Theoretical Statistics and Mathematics Unit

Seminar and Colloquium of the week


Title :Ihara-Bass Theorem for the Zeta Function of a Graph and the Ramanujan Property
Speaker : Bharatram Rangarajan (Chennai Mathematical Institute - Chennai)
Time : November 21, 2017 (Tuesday), 03:15 PM
Venue : Auditorium
Abstract : We give an elementary combinatorial proof of Bass's determinant formula for the zeta function of a finite regular graph. This is done by expressing the number of non-backtracking cycles of a given length in terms of Chebyshev polynomials in the eigenvalues of the adjacency operator of the graph. A related observation of independent interest is that the Ramanujan property of a regular graph is equivalent to tight bounds on the number of non-backtracking cycles of every length.


Title : Flops as blowing-ups in maximal Cohen-Macaulay modules
Speaker : Runar Ile (Norwegian Business School)
Time : November 27, 2017 (Monday), 11:30 AM
Venue : Auditorium
Abstract : In dimension 3 minimal models for birational equivalent varieties exist, but are in general singular and not unique. Two birational minimal models are connected by a chain of birational surgery operations called flops. A flop gives a pair of small (partial) resolutions of a singularity. C. Curto and D. Morrison conjectured (arXiv 2008, J. Algebraic Geom. 2013) that certain 3-dimensional flops locally are given by a blowing-up procedure where input is a matrix factorisation of the polynomial defining the singularity in the flop. They verified their conjectures in the $A_n$ and $D_n$-cases by explicit calculations. We generalise the conjectures by employing Gruson-Raynaud blowing-up in maximal Cohen-Macaulay (MCM) modules and show that the conjectures follow from a result which relates deformations of the pair (rational surface singularity, MCM) and deformations of the corresponding pair (partial resolution, locally free strict transform), and a result of H. Knörrer. This is joint work with Trond S. Gustavsen; arXiv:1609.01033 (v2 July 2017).


Title : Betti Numbers of Gaussian Excursions in the Sparse Regime
Speaker : 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.


Title : Longest Increasing Subsequence Under Curvature Constraint
Speaker : 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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