@article{14278, abstract = {The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely, integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of almost every ellipse that preserves integrability near the boundary, is itself an ellipse. We apply this result to study local spectral uniqueness of ellipses using the connection between the wave trace of the Laplacian and the dynamics near the boundary and establish local uniqueness for almost all of them.}, author = {Koval, Illya}, issn = {1432-1297}, journal = {Inventiones Mathematicae}, pages = {221--298}, publisher = {Springer Nature}, title = {{Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse}}, doi = {10.1007/s00222-025-01397-y}, volume = {244}, year = {2026}, } @article{20838, abstract = {For any 1 ≤ r ≤ ∞, we show that every diffeomorphism of a manifold of the form R/Z × M is a total renormalization of a Cr-close to identity map. In other words, for every diffeomorphism f of R/Z×M, there exists a map g arbitrarily close to identity such that the first return map of g to a domain is conjugate to f and moreover the orbit of this domain is equal to R/Z×M. This enables us to localize near the identity the existence of many properties in dynamical systems, such as being Bernoulli for a smooth volume form.}, author = {Berger, Pierre and Gourmelon, Nicolaz and Helfter, Mathieu}, issn = {1432-1297}, journal = {Inventiones mathematicae}, number = {2}, pages = {431--468}, publisher = {Springer Nature}, title = {{Every diffeomorphism is a total renormalization of a close to identity map}}, doi = {10.1007/s00222-024-01305-w}, volume = {239}, year = {2024}, } @article{22065, abstract = {The Benjamin–Ono equation is shown to be well-posed, both on the line and on the circle, in the Sobolev spaces H^s for s > -1/2. The proof rests on a new gauge transformation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional dividends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of Gérard’s explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy.}, author = {Killip, Rowan and Laurens, Thierry and Visan, Monica}, issn = {1432-1297}, journal = {Inventiones mathematicae}, number = {3}, pages = {999--1054}, publisher = {Springer Nature}, title = {{Sharp well-posedness for the Benjamin–Ono equation}}, doi = {10.1007/s00222-024-01250-8}, volume = {236}, year = {2024}, } @article{12877, abstract = {We consider billiards obtained by removing from the plane finitely many strictly convex analytic obstacles satisfying the non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift, which provides a natural labeling of periodic orbits. We show that under suitable symmetry and genericity assumptions, the Marked Length Spectrum determines the geometry of the billiard table.}, author = {De Simoi, Jacopo and Kaloshin, Vadim and Leguil, Martin}, issn = {1432-1297}, journal = {Inventiones Mathematicae}, pages = {829--901}, publisher = {Springer Nature}, title = {{Marked Length Spectral determination of analytic chaotic billiards with axial symmetries}}, doi = {10.1007/s00222-023-01191-8}, volume = {233}, year = {2023}, } @article{10704, abstract = {We define and study the existence of very stable Higgs bundles on Riemann surfaces, how it implies a precise formula for the multiplicity of the very stable components of the global nilpotent cone and its relationship to mirror symmetry. The main ingredients are the Bialynicki-Birula theory of C∗-actions on semiprojective varieties, C∗ characters of indices of C∗-equivariant coherent sheaves, Hecke transformation for Higgs bundles, relative Fourier–Mukai transform along the Hitchin fibration, hyperholomorphic structures on universal bundles and cominuscule Higgs bundles.}, author = {Hausel, Tamás and Hitchin, Nigel}, issn = {1432-1297}, journal = {Inventiones Mathematicae}, pages = {893--989}, publisher = {Springer Nature}, title = {{Very stable Higgs bundles, equivariant multiplicity and mirror symmetry}}, doi = {10.1007/s00222-021-01093-7}, volume = {228}, year = {2022}, } @article{7901, abstract = {We derive rigorously the leading order of the correlation energy of a Fermi gas in a scaling regime of high density and weak interaction. The result verifies the prediction of the random-phase approximation. Our proof refines the method of collective bosonization in three dimensions. We approximately diagonalize an effective Hamiltonian describing approximately bosonic collective excitations around the Hartree–Fock state, while showing that gapless and non-collective excitations have only a negligible effect on the ground state energy.}, author = {Benedikter, Niels P and Nam, Phan Thành and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert}, issn = {1432-1297}, journal = {Inventiones Mathematicae}, pages = {885--979}, publisher = {Springer}, title = {{Correlation energy of a weakly interacting Fermi gas}}, doi = {10.1007/s00222-021-01041-5}, volume = {225}, year = {2021}, } @article{22054, abstract = {We consider the Korteweg–de Vries equation with white noise initial data, posed on the whole real line, and prove the almost sure existence of solutions. Moreover, we show that the solutions obey the group property and follow a white noise law at all times, past or future. As an offshoot of our methods, we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting.}, author = {Killip, Rowan and Murphy, Jason and Visan, Monica}, issn = {1432-1297}, journal = {Inventiones mathematicae}, number = {1}, pages = {203--282}, publisher = {Springer Nature}, title = {{Invariance of white noise for KdV on the line}}, doi = {10.1007/s00222-020-00964-9}, volume = {222}, year = {2020}, } @article{209, author = {Browning, Timothy D and Heath Brown, Roger}, issn = {1432-1297}, journal = {Inventiones Mathematicae}, number = {3}, pages = {553 -- 573}, publisher = {Unknown}, title = {{Equal sums of three powers}}, doi = {10.1007/s00222-004-0360-9}, volume = {157}, year = {2004}, } @article{1457, abstract = {Among the major mathematical approaches to mirror symmetry are those of Batyrev-Borisov and Stromdnger-Yau-Zaslow (SYZ). The first is explicit and amenable to computation but is not clearly related to the physical motivation; the second is the opposite. Furthermore, it is far from obvious that mirror partners in one sense will also be mirror partners in the other. This paper concerns a class of examples that can be shown to satisfy the requirements of SYZ, but whose Hodge numbers are also equal. This provides significant evidence in support of SYZ. Moreover, the examples are of great interest in their own right: they are spaces of flat SLr-connections on a smooth curve. The mirror is the corresponding space for the Langlands dual group PGLr. These examples therefore throw a bridge from mirror symmetry to the duality theory of Lie groups and, more broadly, to the geometric Langlands program.}, author = {Hausel, Tamas and Thaddeus, Michael}, issn = {1432-1297}, journal = {Inventiones Mathematicae}, pages = {197 -- 229}, publisher = {Springer Nature}, title = {{Mirror symmetry, langlands duality, and the Hitchin system}}, doi = {10.1007/s00222-003-0286-7}, volume = {153}, year = {2003}, }