Georgia Institute of Technology College of Sciences

This talk will serve as an introduction to the algebra of hyperfields—fields with a multivalued addition. For example the sign hyperfield which is the arithmetic of real numbers modulo their absolute value (e.g. positive + positive = positive, positive + negative = any possibility). We will also introduce valued fields which capture the idea of how many times a fixed prime p divides the numerator or denominator of a rational number.

Using this arithmetic we will consider the combinatorial question of factoring a polynomial over a hyperfield. This will present a unified and conceptual way of looking at Descartes's rule of signs (how many positive roots does a real polynomial have) and the Newton polygon rule (how many roots are there which are divisible by p or p^2).

I will define and discuss the tropicalization and analytification of semialgebraic sets. We show that the real analytification is homeomorphic to the inverse limit of real tropicalizations, analogously to a result of Payne. We also show a real analogue of the fundamental theorem of tropical geometry. This is based on joint work with Philipp Jell and Claus Scheiderer.

The Hot Spots conjecture (due to J. Rauch from the 1970s) is one of the most interesting open problems in elementary PDEs: it basically says that if we run the heat equation in an insulated domain for a long period of time, then the hottest and the coldest spot will be on the boundary. What makes things more difficult is that the statement is actually false but that it's extremely nontrivial to construct counterexamples. The statement is widely expected to be true for convex domains but even triangles in the plane were only proven recently. We discuss the problem, show some recent pictures of counterexample domains and discuss some philosophically related results: (1) the hottest and the coldest spots are at least very far away from each other and (2) whenever the hottest spot is inside the domain, it is not that much hotter than the hottest spot on the boundary. Many of these questions should have analogues on combinatorial graphs and we mention some results in that direction as well.

The seminar will be held on Zoom and can be found at the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

Multiple cellular processes such as replication, recombination, and packing change the topology of nucleic acids. The genetic code of viruses and of living organisms is encoded in very long DNA or RNA molecules, which are tightly packaged in confined environments. Understanding the geometry and topology of nucleic acids is key to understanding the mechanisms of viral infection and the inner workings of a cell. We use techniques from knot theory and low-dimensional topology, aided by discrete methods and computational tools, to ask questions about the topological state of a genome. I will illustrate the use of these methods with examples drawn from recent work in my group.

Recording link: https://bluejeans.com/s/bQ3pI0YI2f5