Applications of decoupling inequality in Vinogradov problems and discrete Strichartz estimates

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The HADES seminar on Wednesday, May 6th, will be at 3:30pm in Room 736.

Speaker: Yuda Chen

Abstract: The decoupling inequality, introduced by Bourgain and Demeter in their celebrated proof of the ℓ² decoupling theorem for the paraboloid, has become a fundamental tool in modern harmonic analysis.

This talk will survey its key applications. We first discuss its pivotal role in the Vinogradov Mean Value Theorem. We then focus on more recent applications to proving discrete Strichartz estimates. Finally, we conclude by proposing several conjectures concerning potential extensions of these estimates.

Accelerated Shock Formation for the Energy-Critical Euler-Poisson System

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The HADES seminar on Tuesday, April 28st, will be at 3:30pm in Room 740.

Speaker: Ely Sandine

Abstract: The Euler-Poisson system of partial differential equations describes the dynamics of a self-gravitating gas. For the energy-critical polytropic pressure law, there is an explicit steady-state solution describing an isolated star. I will discuss recent work which describes the nonlinear phase space around this solution and proves existence of a new instability mechanism: accelerated shock formation. This talk is based on forthcoming work done in collaboration with Mahir Hadžić, Juhi Jang and Sung-Jin Oh.

Unique maximal globally hyperbolic developments for the 3D compressible Euler equations in spherical symmetry

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The HADES seminar on Tuesday, April 21st, will be at 3:30pm in Room 740.

Speaker: Dongxiao Yu

Abstract: We study the 3D non-relativistic compressible Euler equations. We assume that the flow is irrotational and isentropic. We treat all equations of state except that of the Chaplygin gas, for which shocks are not expected to form. For an open set of initial data with tails at infinity, we provide a complete and precise description of the maximal globally hyperbolic development (MGHD). The boundary of this MGHD consists of an initial singularity known as the crease, a singular boundary where gradient blow-up occurs, and a Cauchy horizon emanating from the crease. The analysis involves delicate competition between dispersion and resonant nonlinear terms. Moreover, we prove that this MGHD is unique by applying a uniqueness theorem of Eperon-Reall-Sbierski. This is joint work with Leonardo Abbrescia and Jared Speck.

Inverse theorems in additive combinatorics

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The HADES seminar on Tuesday, April 14th, will be at 3:30pm in Room 740.

Speaker: James Leng

Abstract: Let $r_k(N)$ denote the largest subset of the integers from $1$ to $N$ without a $k$-term arithmetic progression. A famous open problem is to find optimal upper bounds on $r_k(N)$. In this talk, I will survey work leading up to the current best upper bounds. Of particular note is the inverse theory of Gowers norms, which I will motivate through both finitary combinatorics and ergodic theory. Time permitting, I will end off by describing my favorite open problem in this area.

Overdamped quasi-normal modes for Schwarzschild black holes

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The HADES seminar on Tuesday, May 5th, will be at 3:30pm in Room Evans 740.

Speaker: Tyler Guo

Abstract: Quasi-normal modes (QNM) of black holes are supposed to describe the ringdown of decaying gravitational waves. This ringdown is dominated by the modes closest to the real axis but the overdamped modes (deeper in the complex) remain of interest to physicists and mathematicians.

In this talk, I will give a rigorous definition of QNM as scattering resonances. I will then explain how, in the Schwarzschild case, the problem of finding QNM can be reduced to a study of eigenvalues of a family of non-self-adjoint 1D operators using the method of complex scaling. To explain the structure of the resulting spectral problems I will consider the simpler self-adjoint case and give a proof of the emergence of Bohr–Sommerfeld quantisation rules. If time permits, I will discuss the modifications needed to treat the non-self-adjoint case.

Late-time tails of the Einstein vacuum equation near Minkowski spacetime in wave gauges

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The HADES seminar on Tuesday, March 31th, will be at 3:30pm in Room 740.

Speaker: Yuchen Mao

Abstract: It has been expected that wave gauges lead to weak time decay of solutions of the Einstein vacuum equation near Minkowski spacetime, even though Lindblad-Rodnianski used the standard wave gauge to prove global nonlinear stability of Minkowski spacetime in their renowned work. Lindblad further observed the weak $\tau^{-1}$ decay by identifying the source of the time asymptotics. In this talk, I will introduce a new result that proves i) the late time tail; i.e. the leading term in the time asymptotic expansion, is indeed $\tau^{-1}$ in the standard wave gauge, and ii) by slightly modifying the wave gauge condition, one can achieve decay better than $\tau^{-1}$, hence better than expected before. The proof adapts the iteration scheme developed in Luk-Oh to the weak null system of wave equations under wave gauges in order to identify the tail, and chooses the wave gauge so that the zeroth spherical harmonic in the semilinear weak null structure that causes the weak decay is canceled.

On ODE blow-up surfaces for the focusing NLW

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The HADES seminar on Tuesday, March 17th, will be at 3:30pm in Room 740.

Speaker: Warren Li

Abstract: We consider the focusing wave equation for all powers in all dimensions. It is well-known that the equation admits spatially homogeneous blow-up solutions, often dubbed ODE blow-up, terminating in a singular hypersurface at {t=T}. In this talk, we show both that we can construct solutions that (locally) blow-up on an arbitrary spacelike hypersurface, unique up to the choice of a function we call auxiliary scattering data, and that such blow-up hypersurfaces and auxiliary scattering data is stable to perturbations away from the singularity. For instance, we show smooth perturbations of the ODE blow-up solution yields a smooth spacelike blow-up hypersurface. This is based on joint work with Isti Kadar (ETH).

Existence of weak solutions to a model of the geodynamo

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The HADES seminar on Tuesday, March 10th, will be at 3:30pm in Room 740.

Speaker: Tom Schang

Abstract: In this talk, I will discuss a model of the earth’s magnetic field that has previously been simulated numerically, but has not been shown to be well-posed. This model couples solid physics for the electrically conducting inner core with magnetohydrodynamic (MHD) equations in the liquid outer core, as well as the magnetic field outside of the core, which is taken to be non-conducting. I will define and prove existence of Leray-Hopf-type weak solutions for this system. Particular challenges include the transmission conditions of the magnetic field coming from non-constant physical parameters and extending the magnetic field to a non-conducting exterior. To address these problems, we must carefully define a function space and prove appropriate embeddings.

Cohomogeneity One Expanding Ricci Solitons and the Expander Degree

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The HADES seminar on Tuesday, March 3rd, will be at 3:30pm in Room 740.

Speaker: Abishek Rajan

Abstract: We consider the space of smooth gradient expanding Ricci soliton structures on S^1\times \mathbb R^3 and S^2\times \mathbb R^2 which are invariant under the action of SO(3)\times SO(2). In the case of each topology, there exists a 2-parameter family of cohomogeneity one solitons asymptotic to cones over the link S^2\times S^1, as constructed by Nienhaus-Wink and Buzano-Dancer-Gallaugher-Wang. Analogous to work of Bamler and Chen, we define a notion of expander degree for these cohomogeneity one solitons through a properness result. We then proceed to calculate this cohomogeneity one expander degree in the cases of the specific topologies

Enhanced lifespan bounds for 1D quasilinear Klein-Gordon flows

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The HADES seminar on Tuesday, February 24th, will be at 3:30pm in Room 740.

Speaker: Hongjing Huang

Abstract:

We consider one-dimensional scalar quasilinear Klein–Gordon equations with general nonlinearities, on both $\mathbb R$ and $\mathbb T$.
By employing a refined modified-energy framework of Ifrim and Tataru, we investigate long time lifespan bounds for small data solutions.
Our main result asserts  that solutions with small initial data of size $\epsilon$ persist on the improved cubic timescale $|t| \lesssim \epsilon^{-2}$ and satisfy sharp cubic energy estimates throughout this interval. We also establish difference bounds on the same time scale. In the case 
of $\mathbb R$, we are further able 
to use dispersion in order to extend the lifespan to $\epsilon^{-4}$. This generalizes earlier results 
obtained by Delort, \cite{Delort1997_KG1D}  in the semilinear case.
This joint work with Mihaela Ifrim and Daniel Tataru.