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.

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.

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.

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 and which are invariant under the action of . In the case of each topology, there exists a 2-parameter family of cohomogeneity one solitons asymptotic to cones over the link , 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

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

Speaker: Hongjing Huang

Abstract:

The HADES seminar on Tuesday, February 10th, will be at 3:30PM on Zoom.

Speaker: Yongming Li

Abstract: In this talk we present a perturbative proof of the asymptotic stability of the solitary wave solutions for the 1D focusing cubic Schrödinger equation under small perturbations in weighted Sobolev spaces. The strategy of our proof is based on the space-time resonances approach, using the distorted Fourier transform and modulation techniques to capture the asymptotic behavior of the solution. A major difficulty throughout the nonlinear analysis is the slow local decay of the radiation term caused by the threshold resonances in the spectrum of the linearized operator around the solitary wave. The presence of favorable null structures in the quadratic terms mitigates this problem through the use of normal form transformations.

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

Speaker: Felix Brandt

Abstract: The CAO-problem concerns a system of two fluids described by two primitive equations coupled by nonlinear interface conditions. Lions, Temam and Wang proved in their pioneering work the existence of a weak solution to the CAO-system. Its uniqueness remained an open problem.

In this talk, we show that this coupled CAO-system is globally strongly well-posed for large data in critical Besov spaces. The approach presented relies on an optimal data result for the boundary terms in the linearized system in terms of time-space Triebel-Lizorkin spaces. Boundary terms are controlled by paraproduct methods.

This talk is based on joint work with Tim Binz, Matthias Hieber and Tarek Zöchling.