Friday Apr 13 |
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| 4:00-4:50 UCLA, MS 6627 Ákos Magyar (U Georgia) Distance graphs in sets of positive density A distance graph $\Gamma$ is a graph with vertices in a Euclidean space, with edges made of rigid rods that can freely turn around the vertices. An isometric embedding of $\Gamma$ into a set $A$ can be visualized as a folding of $\Gamma$ such that all of its vertices are supported by $A$. We show that if $\Gamma$ is $k$-degenerate, i.e. if all of its sub-graphs has a vertex of degree at most $k$, then all its large dilates can be isometrically embedded into any set $A\subseteq \mathbb{R}^d$ of positive upper density, when $d>k$. We also discuss isometric embeddings of distance graphs into subsets of the integer lattice $\mathbb{Z}^d$. |
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| 5:00-5:50 UCLA, MS 6627 Oliver Dragičević (U Ljubljana) On Generalized Convexity of Power Functions We present a condition on accretive matrix functions, called $p$-ellipticity, and discuss its applications to the $L^p$ theory of elliptic PDE with complex coefficients. The examples we consider concern: (1) generalized convexity of power functions (Bellman functions), (2) dimension-free bilinear embeddings, (3) $L^p$-contractivity of semigroups, (4) holomorphic functional calculus, (5) regularity theory of elliptic PDE with complex coefficients, (6) maximal $L^p$ regularity for divergence-form operators with Neumann boundary conditions. Example (5) is due to Dindoš and Pipher. The condition arises from studying uniform positivity of a quadratic form associated with the matrix in question on one hand, and the Hessian of a power function $|z|^p$ on the other. The talk is based on joint work with Andrea Carbonaro. |
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Friday Apr 27 |
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| 4:00-4:50 UCLA, MS 6627 Barry Simon (Caltech) Heinävarra's Proof of the Dobsch-Donoghue Theorem In 1934, Loewner proved a remarkable and deep theorem about matrix monotone functions. Recently, the young Finnish mathematician, Otte Heinävarra settled a 10 year old conjecture and found a 2 page proof of a theorem in Loewner theory whose only prior proof was 35 pages. I will describe his proof and use that as an excuse to discuss matrix monotone and matrix convex functions, including, if time allows, my own recent proof of Loewner's original theorem. |
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| 5:00-5:50 UCLA, MS 6627 Virginia Naibo (KSU) Leibniz-type rules associated to bilinear pseudodifferential operators We will present various Leibniz-type rules for Coifman-Meyer multiplier operators as well as for bilinear pseudodifferential operators with homogeneous symbols and with symbols in the bilinear Hörmander classes. Such estimates will be discussed in the settings of Triebel-Lizorkin and Besov spaces based on quasi-Banach spaces such as weighted Lebesgue, Lorentz and Morrey spaces. The talk is based on joint works with Joshua Brummer and Alexander Thomson. |
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Friday May 11 |
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| 4:30-5:20 Caltech, Downs 103 Malik Younsi (U Hawaii) Removability, rigidity of circle domains and Koebe's conjecture A domain in the Riemann sphere is called a circle domain if every connected component of its boundary is either a round circle or a point. The famous Koebe uniformization conjecture states that every planar domain is conformally equivalent to a circle domain. This has been proved only in some special cases, such as domains with at most countably many boundary components, thanks to the major progress of He and Schramm in the 1990's. In this talk, I will discuss uniqueness of the Koebe conformal map, which is closely related to the notion of conformal rigidity. More precisely, we say that a circle domain is (conformally) rigid if every conformal map of the domain onto another circle domain is the restriction of a Möbius transformation. It is well-known that some circle domains are rigid while some are not, but both sufficient and necessary conditions are yet to be found. I will survey recent results on a conjecture of He and Schramm relating rigidity to the notion of conformal removability. This is partly based on joint work with Dimitrios Ntalampekos. |
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| 5:30-6:20 Caltech, Downs 103 Jonas Lührmann (Johns Hopkins) Probabilistic local well-posedness and scattering for the 4D cubic NLS We consider the Cauchy problem for the defocusing cubic nonlinear Schrodinger equation (NLS) in four space dimensions. It is known that for initial data at energy regularity, the solutions exist globally in time and scatter. However, the problem is ill-posed for initial data at super-critical regularity, i.e. for regularities below the energy regularity. In this talk we study the super-critical data regime for this Cauchy problem from a probabilistic point of view, using a randomization procedure that is based on a unit-scale decomposition of frequency space. In the first part of the talk we will explain how the problem of establishing almost sure local existence for the cubic NLS for such random data has some features in common with proving local existence for a derivative NLS equation. Our method is inspired by the local smoothing estimates and functional frameworks from the Schrodinger maps literature. In the second part of the talk we will turn to the long-time dynamics of the solutions. We will present a conditional almost sure scattering result and an almost sure scattering result for randomized radial data. This is joint work with Ben Dodson and Dana Mendelson. |
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Friday May 25 |
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| 4:00-4:50 UCLA, MS 6627 Antoine Gloria (ULB) Approximate spectral theory and long-time homogenization of the wave equation In this talk I will consider the wave equation in heterogeneous media and shall discuss long-time transport properties of localized initial conditions, and more precisely the regime of asymptotic ballistic transport. To this aim I will introduce an approximate spectral theory inspired by the Floquet theory for periodic media that allows to diagonalize the wave operator up to an error that we control by energy estimates combined with estimates on "correctors". This yields new long time homogenisation results. This is based on joint works with A. Benoit, M. Duerinckx, and C. Shirley. |
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| 5:00-5:50 UCLA, MS 6627 Diogo Oliveira e Silva (U Birmingham) Sharp Strichartz inequalities for fractional and higher order Schrödinger equations It has long been understood that Strichartz estimates for the homogeneous Schrödinger equation correspond to adjoint Fourier restriction estimates on the paraboloid. The study of extremizers and sharp constants for the corresponding inequalities has a short but rich history. In this talk, I will summarize it briefly, and then specialize to the case of certain planar power curves. A geometric comparison principle for convolution measures can be used to establish the corresponding sharp Strichartz inequality, and to decide whether extremizers exist. The mechanism underlying the possible lack of compactness is explained by the behaviour of extremizing sequences and will be described via concentration-compactness. Time permitting, I will show how this resolves a dichotomy from the recent literature concerning the existence of extremizers for the fourth order Schrödinger equation in one spatial dimension. This talk is based on joint work with Gianmarco Brocchi and René Quilodrán. |
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Friday June 8 |
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| 4:30-5:20 Caltech, Downs 103 Nikolai Makarov (Caltech) Calculus on Riemann surfaces I will give "physical" interpretations of various formulas in function theory (e.g. analogues of addition theorems). Joint work with Nam-Gyu Kang. |
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| 5:30-6:20 Caltech, Downs 103 Alpár Mészáros (UCLA) Master equations in the theory of Mean Field Games The theory of Mean Field Games (MFG) was introduced simultaneously roughly a decade ago by J.-M. Lasry and P.-L. Lions on the one hand and P. Caines, M. Huang and R. Malhamé on the other hand. The aim of both groups was to study limits of Nash equilibria of differential games when the number of players tends to infinity. A fundamental object - introduced by Lions in his lectures - that fully characterizes these limit equilibria is the so-called master equation. This is an infinite dimensional Hamilton-Jacobi equation on the Wasserstein space (of Borel probability measures endowed with a distance arising in the Monge-Kantorovich optimal transport problem). A central question in the theory of MFG is the well-posedness of this equation in various settings. In this talk, we will focus on the first order equation, i.e. without any noise in the dynamics of the agents. Because of the lack of a smoothing effect, only a short time existence result of classical solutions (due to W. Gangbo and A. Swiech) is available. The highly nonlocal nature of the equation together with the non-flat geometry of the Wasserstein space prevent us from developing a theory of viscosity solutions in this setting. After an overview of the subject, in the second half of the talk - as part of an ongoing joint work with W. Gangbo - we will present some connections to the weak KAM theory on the Wasserstein space (developed recently by W. Gangbo and A. Tudorascu). |
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Last update: June 1, 2018 | |