01:00 PM  02:00 PM


Undergraduate Colloquium:Easytoexplain but hardtosolve problems in Convex Geometry
Jesus de Loera (University of California, Davis)

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 Abstract
 Examples of convex sets are balls, cubes, triangles. These are figures that do not have "holes" or "valleys". Unknown to most people convex bodies are finding more and more applications in such diverse fields as optimization, statistics, algebra, or computer science. In this talk we will convince the audience that there is life after calculus and that even the most seasoned of mathematicians can't solve easy questions about convex bodies.
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02:00 PM  02:30 PM


Introduction to MSRI

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02:00 PM  05:00 PM


Undergraduate minicourse on the Calculus of Analytic Critical Points
Brendan Hassett (Brown University)

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 Abstract
 Every calculus student learns the secondderivative test for critical points: Depending on the sign of the second derivative, we can decide whether the critical point is a maximum or minimum. However, the test is inconclusive if the second derivative is zero. For functions in two variables, there is a secondderivative test for critical points using the matrix of secondorder partial derivatives. This works well when the determinant of this matrix is nonzero; these are called nondegenerate critical points.
Over the last three years, undergraduate students at Rice University have been doing research on critical points of analytic functions in two variables, classifying degenerate critical points using invariants generalizing those from multivariable calculus. Their work revolves around developing computational techniques for evaluating these invariants efficiently. We will present these techniques, as well as the mathematical principles underlying them.
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02:30 PM  03:00 PM


Break

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03:00 PM  04:00 PM


Most Hard Equations are Easy
J. Maurice Rojas

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 While randomization has long been used profitably in optimization and computer science, it's importance in algebraic geometry has only recently been realized. In particular, the study of random systems of equations leads to some beautiful interactions between geometry, applications, and algorithmic complexity.
We survey some of these developments from the point of view of polynomial system solving, focusing particularly on real (as opposed to complex) solutions. We will also see how centuriesold questions on counting real solutions can be solved with modern ideas from algebraic geometry (and a little randomization).
No background in algebraic geometry or algorithms is assumed.
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04:00 PM  05:00 PM


Ergodic Ramsey Theory
Vitaly Bergelson

 Location
 
 Video


 Abstract
 The main focus of this introductory lecture is the fascinating interplay between ergodic theory, Ramsey theory and Diophantine analysis.
Ergodic theory has its roots in statistical and celestial mechanics and studies such phenomena as recurrence and uniform distribution of orbits.
Ramsey theory, a branch of combinatorics, is concerned with the phenomenon of preservation of highly organized structures under finite partitions.
Diophantine analysis concerns itself with integer and rational solutions of systems of polynomial equations.
We will start with some examples which demonstrate the usefulness of the ergodic approach to combinatorics and number theory. The discussion will naturally lead us to some fascinating recent developments such as the celebrated GreenTao theorem on arithmetic progressions in primes. We will conclude by formulating and discussing some natural open problems.
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