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Vladimir C

- Research Program Mentor

PhD at Uppsala University-Sweden

Expertise

Constructive Mathematical Methods in Economics

Bio

I am a tenured professor in topology and geometry (mathematics) at Dartmouth College. That said, I have side interests in knot theory, low dimensional and geometric topology, and the interactions of contact, symplectic, and Lorentz geometry (with General relativity and the theory of causality in spacetimes). As a beginning student I learned constructive mathematics from my father, which is a belief that only objects and numbers can be obtained as an output of some computer program. Strangely enough most numbers are not constructive, but all the numbers one can think of are in fact constructive. Through Polygence, I mainly offer projects in mathematical constructive economics, a side interest of mine. As for the hobbies I like to read and collect stamps. :-) A brief note on Topology: Topology is the branch of the mathematics in which (as opposed to Geometry) the name of the shape does not change when you squeeze, stretch, or bend it without gluing or cutting. So the earth is indeed a sphere from the view point of Topology even though it is slightly squeezed, coffee cup is the same as a doughnut etc. One of the big questions topology allows one to pose is: what is the shape of the universe we live in?

Project ideas

Project ideas are meant to help inspire student thinking about their own project. Students are in the driver seat of their research and are free to use any or none of the ideas shared by their mentors.

Can you always verify whether the arithmetic mean of CRNs equals to one of the numbers?

Constructive Real Numbers CRNs were introduced by the founder of Computer Science Alan Turing. Essentially a CRN is a computer generated sequence of rational numbers about which you know how fast it converges. Constructive Mathematics was developed in two schools founded by Bishop in the USA and by Markov and Shanin in Russia. You are give a finite set of numbers. The easy part of the project is to show that when the numbers are rational it is easy to create an algorithm that verifies if the arithmetic mean of these numbers equals to one of them. The challenging part is to show that such an algorithm does not exist in general when the numbers are CRNs.

Can you always verify whether a Geometric Mean of a finite collection of numbers equals to one of them?

Constructive Real Numbers CRNs were introduced by the founder of Computer Science Alan Turing. Essentially a CRN is a computer generated sequence of rational numbers about which you know how fast it converges. Constructive Mathematics was developed in two schools founded by Bishop in the USA and by Markov and Shanin in Russia. You are give a finite set of numbers. The easy part of the project is to show that when the numbers are rational it is easy to create an algorithm that verifies if the geometric mean of these numbers equals to one of them. The challenging part is to show that such an algorithm does not exist in general when the numbers are CRNs.

Can you always verify whether the arithmetic mean of a finite set of numbers equals to its geometric mean?

Constructive Real Numbers CRNs were introduced by the founder of Computer Science Alan Turing. Essentially a CRN is a computer generated sequence of rational numbers about which you know how fast it converges. Constructive Mathematics was developed in two schools founded by Bishop in the USA and by Markov and Shanin in Russia. You are give a finite set of numbers. It is well known that the geometric mean of them does not exceed the arithmetic mean. The easy part of the project is to show that when the numbers are rational it is easy to create an algorithm that verifies if the arithmetic mean equals to the geometric mean. The challenging part is to show that such an algorithm does not exist in general when the numbers are CRNs.

Languages I know

Russian

Teaching experience

I was a graduate student in 1992-1998, taught at University of Zurich Switzerland, then I was employed by Dartmouth College and stayed there working from the Assistant Professor to the Full Professor rank. During my teaching experience I did have classes in Calculus 2 and 3, Differential Equation, Linear Algebra, Abstract Algebra, Real Analysis, Complex Analysis, Graduate Analysis, and Topology courses.

Credentials

Work experience

Dartmouth College (2001 - Current)
Assistant, Associate Tenured Professor and Full Tenured Professor
Zurich University (2000 - 2001)
PostDoc
Max Planck Institute for Mathematics (1999 - 2000)
Scientific Visitor
ETH Zurich (1998 - 1999)
PostDoc

Education

St Petersburg State University-Russia
MS Master of Science (1996)
Mathematics
University of California Riverside (UCR)
MS Master of Science (1994)
Mathematics
Uppsala University-Sweden
PhD Doctor of Philosophy (1998)
Mathematics

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