The HADES seminar on Tuesday, October 12th, will be given by Ovidiu-Neculai Avadanei at 5 pm in 740 Evans.

Speaker: Ovidiu-Neculai Avadanei

Abstract: This talk represents a first step towards understanding the well-posedness for the dispersive Hunter-Saxton equation. This problem arises in the study of nematic liquid crystals, and its non-dispersive version is known to be completely integrable. Although the equation has formal similarities with the KdV equation, the lack of $L^2$ control gives it a quasilinear character, with only continuous dependence on initial data. Here, we prove the local and global well-posedness of the Cauchy problem using a normal form approach to construct modified energies, and frequency envelopes in order to prove the continuous dependence with respect to the initial data. This is joint work with Albert Ai.

The HADES seminar on Tuesday, October 5th, will be given by Ruixiang Zhang at 5 pm in 740 Evans.

Speaker: Ruixiang Zhang (UC Berkeley)

Abstract: Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a “stationary set” method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry’s problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.

The HADES seminar on Tuesday, September 28th, will be given by Joey Zou at 5 pm in 740 Evans.

Speaker: Joey Zou (University of California, Santa Cruz)

Abstract: In X-ray CT scans with metallic objects, streak artifacts in the computed image may arise due to beam hardening effects, where the attenuation coefficient of metallic objects vary strongly with energy. A mathematical description of these artifacts using the notion of wavefront sets was given by Choi, Park, and Seo in 2014, followed by the work of Palacios, Uhlmann, and Wang, who gave quantitative descriptions of the artifacts that recovered qualitative observations from CT scans when the metallic objects are strictly convex. In this talk, I will discuss joint work with Yiran Wang which builds on the previous work by using microlocal analysis to study artifacts generated by non-convex metallic objects, as well as artifacts associated to a broader class of attenuation variations than was considered before. The problem relies on the analytic behavior of a nonlinear function composed with the image of the X-ray transform applied to certain functions, for which we use the work of Melrose, Ritter, Sa Barreto et al. on semilinear wave equations via the usage of iterated regularity spaces in which both the X-ray transform image and its nonlinear composition live.

The HADES seminar on Tuesday, September 21st, will be given by James Rowan from at 5 pm in 740 Evans.

Speaker: James Rowan (University of California, Berkeley)

Abstract: It is well-known that the cubic nonlinear schrodinger equation gives a good approximation for frequency-localized solutions to the irrotational 2D gravity water waves equations, at least on a cubic timescale.  Replacing the assumption of irrotationality with one of constant vorticity allows the model to apply to waves in settings with countercurrents, but the new terms introduced by the vorticity break the scaling symmetry, and in the low-frequency regime, they should have a large effect.  We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good approximation to the 2D gravity water waves equations with constant vorticity.  This work is joint with Mihaela Ifrim, Daniel Tataru, and Lizhe Wan.  Along the way to this result, I will give a brief introduction to some topics in nonlinear dispersive PDE and fluid dynamics.

The HADES seminar on Tuesday, May 11 will be given by Wenjie Lu via Zoom from 3:40 to 5 pm.

Speaker: Wenjie Lu (University of Minnesota)

Abstract: Hydrodynamic stability is one of the oldest problems studied in PDEs. In this talk, I will introduce results related to the linear stability of monotone shear flows with boundaries. If the vorticity vanishes near boundaries, one can obtain optimal decay estimates in Gevery spaces. However, the boundary effect is significant and can be an obstruction for the scattering of the vorticity in high regularity spaces. In order to understand the asymptotic behavior more clearly, we need to have a full picture of the singularity structure of the generalized eigenfunctions. It turns out that we can actually track singularities of arbitrary derivatives of the generalized eigenfunctions. With this, we can get arbitrary many terms in the asymptotic, not only the main term. This is a recent joint work with Hao Jia.

The HADES seminar on Tuesday, April 27 will be given by Federico Pasqualottovia Zoom from 3:40 to 5 pm.

Speaker: Federico Pasqualotto

Abstract: Shock waves are a fundamental phenomenon which appears in the context of compressible fluid flow.In this talk, we will review the problem of shock formation, focusing on various techniques which are suitable to study the problem in one and several space dimensions.

The HADES seminar on Tuesday, March 30 will be given by Georgios Moschidis via Zoom from 3:40 to 5 pm.

Speaker: Georgios Moschidis

Abstract: The AdS instability conjecture provides an example of weak turbulence appearing in the dynamics of the Einstein equations in the presence of a negative cosmological constant. According to this conjecture, there exist arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting  boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time.  In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric  Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams. 

The HADES seminar on Tuesday, February 16 will be given by Michael McNulty via Zoom from 3:40 to 5 pm.

Speaker: Michael McNulty (UC Riverside)

Abstract: The strong-field Skyrme model is a particular limiting case of the Skyrme model, a geometric field theory from nuclear physics and a quasilinear modification of the nonlinear sigma model (wave maps). Singularity formation in these models is known to serve as great toy models for physically realistic situations like gravitational collapse in Einstein’s equation of general relativity. This limit restores the scale invariance of the equation of motion allowing for the existence of self-similar solutions, i.e., singularity formation. In this talk, we discuss work in progress toward establishing the nonlinear stability of an explicitly known self-similar solution to the equation of motion for the strong-field Skyrme model. In addition, we discuss the current strategy used for studying nonlinear stability of self-similar blowup in the broader context of energy supercritical wave equations and the new challenges one faces when applying this to the strong-field Skyrme model. 

The HADES seminar on Tuesday, February 9 was given by Calum Rickard via Zoom from 3:40 to 5 pm.

Speaker: Calum Rickard (USC)

Abstract: The compressible Euler equations describe the flow of an inviscid ideal gas. The global-in-time existence of strong solutions is proven for three distinct compressible Euler systems in the presence of vacuum states which describe different physical and mathematical situations. Our results are obtained through perturbations around various forms of expanding background affine motions. The particular properties of the different affine motions present new mathematical challenges to the stability analysis in each case.

The HADES seminar on Tuesday, November 17th will be given by Kevin O’Neill via Zoom from 3:40 to 5 pm.

Speaker: Kevin O’Neill (UC Davis)

Abstract: Whitney’s Extension Problem asks the following: Given a compact set $E\subset\mathbb{R}^n$ and a function $E\to\mathbb{R}$, how can we tell if there exists $F\in C^m(\mathbb{R}^n)$ such that $F|_E=f$? The classical Whitney Extension theorem tells us that, given potential Taylor polynomials $P^x$ at each $x\in E$, there is such an extension $F$ if and only if the $P^x$’s are compatible under Taylor’s theorem. However, this leaves open the question of how to tell solely from $f$. A 2006 paper of Charles Fefferman answers this question. We explain some of the concepts of that paper, as well as recent work of the speaker, joint with Fushuai Jiang and Garving K. Luli, which establishes the analogous result when $f\geq 0$ and we require $F\geq 0$.