We study stability of a spherical vortex introduced by M. Hill in 1894, which is an explicit solution of the three-dimensional incompressible Euler equations. The flow is axi-symmetric with no swirl, the vortex core is simply a ball sliding on the axis of symmetry with a constant speed, and the vorticity in the core is proportional to the distance from the symmetry axis. We use the variational setting introduced by A. Friedman and B. Turkington (Trans. Amer. Math. Soc., 1981). As a consequence, the stability up to a translation is obtained by using a concentrated compactness method. As an application, we prove linear in time filamentation near Hill’s vortex: there exists an arbitrary small outward perturbation growing linearly for all times. These results rigorously confirm numerical simulations by Pozrikidis in 1986. The second part is joint work with In-Jee Jeong(SNU).

---

ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085

Ordinary differential equations are useful in modeling the periodic behavior of organisms, such as circadian rhythm, based on known biological knowledge and researchers' hypotheses. The theoretical mathematical models are calibrated to the experimental measurements by estimating a set of unknown model parameters. Traditional parameter estimation with mathematical models often focuses only on the point estimation relying on an optimization method such as simulated annealing, but it often neglects the uncertainty in point estimates and suffers from the local trap issue. This talk provides a gentle introduction to Bayesian analysis focusing on its usefulness in uncertainty quantification; introduces a Bayesian computing method with an advanced Markov chain Monte Carlo called the generalized multiset sampler; and illustrates the proposed Bayesian inference with circadian oscillations observed in a model filamentous fungus, Neurospora crassa.

---

ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085

Random simple and multigraph models based on exchangeable random measures, sometimes named graphexprocesses or generalisedgraphonmodels, have recently been proposed as a flexible class of sparse random graph models. This class of models can be seen as a generalisationof the popular graphonmodels. I will present this class of models, discuss some of their asymptotic properties, in particular the asymptotic behaviourof the degree distribution and of the clustering coefficients. I will also present some particular parametric models within this class and their use for discovering latent communities in sparse real-world networks.

When a biological system is modeled using a mathematical procedure, the following step is normally to estimate the system parameters. Despite the numerous computational and statistical techniques, estimating parameters for complex systems can be a difficult task. As a result, one can think of revealing parameter-independent dynamical properties of a system. More precisely, rather than estimating parameters, one can focus on the underlying structure of a biochemical system to derive the qualitative behavior of the associated mathematical process. In this talk, we will discuss introduce reaction network theory. A reaction network is a graphical configuration of a biochemical system. One of the most important problems in this field is to relate dynamical properties and the underlying reaction network structure. When abundances of biochemical species (variables) in the system are small, then the randomness inherent in the molecular interactions is crucial to the system dynamics, and the abundances are modeled stochastically as a jump by jump fashion continuous-time Markov chain. The goal of this talk is to 1. walk you through the basic modeling aspect of the stochastically modeled reaction networks, and 2. to show how to derive stability (ergodicity) of the associated Markov process solely based on the underlying network structure.

---

ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085

The Gordon-Bender-Knuth identities are determinant formulas for the sum of Schur functions of partitions with bounded length. There are interesting combinatorial consequences of the Gordon-Bender-Knuth identities, for instance, connections between standard Young tableaux of bounded height, lattice walks in a Weyl chamber, and noncrossing matchings. In this talk we prove an affine analog of the Gordon-Bender-Knuth identities and study their combinatorial properties. As a consequence we obtain an unexpected connection between cylindric standard Young tableaux and r-noncrossing and s-nonnesting matchings. This is joint work with JiSun Huh, Christian Krattenthaler, and Soichi Okada.

---

ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085