Title: Global attractors for a nonlinear boundary damped wave equation with nonlinear critical source.

Speaker:  Dr. Madhumita Roy

Abstract: In this talk we shall consider a wave model in 3D on a bounded domain which contains nonlinear sources with critical exponent in the interior and nonlinear feedback dissipation on the boundary. Similar models with simpler nonlinear boundary terms have been already studied broadly whereas the generosity of our model is not only the presence of nonlinear damping but also nonlinear boundary source. Boundary actuators are easily accessible to external manipulations-hence feasible from the engineering point of view and practically implementable. On the other hand, the underlying mathematics is challenging. Boundary actions are represented by unbounded, unclosable operators, hence not treatable by perturbation theory(even from the point of view of well-posedness theory.)

Our main result shows that a suitably calibrated boundary damping prevents the blow up of the waves and allows to contain these asymptotically (in time) in a suitable attracting set which is compact.

Title: The Lightning Model

Abstract: We introduce a non-standard model for percolation on the integer lattice $\Z^2$. Randomly assign to each vertex $a \in \Z^2$ a potential, denoted $\phi_a$,  chosen independently and uniformly from the interval $[0, 1]$. For fixed $\epsilon \in [0,1]$, draw a directed edge from vertex $a$ to a nearest-neighbor vertex $b$ if $\phi_b < \phi_a + \ep$, yielding a directed subgraph of the infinite directed graph $\overrightarrow{G}$ whose vertex set is $\Z^2$, with nearest-neighbor edge set. We define notions of weak and strong percolation for our model, and observe that when $\ep = 0$ the model fails to percolate weakly, while for $\ep = 1$ it percolates strongly. We show that there is a positive $\epsilon_0$ so that for $0 \le \ep \le \ep_0$, the model fails to percolate weakly, and that when $\ep > p_\text{site}$, the critical probability for standard site percolation in $\Z^2$, the model percolates strongly. We study the number of infinite strongly connected clusters occurring in a typical configuration. We show that for these models of percolation on directed graphs, there are some subtle issues that do not arise for undirected percolation. Although our model does not have the finite energy property, we are able to show that, as in the standard model, the number of infinite strongly connected clusters is almost surely 0, 1 or $\infty$.

Location: DH 109

Time: 4/14/2022 1:00 pm.

There are many generalizations of this result, including those quantifying just how large a prime power $p^m$ divides the cardinality $|V|$. We will discuss  these results and illustrate them by giving some examples of simple proofs of otherwise very difficult theorems. We will then talk about a seemingly un-noticed generalization of  the Chevalley-Warning Theorem. By a rather straightforward   modification of a  proof of R. Wilson, one can obtain a version of the Chevalley-Warning Theorem applicable when the equations $f_1(\textbf x)\equiv 0\mod p^{m_1},\;\ldots,\;f_n(\textbf x)\equiv 0\mod p^{m_s}$ are considered modulo \emph{differing prime powers} of the  common prime $p$. Using Hensel’s Lemma, which we will discuss in detail for those unfamiliar, we will explain how this result means any typical Chevalley-Warning argument modulo $p$ can effectively be implemented modulo varying primes powers $p^{m_1},\ldots,p^{m_s}$ of a common prime, with tight bounds for how large a prime power $p^m$ divides the cardinality of the corresponding set $V$.   As one simple example, this gives a new proof of the Davenport Constant of a finite abelian $p$-group without need of using group algebras.