In this talk we will survey some old and new results on maximum principles for P-functions and their applications to the study of partial differential equations. More precisely, we will show how one can employ the maximum principle in problems of physical or geometrical interest, in order to get the shape of some free boundaries, isoperimetric inequalities, symmetry results, convexity results and Liouville type results. In the first part of the talk  we'll be mainly focused on some overdetermined problems, while in the second part of the talk we'll present our contributions to some Monge-Ampere type problems and eventually discuss some open problems.

The problem of computing invariants of natural stacks of curves has a long history, starting from Mumford's seminal paper on the Picard group of the stack of 1-pointed elliptic curves. The Picard group of the stack \(\mathcal{M}_{g,n}\) of \(n\)-pointed smooth curves of genus \(g\geq3\) was later computed over \(\mathbb{C}\) by Harer.

We study the closed substack \(\mathcal{H}_{g,n}\) in \(\mathcal{M}_{g,n}\) of \(n\)-pointed smooth hyperelliptic curves of genus \(g\), and compute its Picard group. As a corollary, taking \(g=2\) and recalling that \(\mathcal{H}_{2,n}=\mathcal{M}_{2,n}\), we obtain \(\mathrm{Pic}(\mathcal{M}_{2,n})\) for all \(n\).

Moreover, we give a very explicit description of the generators of the Picard group, which have evident geometric meaning.

We discuss various results on singular integrals adapted to entangled dilations from the past two years. The existing results are mostly on the so-called Zygmund dilations that constitute the simplest intermediate dilation structure lying in between the classical one-parameter setting and the multi-parameter setting. We start with an overview of the subtle optimal weighted theory in the Zygmund case, the techniques behind that, and the implications these have for further results, such as, commutator estimates.  We then discuss the more recent multilinear versions of this theory, the current limitations and, time permitting, some possible further directions and challenges in the area.

We prove a wavelet $T(1)$ theorem for compactness of multilinear Calder\'{o}n-Zygmund (CZ) operators. Our approach characterizes compactness in terms of testing conditions and yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator. This talk is based on joint work with Walton Green and Brett Wick.

Motivated by recent result of F. Perez and R.R.G. on equality of test ideal of module closure operation and trace ideal, and the well-known result by K.E. Smith that parameter test ideal can never be contained in parameter ideals, we study the obstruction of containment of trace ideals in ideals of finite projective (or injective) dimension. As consequences of our results , we give upper bounds on m-adic order of trace ideals of certain modules. We also prove analogous results for ideal of entries of maps in a free resolution of certain modules. This is joint work with Souvik Dey.

The Camassa-Holm-Kadomtsev-Petviashvili equation (CH-KP) is a two dimensional generalization of the Camassa-Holm equation which has been recently derived in the context of shallow water waves and nonlinear elasticity. In this talk we will discuss the stability of the one-dimensional traveling waves, solitary or periodic, with respect to two dimensional perturbations which are periodic in the transverse direction. We show that the stability or instability depends on a sign parameter of the transverse dispersion term. In particular, a nonlinear instability of the one-dimensional solitary waves of any size can be proved for the so-called CH-KP-I model, while for one-dimensional periodic waves we are able to obtain spectral instability for small amplitude CH-KP-I waves.  This is a joint work with Lili Fan, Jie Jin, Xingchang Wang and Runzhang Xu.

This talk is based on the degenerate semi-linear Schrödinger and Korteweg-de Vries equations in one spatial dimension. We construct variationally special solutions of the two models, that is,  standing wave solutions of NLS and traveling waves for KDV, which turn out to have compact support, hence compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDE's and for appropriate variational problems as well.

 We also provide a complete spectral characterization of such waves, for all values of \(p\). Namely, we show that all waves are spectrally stable for \(2<p\leq 8\),  while a single mode

instability occurs for \(p>8\).  This extends the previous work of Germain, Harrop-Griffiths and Marzuola, who have previously established orbital stability for some specific waves, in the range \(p<8\). This is a joint work with Atanas stefanov and Sevdhan Hakkaev.

It has been known for quite some time that various conditions on the oscillation of the matrix $A$ in elliptic/parabolic operators of the form 

 $L = -div A \nabla$ or $L = \partial_t - \div A \nabla$ 

are sufficient to guarantee either the $L^p$ solvability of the Dirichlet problem or the logarithm of the density of the elliptic/parabolic measure, $k$,  is in BMO or (locally) Hölder continuous. Conditions such as the global Hölder continuity of $A$ are classical, whereas other conditions quantifying the oscillation of $A$ in terms of Carleson measures are more recent. Only very recently in joint works with Toro and Zhao, and later with Egert and Saari, was it shown that the BMO norm of $\log k$ could be controlled by these Carleson conditions. These new results are elliptic and heavily relied on work of David, Li and Mayboroda.

In forthcoming work with Egert and Saari, we adapt these new results (B., Toro, Zhao and B. Egert, Saari) to the parabolic setting. Therefore we needed to also adapt the work of David, Li and Mayboroda to the parabolic setting and, in doing so, sharpened their results. It seems very likely that these sharper estimates will allow one to treat the classical results (Hölder continuous coefficients) and the modern results (Carleson conditions) with a unified method. 

I will delve into the ideas underpinning these results.

Thanks to the Direct Summand Theorem, splinter conditions have emerged as a way of studying singularities in commutative algebra and algebraic geometry. In characteristic zero, work of Kovács (2000) and Bhatt (2012) characterizes rational singularities as derived splinters. In this talk, I will present an analogous characterization of klt singularities by imposing additional conditions on the derived splinter property. This follows from a new characterization of the multiplier ideal, an object that measures the severity of the singularities of a variety, viewing it as a sum of trace ideals. This perspective also gives rise analogous description of the test ideal in characteristic as a corollary to a result of Epstein-Schwede (2014).