Mathematical structures from studying and counting points

29.05.2024, 14:00 - 16:00 (Special session)  –  Campus Golm, Building 9, Room 2.22 and via Zoom

Abhiram Kidambi (Leipzig), Noah Arbesfeld (Vienna)

14:00  Abhiram M. Kidambi (MPI, Leipzig):  Computer assisted number theory: Past, Present and Future.
  Tea and Coffee Break
15:15  Noah Arbesfeld (University of Vienna):  Structures from points on complex surfaces.


Abhiram M. Kidambi (MPI, Leipzig) :  Computer assisted number theory: Past, Present and Future.

Abstract:     Number theory at its very core starts out with searching for patterns/relations between numbers and then quantifying them/abstracting them. While simple patterns/relations are easy to spot by human mathematicians, one needs to employ powerful techniques of either abstraction or computation to see more complicated patterns in number theory. This often has to do with the fact that the numbers arise as values of arithmetic functions which are not as predictive. For example, Can we predict what comes next in this sequence: 4,7,10,13,..? Quite easily, yes. What about the location of primes on the real number line? Yes, but this is much harder and can be done only up to a maximal error term. What about the values of the imaginary part of the zeroes of the Riemann zeta function that lie on the critical line? This is even harder still.
     Machines are not usually good at pattern recognition, but with the advances of powerful computational techniques, we can use them to study more complicated patterns that arise in number theory. I will go over how machines and computers have assisted us in problems of number theory, and how they are being used now, and what might lie ahead for the intersection of number theory and mathematical metrology.
     No background on number theory or computing will be assumed..


Noah Arbesfeld (University of Vienna):  Structures from points on complex surfaces.

Abstract:     Geometry and topology provide numerous ways to study the geometry of configurations of points. Differences between these approaches emerge when points are allowed to collide. One approach in algebraic geometry uses ideals in a polynomial ring to encode point collisions. The resulting space of configurations is called the Hilbert scheme of points and is of fundamental importance to modern algebraic geometry.
     In the special case of points on a complex surface, the resulting Hilbert scheme exhibits remarkable properties. I’ll give an introduction to this space and its appearances across subfields of mathematics. In particular, I’ll explain how structures in number theory, including those from Abhiram's talk, emerge from the geometry of the Hilbert scheme of points and its generalizations.


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