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1. Degenerate linear parabolic equations in divergence form on the upper half space

   https://doi.org/10.1090/tran/8892  

   Dong, Hongjie; Phan, Tuoc; Tran, Hung (June 2023, Transactions of the American Mathematical Society)

   We study a class of second-order degenerate linear parabolic equations in divergence form in ( − ∞ , T ) × R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on ( − ∞ , T ) × ∂ R + d (-\infty , T) \times \partial {\mathbb {R}}^d_+ , where R + d = { x ∈ R d : x d > 0 } {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T ∈ ( − ∞ , ∞ ] T\in {(-\infty , \infty ]} is given. The coefficient matrices of the equations are the product of μ ( x d ) \mu (x_d) and bounded uniformly elliptic matrices, where μ ( x d ) \mu (x_d) behaves like x d α x_d^\alpha for some given α ∈ ( 0 , 2 ) \alpha \in (0,2) , which are degenerate on the boundary { x d = 0 } \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems. 

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2. Fast Regression for Structured Inputs

   Meyer, Raphael; Musco, Cameron; Musco, Christopher; Woodruff, David P.; Zhou, Samson (April 2022, International Conference on Learning Representations (ICLR))

   We study the $$\ell_p$$ regression problem, which requires finding $$\mathbf{x}\in\mathbb R^{d}$$ that minimizes $$\|\mathbf{A}\mathbf{x}-\mathbf{b}\|_p$$ for a matrix $$\mathbf{A}\in\mathbb R^{n \times d}$$ and response vector $$\mathbf{b}\in\mathbb R^{n}$$. There has been recent interest in developing subsampling methods for this problem that can outperform standard techniques when $$n$$ is very large. However, all known subsampling approaches have run time that depends exponentially on $$p$$, typically, $$d^{\mathcal{O}(p)}$$, which can be prohibitively expensive. We improve on this work by showing that for a large class of common \emph{structured matrices}, such as combinations of low-rank matrices, sparse matrices, and Vandermonde matrices, there are subsampling based methods for $$\ell_p$$ regression that depend polynomially on $$p$$. For example, we give an algorithm for $$\ell_p$$ regression on Vandermonde matrices that runs in time $$\mathcal{O}(n\log^3 n+(dp^2)^{0.5+\omega}\cdot\text{polylog}\,n)$$, where $$\omega$$ is the exponent of matrix multiplication. The polynomial dependence on $$p$$ crucially allows our algorithms to extend naturally to efficient algorithms for $$\ell_\infty$$ regression, via approximation of $$\ell_\infty$$ by $$\ell_{\mathcal{O}(\log n)}$$. Of practical interest, we also develop a new subsampling algorithm for $$\ell_p$$ regression for arbitrary matrices, which is simpler than previous approaches for $$p \ge 4$$. 

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3. Central Extensions and Almost Representations

   https://doi.org/10.1093/imrn/rnaf261  

   Dadarlat, Marius; Glebe, Forrest (September 2025, International Mathematics Research Notices)

   Abstract For a sequence of unital tracial $$C^{*}$$-algebras $$(A_{n},\tau _{n}),$$ we construct a canonical central extension of the unitary group $$U(\ell ^\infty (\mathbb{N},A_{n})/c_{0}(\mathbb{N},A_{n}))$$ by $$Q(\mathbb{R})=c_{0}(\mathbb{N},\mathbb{R})/\mathbb{R}^\infty ,$$ using de la Harpe–Skandalis pre-determinant. For an asymptotic group homomorphism $$\rho _{n}: \Gamma \to U(A_{n}),$$ the corresponding pullback of the canonical central extension gives a 2-cohomology class in $$H^{2}(\Gamma ,Q(\mathbb{R})),$$ which obstructs the perturbation of $$(\rho _{n})$$ to a sequence of true homomorphisms of groups $$\pi _{n}:\Gamma \to GL(A_{n})$$. The pairing of the obstruction class with elements of $$H_{2}(\Gamma ,\mathbb{Z})$$ yields numerical invariants in $$\tau _{n\,*} (K_{0}(A_{n}))$$ that subsume the winding number invariants of Kazhdan, Exel, and Loring. For generality, we allow bounded asymptotic homomorphisms to map the group $$\Gamma $$ into the general linear group of any sequence of tracial unital Banach algebras. In that case, the obstruction class belongs to $$H^{2}(\Gamma ,Q(\mathbb{C})),$$ where $$Q(\mathbb{C})=c_{0}(\mathbb{N},\mathbb{C})/\mathbb{C}^\infty .$$ As an application, we show that 2-cohomology obstructs various stability properties under weaker assumptions than those found in existing literature. In particular, we show that the full group $$C^{*}$$-algebra $$C^{*}(\Gamma )$$ of a discrete group $$\Gamma $$ is not $$C^{*}$$-stable if $$H^{2}(\Gamma ,\mathbb{R})\neq 0$$ and in fact, $$\Gamma $$ is not stable in operator norm with respect to tracial von Neumann algebras. 

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4. The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

   https://doi.org/10.1007/s00208-021-02219-1  

   Hofmann, Steve; Li, Linhan; Mayboroda, Svitlana; Pipher, Jill (July 2021, Mathematische Annalen)

   Grafakos, Loukas (Ed.)

   We establish the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti- symmetric part. In particular, the coefficients are not necessarily bounded. We prove that for some $$p\in (1,\infty)$$, the Dirichlet problem for the elliptic equation $$Lu= \dv A\nabla u=0$$ in the upper half-space $$\mathbb{R}^{n+1},\, n\geq 2,$$ is uniquely solvable when the boundary data is in $$L^p(\mathbb{R}^n,dx)$$, provided that the coefficients are independent of the vertical variable. This result is equivalent to saying that the elliptic measure associated to $$L$$ belongs to the $$A_\infty$$ class with respect to the Lebesgue measure $dx$, a quantitative version of absolute continuity. 

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5. The Benjamin–Ono initial-value problem for rational data with application to long-time asymptotics and scattering

   https://doi.org/10.4171/aihpc/169  

   Blackstone, Elliot; Gassot, Louise; Gérard, Patrick; Miller, Peter D (October 2025, Annales de l'Institut Henri Poincaré C, Analyse non linéaire)

   We show that the initial-value problem for the Benjamin–Ono equation on\mathbb{R}withL^{2}(\mathbb{R})rational initial data with only simple poles can be solved in closed form via a determinant formula involving contour integrals. The dimension of the determinant depends on the number of simple poles of the rational initial data only and the matrix elements depend explicitly on the independent variables(t,x)and the dispersion coefficient\epsilon. This allows for various interesting asymptotic limits to be resolved quite efficiently. As an example, and as a first step towards establishing the soliton resolution conjecture, we prove that the solution with initial datum equal to minus a soliton exhibits scattering. 

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