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This theorem says that the true electron density of a molecule has a convergent Taylor expansion away from the positions of the nuclei. We discuss this, and an open problem about its structure at the nuclei.

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Our primary motivation for persistent homology is in its applications to shape similarity measures. Multidimensional or multiparameter persistence comes into play in that context when two objects are to be simultaneously compared according to several features. The ideas go back to early 1900s when Paretoâ s optimal points of multiple functions were studied with applications to economy on mind. In our previous work, we developed an algorithm that produces an acyclic partial matching (A, B, C) on the cells of a given simplicial complex, in the way that it is compatible with a vector-valued function given on its vertices. This implies the construction can be used to build a reduced filtered complex with the same multidimensional persistent homology as of the original one filtered by the sublevel sets of the function. Until now, any simplex added to C by our algorithm has been defined as critical. It was legitimate to do so, because an application- driven extension of Formanâ s discrete Morse theory to multi-parameter functions has not been carried out yet. In particular, no definition of a general combinatorial critical cell has been given in this context. We now propose new definitions of a multidimensional discrete Morse function (for short, mdm function), of its gradient field, its regular and critical cells. We next show that the function f used as input for our algorithm gives rise to an mdm function g with the same order of sublevel sets and the same partial matching as the one produced by our algorithm. This is a joint work with Madjid Allili, Claudia Landi, and Filippo Masoni.

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Finding a maximum clique in a graph is one of the most basic computational problems on graphs. The various applications of this problem has motivated the design of algorithms which today can successfully solve real-world instances with thousands of vertices. However, from a theoretical point a view, it is widely believed that this is a hard problem: in particular that determining whether a graph on n vertices contains a k-clique requires time $n^{\Omega(k)}$. In terms of upper bounds, it is easy to determine this in time roughly $n^k$ by checking if any of the sets of vertices of size k forms a clique.<br><br> We analyse the running time of the most successful algorithms used in practice: colour-based branch-and-bound strategies and Ã stergÃ¥rd's algorithm based on Russian doll search. When analysing such algorithms, it is convenient to view the execution trace as a proof establishing the maximal clique size for the input graph. In particular, if this graph does not have a k-clique, then the trace provides an efficiently verifiable proof in so-called regular resolution of the statement that the graph is k-clique-free. We show that for any $k \ll n^{1/4}$ if the input graph is a k-clique-free random graph sampled from the right distribution then the size of such regular resolution proofs, and hence the running time of these algorithm, is at least $n^{\Omega(k)}$.

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Metastability is a phenomenon that concerns both molecular dynamic (and in particular protein dynamic) and Bayesian inference, especially when the posterior distribution of interest is multimodal. Restricting our focus to Markov processes and Markov chains, we explore this duality and study how Bayesian inference methods can benefit from practices developed in protein dynamic analysis and conversely. Questions of interest include the characterization and consequences of metastability in both perspectives. A particular attention will be given to distinguish between reversible and non-reversible processes/chains.

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We demonstrate that a class of one and two phase free boundary problems can be recast as nonlocal parabolic equations on a codimension one submanifold. The canonical examples would be one-phase Hele-Shaw and Laplacian growth. In the special class of free boundaries that are graphs over $\mathbb{R}^d$, we give a precise characterization that shows their motion is equivalent to that of a solution of a nonlocal (fractional) and nonlinear parabolic equation in Euclidean space. Our main observation is that the free boundary condition defines a nonlocal operator having what we call the Global Comparison Property. A consequence of the connection with nonlocal parabolic equations is that for free boundary problems arising from translation invariant elliptic operators in the positive and negative phases, one obtains, in a uniform treatment for all of the problems, a propagation of modulus of continuity for weak solutions of the free boundary flow. This is based on joint works with Hector Chang-Lara and Russell Schwab.

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In this talk we study small time asymptotic of the heat content for a smoothly bounded domain with non-characteristic boundary in the Heisenberg group, which captures geometric information of the of the boundary including perimeter and the total horizontal mean curvature of the boundary of the domain. We use probabilistic method by studying the escaping probability of the horizon- tal Brownian motion process that is canonically associated to the sub-Riemannian structure of the Heisenberg group. This is a joint work with J. Tyson.

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En los aÃ±os 80, Gromov definiÃ³ una distancia, mÃ³dulo isometrÃas, entre variedades riemaniannas y demostrÃ³ que la clase de variedades riemannianas con curvatura de Ricci acotada inferiormente es precompacta. En los aÃ±os 90, Cheeger y Colding estudiaron propiedades de los lÃmites de sucesiones de variedades riemannianas con curvatura de Ricci acotada inferiormente. En 2006, Lott-Sturm-Villani definieron una nociÃ³n sintÃ©tica de curvatura de Ricci para espacios para espacios que no necesariamente sean variedades. Esta condiciÃ³n estÃ¡ basada en el transporte Ã³ptimo entre medidas de probabilidad y la convexidad de un funcional de entropÃa. Los espacios que satisfacen esta condiciÃ³n son llamados espacios CD(K,N) e incluyen a variedades riemannianas con curvatura de Ricci acotada inferiormente. En esta charla nos centraremos en describir ejemplos y propiedades de estos espacios, asÃ como la estructura de su grupo de isometrÃas.

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