Seminars, for informal dissemination of research results, exploratory work by research teams, outreach activities, etc., constitute the simplest form of meetings at a Mathematics research centre.
CAMGSD has recorded the calendar of its seminars for a long time, this page serving both as a means of public announcement of forthcoming activities but also as a historic record.
We begin with a brief overview of the rapidly developing research area of active matter (a.k.a. active materials). These materials are intrinsically out of equilibrium resulting in novel physical properties whose modeling requires the development of new mathematical tools. We focus on studying the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction. We introduce a minimal two-dimensional free-boundary PDE model that captures the evolution of the cell shape and nonlinear diffusion of myosin.
We first consider a linear diffusion model with two sources of nonlinearity: Keller-Segel cross-diffusion term and the free boundary that models moving/deformable cell membranes. Here we establish asymptotic linear stability and derive the explicit formula for the stability-determining eigenvalue.
Next, we consider the effect of nonlinear myosin diffusion, which results in the change of the bifurcation type from super- to subcritical, and we obtain an asymptotic representation of the bifurcation curve (for small velocities). This allows us to derive an explicit formula for the curvature at the bifurcation point that controls the bifurcation type. In the most recent work in progress with the Heidelberg biophysics group, we study the relation between various types of nonlinear diffusion and bistability.
Finally, we discuss novel mathematical features of this free boundary model with a focus on non-self-adjointness, which plays a key role in the spectral stability analysis. Our mathematics reveals the physical origins of the non-self-adjoint of the operators in this free boundary model.
Joint works with A. Safsten & V. Rybalko (Transactions of AMS 2023, and Phys. Rev. E 2022), with O. Krupchytskyi &T. Laux (Preprint 2024), and with A. Safsten & L. Truskinovsky ( Arxiv preprint 2024). This work has been supported by NSF grants DMS-2404546, DMS-2005262, and DMS-2404546.
I will talk about recent joint work with Daniel Fadel (University of São Paulo) and Luiz Lara (Unicamp), where we study the SU(2) Yang-Mills-Higgs functional with positive coupling constant on 3-manifolds. Motivated by the work of Alessandro Pigati and Daniel Stern (2021) on the U(1)-version of the functional, we also include a scaling parameter.
When the 3-manifold is closed and the parameter is small enough, by adapting to our context the min-max method used by Pigati and Stern, we construct non-trivial critical points satisfying energy upper and lower bounds that are natural from the point of view of scaling.
Then, over 3-manifolds with bounded geometry, we show that, in the limit as the parameter tends to zero, and under the above-mentioned energy upper bound, a sequence of critical points exhibits concentration phenomenon at a finite collection of points, while the remaining energy goes into an $L^2$ harmonic 1-form. Moreover, the concentrated energy at each point is accounted for by finitely many "bubbles", that is, non-trivial critical points on $R^3$ with the scaling parameter set to 1.
We consider the characterization of global attractors $A_f$ for semiflows generated by scalar one-dimensional semilinear parabolic equations of the form $u_t = u_{xx} + f(u,u_x)$, defined on the circle $x\in S^1$, for a class of reversible nonlinearities. We modify a proof developed for nonlinearities of simple type, making it simpler and amenable to generalization. We obtain a classification up to connection equivalence of global attractors for $S^1$-equivariant parabolic equations.
We prove that the space of symplectic embeddings of $n\geq 1$ standard balls, each of capacity at most $\frac{1}{n}$, into the standard complex projective plane $\mathbb{CP}^2$ is homotopy equivalent to the configuration space of $n$ points in $\mathbb{CP}^2$. Our techniques also suggest that for every $n \geq 9$, there may exist infinitely many homotopy types of spaces of symplectic ball embeddings.
In this talk I will review the notion of unbalanced optimal transport, introduced $\sim10$ years ago to handle mass variarions betwenn nonnegative measures. I will discuss the induced pseudo-Riemannian structure and gradient-flow evolution, corresponding to some class of parabolic reaction-diffusion equations. If time permits I will present two applications: a fitness-driven model from population dynamics, and a Hele-Shaw free boundary problem for tumor growth.
