AnàlisiOn the (1/2,+)-caloric capacity of Cantor sets
Resum: In this talk we will present the concept of 1/2-caloric capacity, an object associated with the 1/2-heat equation in $\mathbb{R}^{n+1}$, i.e. the PDE defined via the pseudo-differential operator $\Theta^{1/2} := (-\Delta_x)^{1/2}+\partial_t$. Such capacity is useful to characterize removable subsets for the latter PDE in terms of some of its measure theoretic properties. The main goal of the talk will be to present a characterization of a variant of the 1/2-caloric capacity (defined only using positive measures) of the usual corner-like Cantor set of $\mathbb{R}^{n+1}$. The results obtained for the latter are analogous to those found for Newtonian capacity.
Geometria On the stable norm of flat surfaces
Resum: On a Riemannian manifold, it is known that the systole provides informations on the global geometry of the manifold. Since the shortest length of a non-homologically trivial curve is interesting, it is natural to ask what is the shortest length of a curve inside a fixed homology class, and how it depends on the chosen homology class. This is called the stable norm of the manifold, and to this day there are very few explicit examples.
Plateau problems and asymptotic counting of surfaces subgroups
Resum: We adapt the asymptotic counting result of Calegari-Marques-Neves to the cas of constant extrinsic curvature (CEC) surfaces. In particular, following recent work of Labourie, we show how this result is expressed in a natural manner in terms of an equidistribution property of a certain class of measures over the space of pointed CEC surfaces. This is joint work with Ben Lowe and Sébastien Alvarez.
L'espectre de longituds de varietats tridimensionals hiperbòliques aleatòries
Resum: Per poder conèixer millor les varietats tridimensionals hiperbòliques, podem mirar el comportament dels seus invariants geomètrics, com la longitud de les seves geodèsiques. Una forma d'encarar aquestes questions és utilitzant mètodes probabilístics. És a dir, considerem un conjunt de varietats hiperbòliques, l'equipem amb una mesura de probabilitat, i ens preguntem questions de la forma: quina és la probabilitat de que una varietat aleatòria tingui un certa propietat? Existeixen diferents models de varietats aleatòries. En aquesta xerrada, explicaré un del principals models probabilístics que existeixen en dimensió 3 i presentaré un resultat relatiu a l'espectre de longituds- el conjunt de longituds de totes les geodèsiques tancades- d'una variedad tridimensional construida sota aquest model.
Geometria TropicalEsquemes Saturats per Semianells II
Resum: Seguiré explicant els ingredients necessaris per cuinar una teoria d'esquemes amb ideals primers saturats.
Models Estocàstics i DeterministesSmall-ball probabilities, with applications to the analysis of some exceptional points for parabolic stochastic partial differential equations.
Resum: I will present some ongoing work with Kunwoo Kim (Pohang, S. Korea) and Carl Mueller (Rochester, U.S.) in which we develop asymptotic evaluations of small-ball probabilities for parabolic SPDEs. These probability estimates include not only statements that ensure the existence of a so-called "small-ball constant,” but to our surprise also an evaluation of that constant. I will then discuss applications of these probability estimates in the study of exceptional points in the Chung’s law of the iterated logarithm, first developed by K.L. Chung (1948) in the context of Brownian motion and the simple walk on Z, followed by a large literature thereafter. I will discuss how the present infinite-dimensional setting requires new proof ideas, and in return also allows for the existence of new types of exceptional points that do not appear to have natural finite-dimensional analogues.
Sistemes DinàmicsOn geometry of bilinear discretizations of quadratic vector fields
Resum: We discuss dynamics of birational maps which appear as
Teoria de NombresApplications of the plectic philosophy to the BSD conjecture III
Resum: This talk is part of a series presenting recent progress on the Birch and Swinnerton-Dyer conjecture for higher rank elliptic curves. In this third lecture I will focus on the case of rational elliptic curves of rank 2 to showcase the results obtained in joint works with Darmon, Gehrmann, Guitart and Masdeu. The techniques involved will include (but won't be limited to) Iwasawa theory and a delicate mixture of complex and p-adic integration.
TopologiaHomotopy properties of the poset of non-trivial $p$-subgroups of a group IV
Resum: This is an introduction to spherical and Cohen-Macaulay (CM) simplicial complexes in the context of Quillen's conjecture on subgroup complexes. We will focus on complexes of proper subspaces of a finite dimensional vector space
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