# 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

#### 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.