Theoretical Statistics and Mathematics Unit | ||
All upcoming Seminars and Colloquia |
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BANGALORE PROBABILITY SEMINARTitle : Betti Numbers of Gaussian Excursions in the Sparse RegimeSpeaker : Gugan Thoppe (Technion - Israel Institute of Technology Israel) Time : November 28, 2017 (Tuesday), 02:15 PM Venue : Auditorium Abstract : Excursions of random fields is an increasingly important topic within data analysis in medicine, cosmology, materials science, etc. In this talk, we will discuss some detailed results concerning their Betti numbers. Specifically, we shall consider a piecewise constant Gaussian field whose covariance function is positive and satisfies some local and decay rate conditions. We will discuss a way to model its excursion set using a Cech complex. For each Betti number of this complex, we shall then prove various limit theorems in different regimes based on how fast the window size and excursion level grow to infinity. These shall include asymptotic mean estimates, a vanishing to non-vanishing phase transition with a precise estimate of the transition threshold, and a weak law in the non-vanishing regime. We shall further see a Poisson and a central limit theorem close to the transition threshold. The expected vertex degree asymptotically vanishes in the regimes we shall deal with. This places all our above results in the so-called `sparse' regime. Our proofs combine tools from both extreme value theory and combinatorial topology. BANGALORE PROBABILITY SEMINARTitle : Longest Increasing Subsequence Under Curvature ConstraintSpeaker : Riddhipratim Basu (ICTS Bangalore) Time : November 28, 2017 (Tuesday), 03:30 PM Venue : Auditorium Abstract : Motivated by extremal isoperimetric problems in percolation, I shall describe a model which puts a global curvature constraint on the classical Ulam's problem in the plane, and studies the longest increasing path from $(0,0)$ to $(n,n)$ trapping atypically large area. As is typical in these models, the first order behaviour of this random contour is determined by a variational problem which we explicitly solve. More interesting are exponents related to local fluctuation properties which capture the competition between the global curvature constraint and the behaviour of an unconstrained path governed by KPZ universality. These can be studied via maximal facet lengths of the convex hull of the contour and the Hausdorff distance from the hull for which we identify scaling exponents $3/4$ and $1/2$ respectively. I shall also discuss connections to different models and several open problems.Joint work with Shirshendu Ganguly and Alan Hammond. |
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