(J. Sahasrabudhe): In this talk I will try to motivate the interest and some of the mystery in the Ramsey numbers R(k), which are fundamental quantities in combinatorics. I will go on to discuss some recent progress on our understanding of these numbers and make some connections to problems about the geometry of random variables in high dimensions.
(T. Nguyen): Of great physical interest is to resolve the final state conjecture concerning the large time dynamics of a plasma in a non-equilibrium state, whether the transition to turbulence or relaxation to neutrality will occur. This involves extremely rich underlying physics, including phase mixing, Landau damping, and plasma oscillations, of which the talk will provide an overview on recent mathematical advances. The talk should be accessible to graduate students and the general audience.
(P. Takac): In this talk, we look at a model where population growth behaves differently in two separate areas: one where the growth rate is slower at low population levels (concave), and another where it speeds up at high levels (convex). We examine how this mixed growth pattern affects the existence of stable population levels (especially equilibria) in the presence of diffusion in the model. This problem leads to the study of the existence and multiplicity of positive solutions to a corresponding semilinear elliptic Dirichlet problem involving a spectral parameter $\lambda$ and a variable exponent $q(x)$ in the non-linearity $u \mapsto \lambda, u(x)^{q(x) - 1}$. Using methods such as monotone iterations and the Leray-Schauder degree theory, we find at least two positive solutions. These solutions shed light on how the contrasting growth behaviors interact between the two areas connected by diffusive migration. The interaction by diffusive migration between the two subdomains with convex and concave behaviors is key to the findings, which include novel a priori estimates derived from Young's inequality. If time permits, we will also touch on the uniqueness of solutions for related problems involving $p(x)$-Laplacian equations, which have recently received considerable attention.
(C-W.Shu): In scientific and engineering computing, we encounter time-dependent partial differential equations (PDEs) frequently. When designing high order schemes for solving these time-dependent PDEs, we often first develop semi-discrete schemes paying attention only to spatial discretizations and leaving time $t$ continuous. It is then important to have a high order time discretization to maintain the stability properties of the semi-discrete schemes. In this talk we discuss several classes of high order time discretization, including the strong stability preserving (SSP) time discretization, which preserves strong stability from a stable spatial discretization with Euler forward, the implicit-explicit (IMEX) Runge-Kutta or multi-step time marching, which treats the more stiff term (e.g. diffusion term in a convection-diffusion equation) implicitly and the less stiff term (e.g. the convection term in such an equation) explicitly, for which strong stability can be proved under the condition that the time step is upper-bounded by a constant under suitable conditions, the explicit-implicit-null (EIN) time marching, which adds a linear highest derivative term to both sides of the PDE and then uses IMEX time marching, and is particularly suitable for high order PDEs with leading nonlinear terms, and the explicit Runge-Kutta methods, for which strong stability can be proved in many cases for semi-negative linear semi-discrete schemes. Numerical examples will be given to demonstrate the performance of these schemes.
(E. Titi): In this talk I will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for regularizing and stabilizing certain evolution equations, such as the Euler, Navier-Stokes and Burgers equations. On the other hand, I will also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, I will present some results concerning two- and three-dimensional turbulent flows with high Reynolds numbers in periodic domains, which exhibit enhanced dissipation mechanism due to large spatial average in the initial data -- a phenomenon which is similar to the ``Landau-damping" effect.
(A. Wright): Given a surface, the associated curve graph has vertices corresponding to certain isotopy classes of curves on the surface, and edges for disjoint curves. Starting with work of Masur and Minsky in the late 1990s, curve graphs became a central tool for understanding objects in low dimensional topology and geometry. Since then, their influence has reached far beyond what might have been anticipated. Part of the talk will be an expository account of this remarkable story. Much more recently, non-trivial examples of totally geodesic subvarieties of moduli spaces have been discovered, in work of McMullen-Mukamel-Wright and Eskin-McMullen-Mukamel-Wright. Part of the talk will be an expository account of this story and its connections to dynamics. The talk will conclude with new joint work with Francisco Arana-Herrera showing that the geometry of totally geodesic subvarieties can be understood using curve graphs, and that this is closely intertwined with the remarkably rigid structure of these varieties witnessed by the boundary in the Deligne-Mumford compactification.
(E. King):
(G. Wei):
(M. Liu):
(Y. Chen):
(Y. Kawahigashi):
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