algorithm
    - Adding persistence
    - Deformable Models
    - Testing the algorithms on a variety of datasets

Talk Materials


Schedule:

The typical day's schedule will consist of two lectures in the morning, one after lunch, and informal problem sessions in the late afternoon as shown below.

9:00—10:00 lecture
10:00—10:30 break
10:30—11:30 lecture
11:30—1:30 lunch
1:30 until ? computer demos/problem sessions

Organizer is also trying to work out some Saturday excursions. Perhaps a day trip to Memphis one weekend, and a canoe trip in Noxubee wildlife refuge another. More details to follow.

Lecture room: Swalm Hall, Room 140. Please meet here 9:00 Monday morning

Dorm info: single rooms, shared bathroom at Cresswell Hall (http://www.msstate.edu/dept/housing/creswell.htm)

Abstracts, reading materials, etc. will be posted as it is received.


More information about the Department of Mathematics and Statistics at Mississippi State may be found at our website http://www.msstate.edu/dept/math. We look forward to hosting this event. If you need further information, please contact the organizer.


LIST OF CONFIRMED PARTICIPANTS

NAMEDEPARTMENTAFFILIATION
Paul BendichDepartment of Mathematics Duke University
Gunnar CarlssonDepartment of Mathematics Stanford University
Jer-Chin (Luke) ChuangDepartment of Mathematics Rice University
Vin de SilvaDepartment of Mathematics and Computer Science Pomona College
Ali ElgindiDepartment of Mathematics University of Chicago
Cecilia Gonzalez TokmanDepartment of Mathematics University of Maryland
John HarerDepartment of Mathematics Duke University
William KaliesDepartment of Mathematical Sciences Florida State University
Mary KlocDepartment of Biophysics University of Wisconsin, Madison
Kevin KnudsonDepartment of Mathematics & Statistics Mississippi State University
Eun Joo LeeDepartment of Mathematical Sciences Seoul National University
Juhyun LeeSchool of Mathematical Sciences Seoul National University
Ram MedikonduriDepartment of Applied Mathematics and Computer Science The University of Iowa
Jason Mireles-JamesDepartment of Mathematics University of Texas, Austin
Konstantin MischaikowSchool of Mathematics Georgia Institute of Technology
Son NguyenDepartment of Mathematics Wayne State University
Neil NicholsonDepartment of Mathematics The University of Iowa
Juan Ortiz-NavarroDepartment of Mathematics The University of Iowa
Madhulika PannuriDepartment of Electrical and Computer Engineering Mississippi State University
Kyung Bae ParkDepartment of Mathematical Sciences Seoul National University
Lance Pittmanbad address University of Illinois at Urbana-Champaign
Reza RezazadeganDepartment of Mathematics Rutgers, The State University Of New Jersey
Joshua RobertsDepartment of Mathematics University of Kentucky
Sundararajan SrinivasanDepartment of Electrical and Computer Engineering Mississippi State University
Nathaniel StrawnDepartment of Mathematics Texas A & M University
Courtney TaylorDepartment of Mathematics Purdue University
Jonathan WhitehouseDepartment of Mathematics University of Minnesota, Twin Cities
Xuefeng ZhangDepartment of Mechanical Engineering Mississippi State University
Alexander ZorachDepartment of Mathematics University of Delaware
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