Georgia Institute of Technology College of Sciences

The mean curvature flow is to evolve a hypersurface in Euclidean space using the mean curvatures at each point as the velocity field. The flow has good smoothing property, but also develops singularities. The singularities are modeled on an object called shrinkers, which give homothetic solutions to the flows. As there are infinitely many shrinkers that seem impossible to classify, it is natural to explore the idea of generic mean curvature flows that is to introduce a generic perturbation of the initial conditions. In this talk, we shall explain our work on this topic, including perturbing away nonspherical and noncylindrical shrinkers, and generic isolatedness of cylindrical singularities. The talk is based on a series of works jointly with Ao Sun.

Equidistribution problems, originating from the classical works of Kronecker, Hardy and Weyl about equidistribution of sequences mod 1, are of major interest in modern number theory. 

We will discuss how some of those problems relate to unipotent flows and present a conjecture by Margulis, Sarnak and Shah regarding an analogue of these results for the case of the horocyclic flow over a Riemann surface. Moreover, we provide evidence towards this conjecture by bounding from above the Hausdorff dimension of the set of points which do not equidistribute.

The talk will be accessible, no prior knowledge is assumed.

This presentation introduces a methodology for generating computer-assisted proofs (CAPs) aimed at establishing the existence of solutions for nonlinear differential equations featuring non-polynomial analytic nonlinearities. Our approach combines the Fast Fourier Transform (FFT) algorithm with interval arithmetic and a Newton-Kantorovich argument to effectively construct CAPs. A key highlight is the rigorous management of Fourier coefficients of the nonlinear term Fourier series, achieved through insights from complex analysis and the Discrete Poisson Summation Formula. We demonstrate the effectiveness of our method through two illustrative examples: firstly, proving the existence of periodic orbits in the Mackey-Glass (delay) equation, and secondly, establishing the existence of periodic localized traveling waves in the two-dimensional suspension bridge equation.

This is joint work with Jan Bouwe van den Berg (VU Amsterdam, The Netherlands), Maxime Breden (École Polytechnique, France) and Jason D. Mireles James (Florida Atlantic University, USA)

A conjecture of Buchanan and Lillo states that all nontrivial oscillatory solutions of
\begin{equation*}
x'(t)=p(t)x(t-\tau(t)),
\end{equation*}
 with $0\leq p(t)\leq 1,0\leq \tau(t)\leq 2.75+\ln2 \approx 3.44$ tend to a known function $\varpi$, which is antiperiodic:
 \begin{equation*}
 \varpi(t+T/2)\equiv - \varpi(t)
 \end{equation*}
 where $T$ is its minimal period. We discuss recent developments on this question, focusing on the periodic solutions characterizing the threshold case. We consider the case of positive feedback ($0\leq p(t)\leq 1$) with $\sup\tau(t)= 2.75+\ln2$, the well-known $3/2$-criterion corresponding to negative feedback ($0\leq -p(t)\leq 1$) with $\sup\tau(t)=1.5$, as well as higher order equations. 

 We investigate the behavior of the threshold periodic solutions under perturbation together with the symmetry (antiperiodicity) which characterizes them. This problem is set within the broader background of delay effects on stability for autonomous and nonautonomous equations, taking into account the fundamental relation between oscillation speed and dynamics of delay equations. We highlight the crucial role of symmetry in both the intuitions behind this vein of research, as well as the relevant combinatorial-variational problems.