Jiaye Wei
Robustness of Matching and the Backup Nodes Problem abstract
Abstract:
We study the following problem on the robustness of matching. Consider a bipartite graph with parts A and B. We assume that B has a larger size than A and that the graph has an A-perfect matching. The goal is to find an A-perfect matching such that the number of nodes in A which has a neighbor in the unsaturated nodes of B, is maximized; in other words, we want the maximum number of A nodes to have “backup nodes”. A related problem is studied in the setting of second bidder auctions with binary bids by Azar, Birnbaum, Karlin, and Nguyen in 2009. In that problem, the condition of finding a perfect matching is relaxed but every A nodes in the matching is required to have backup nodes. In this talk, I will first explain a tight (1-1/e) approximation for the general Backup Nodes problem via submodular maximization. I will also introduce the results for the degree-constrained case, including a polynomial-time exact algorithm for (d,2)-regular graphs and the APX-hardness for graphs with small degrees. This is joint work with Rom Pinchasi, Neta Singer, and Lukas Vogl.
11:00 • EPF Lausanne, INJ114
Eugenia Malinnikova (Stanford University)
Carleman estimates, unique continuation, and Landis conjecture
13:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Vladimir Dotsenko (University of Strasbourg)
Higher-dimensional Givental symmetries abstract
Abstract:
Some 20 years ago, Chen, Gibney and Krashen introduced beautiful algebraic varieties parametrizing "pointed trees of projective spaces"; these varieties generalize the celebrated Deligne-Mumford compactifications of moduli spaces of genus zero curves with marked points. In the latter case, the homology operad encodes the tree level part of a cohomological field theory. It follows from the work of Givental and many others that the space of all such structures on a given graded vector space has a very rich symmetry group; this fact has been used in many different research areas. In this talk, I shall prove that the homology of the operad made of Chen-Gibney-Krashen spaces possesses many interesting properties, and in particular, there are higher-dimensional Givental symmetries that emerge in this story. Curiously enough, this leads to new results on classical Givental symmetries as well. This is joint work with Eduardo Hoefel, Sergey Shadrin and Grigory Solomadin.
13:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Bruno Vallette (Université Sorbonne Paris Nord)
Operadic Calculus for Higher Colour-Kinematics Duality abstract
Abstract:
This talk aims at providing gauge field theory, in particular colour–kinematics duality and the double copy construction, with the required higher algebraic structures. From the algebro-homotopical perspective, a classical field theory is encoded by a cyclic homotopy Lie algebra whose Maurer–Cartan functional defines the action. For many gauge theories of interest, including Chern–Simons and Yang–Mills, this algebra splits as a tensor product of a cyclic Lie algebra, the colour Lie algebra, and a cyclic homotopy-commutative algebra, the kinematic algebra.The duality between colour and kinematics, first observed by Bern–Carrasco–Johansson in the study of Yang–Mills amplitudes, suggests that the kinematic algebra carries a Lie-type structure. For theories with at most cubic interactions, this structure is captured by coexact Batalin-Vilkovisky (BV) algebras, algebraic objects dual to the exact BV algebras arising in Poisson geometry. To incorporate higher-order interactions, following ideas of M. Reiterer, a homotopy refinement of this notion becomes necessary.The purpose of this talk is to provide a conceptual definition of homotopy coexact BV algebras, expressed in relation to homotopy commutative and BV algebras, together with a concrete operadic model. Our framework gives explicit presentations in terms of generating operations and relations and enables the systematic application of homotopical methods—including homotopy transfer, rectification, infinity-morphisms, and deformation theory—to the resulting algebras. The quartic-level structures recently identified in Yang–Mills theory fits naturally into this framework. This is a joint work with Anibal Medina-Mardones.
14:45 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Ipsita Datta (ETH)
Geometry of Lagrangian Tangles abstract
Abstract:
Lagrangian tangles are cobordisms between smooth links that generalize the classical Arnol\'d theory of Lagrangian cobordism and the theory of Lagrangian cobordisms between Legendrian links. In this talk, we will explore the symplectic geometry of Lagrangian tangle links in the product of a surface with the complex numbers. The main tool is a novel Floer theory for Lagrangian tangles inspired by Morse theory for manifolds with a gradient field tangent to the boundary. We show the existence of an LES of persistence modules giving quantitative obstructions to the existence of Lagrangian tangle links.This is joint work with Josh Sabloff (Haverford College).
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Aleksei Kulikov (University of Copenhagen)
Time-frequency localization operator and its eigenvalues abstract
Abstract:
For a set A ⊂ R^d we denote by P_A the projection onto A and by Q_A = F⁻¹P_A F the Fourier projection onto A, where F is the Fourier transform. For a pair of sets A, B ⊂ R^d the time-frequency localization operator associated with A and B is the operator S_{A,B} = P_A Q_B P_A.This is a self-adjoint operator satisfying 0 ≤ S_{A,B} ≤ Id. If A and B have finite measures then S_{A,B} is a Hilbert–Schmidt operator, in particular it is compact and so we have a sequence of eigenvalues 1 > λ₁(A, B) ≥ λ₂(A, B) ≥ ... ≥ 0.If A = cA₁, B = cB₁ for some fixed nice sets A₁, B₁ and a big parameter c then the eigenvalues exhibit a phase transition: first ≈ c²ᵈ|A₁||B₁| of the eigenvalues are very close to 1, then there are only ≈ c²ᵈ⁻¹ log(c) intermediate eigenvalues, and the remaining eigenvalues tend to 0 extremely fast. Moreover, for fixed ε > 0 if we consider the counting function N_{c,ε} = #{n : ε < λₙ(c) < 1 − ε} then the dependence on 1/c is logarithmic as well. In this talk we will discuss the uniformity of the estimates on N_{c,ε} when ε is small. For a wide range of parameters we show a uniform bound c²ᵈ⁻¹ log(c)/ log(1/ε). Moreover, it turns out that if ε is extremely small, for example if ε < c⁻ᵃ, then the log(c) term disappears and the estimate becomes even better than expected.The talk is based on joint works with Fedor Nazarov and with Martin Dam Larsen.
16:15 • Université de Genève, Conseil Général 7-9, Room 1-15
Prof. Dr. Maria Colombo (EPFL)
Abstract:
In the last 20 years, there have been remarkable progress in understanding non-uniqueness of solutions to the fundamental partial differential equations of incompressible fluid dynamics, namely, the Euler and Navier-Stokes equations. They stem from different viewpoints: just to mention a few, they include convex integration, instability in self-similar variables, numerical evidences. The talk will provide an overview of these developments, focussing on results about the 2 dimensional Euler equations. It will also highlight new results showing that solutions obtained in the vanishing viscosity limit from the (well-posed) Navier-Stokes equations can be nonunique.
17:15 • Universität Bern, Sidlerstrasse 5, 3012 Bern, Room B6