Please Note: The seminar is held in BlueJeans: https://bluejeans.com/900271747
A central question in ergodic theory is whether sequences obtained by sampling along the orbits of a given dynamical system behave similarly to sequences of i.i.d. random variables. Here we consider this question from a spectral-theoretic perspective. Specifically, we study large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift on the 2-torus with irrational frequency. We prove that their global eigenvalue distribution converges to the Wigner semicircle law, a hallmark of random matrix statistics, which evidences the quasi-random nature of the skew-shift dynamics. This is joint work with Arka Adhikari and Horng-Tzer Yau.
An edge-colored graph $G$ is called \textit{rainbow} if every edge of $G$ receives a different color. The \textit{anti-Ramsey} number of $t$ edge-disjoint rainbow spanning trees, denoted by $r(n,t)$, is defined as the maximum number of colors in an edge-coloring of $K_n$ containing no $t$ edge-disjoint rainbow spanning trees. Jahanbekam and West [{\em J. Graph Theory, 2016}] conjectured that for any fixed $t$, $r(n,t)=\binom{n-2}{2}+t$ whenever $n\geq 2t+2 \geq 6$. We show their conjecture is true and also determine $r(n,t)$ when $n = 2t+1$. Together with previous results, this gives the anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees for all values of $n$ and $t$. Joint work with Linyuan Lu.
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Continuing the theme of Hopf algebras, we will discuss a recipe by Reshetikhin and Turaev for link invariants using representations of quantum groups, which are non-commutative, non-cocommutative Hopf algebras. In the simplest case with the spin 1/2 representation of quantum sl2, we recover the Kauffman bracket and the Jones polynomial when combined with writhe. Time permitting, we will also talk about colored Jones polynomials and connections to 3-manifold invariants.
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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.
