- Analysis Seminar
- Wednesday, April 8, 2020 - 13:55 for 1 hour (actually 50 minutes)
- Skiles 005
- Vlad Yaskin – University of Alberta
Zooplankton is an immensely numerous and diverse group of organisms occupying every corner of the oceans, seas and freshwater bodies on the planet. They form a crucial link between autotrophic phytoplankton and higher trophic levels such as crustaceans, molluscs, fish, and marine mammals. Changing environmental conditions such as rising water temperatures, salinities, and decreasing pH values currently create monumental challenges to their well-being.
A signi cant subgroup of zooplankton are crustaceans of sizes between 1 and 10 mm. Despite their small size, they have extremely acute senses that allow them to navigate their surroundings, escape predators, find food and mate. In a series of joint works with Rudi Strickler (Department of Biological Sciences, University of Wisconsin - Milwaukee) we have investigated various behaviors of crustacean zooplankton. These include the visualization of the feeding current of the copepod Leptodiaptomus sicilis, the introduction of the "ecological temperature" as a descriptor of the swimming behavior of the water flea Daphnia pulicaria and the communication by sex pheromones in the copepod Temora longicornis. The tools required for the studies stem from optics, ecology, dynamical systems, statistical physics, computational fluid dynamics, and computational neuroscience.
The main goal of this talk is to discuss my proof of a multiplicity formula for polynomials over a real valued field. I also want to talk about some of the raisons d’être for hyperfields and polynomials over hyperfields. This talk is based on my paper “A Newton Polygon Rule for Formally-Real Valued Fields and Multiplicities over the Signed Tropical Hyperfield” which is in turn based on a paper of Matt Baker and Oliver Lorscheid “Descartes' rule of signs, Newton polygons, and polynomials over hyperfields.”
Right and left eigenvectors of non-Hermitian matrices form a bi-orthogonal system to which one can associate homogeneous quantities known as overlaps. The matrix of overlaps quantifies the stability of the spectrum and characterizes the joint eigenvalues increments under Dyson-type dynamics. Overlaps first appeared in the physics literature: Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results can be obtained for quaternionic Gaussian matrices, as well as matrices from the spherical and truncated-unitary ensembles.