Special Lectures
Title : Gram matrices: connecting GPS triangulation, Euclidean metric embeddings, and Heron’s formula
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Title : Sobolev type inequalities and related problems.
Title : Distance between walks on graphs
Title: The probabilistic method in analysis
Speaker : Koushik Ramachandran, TIFR CAM, Bangalore
Abstract : Paul Erdos invented the probabilistic method to answer some questions in graph theory and combinatorics. In the last 50 years it has become clear that the probabilistic method is very general and is an useful tool in analysis, algebraic geometry, computer science, etc. In this talk I will introduce the probabilistic method as envisioned by Erdos but quickly move to some applications in harmonic and complex analysis. The talk should be accessible to students with a background in basic graduate level analysis, and discrete probability.
Title: Ballot problem, Young tableaux and Non-intersecting paths
Speaker : Manjunath Krishnapur
Abstract: In the ballot problem, there are k candidates who get votes a_1\ge a_2\ge ...\ge a_k. All the votes are placed in a box and drawn one after another at random while counting. What is the probability that at all times during the counting process, the first candidate is leading over the second candidate who is leading over the third and so on?
It turns out that this problem is equivalent to that of counting the number of standard or semi-standard Young tableau of a given shape (the terms will be defined in the lecture). And both problems are solved by a lemma of Karlin--Mcgregor and Lindstrom-Gessel-Viennot that gives a determinantal formula for the number of non-intersecting collections of paths with given starting and ending points in certain directed graphs. Only very basic probability and linear algebra knowledge will be assumed.
Title: Exact solvability of the dimer model
Speaker : Terrence George
Abstract: The dimer model—originally a toy model for the adsorption of diatomic molecules on surfaces—has grown into a central object in statistical mechanics, combinatorics, discrete geometry, etc. In this talk, I will introduce the model from first principles and explore its exact solvability on planar graphs via Kasteleyn's remarkable determinant method.
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Title: On the depth query problem for convex bodies: a geometric approach
Speaker : Purvi Gupta, Indian Institute of Science
Abstract: A typical example of a geometric query problem is that of polytope membership, the objective of which is to synthesize a given convex polytope K⊂ℝd into a compact data structure that supports an efficient algorithm for determining whether a given query point q∈ℝd is in K or not. More efficient algorithms can be designed if one considers the \emapproximate polytope membership problem where the membership data structure is allowed to answer incorrectly for points lying ε-close to the boundary of the polytope, where ε>0 is a fixed error. Recently, efficient data structures with optimal storage space and query time were constructed for this problem using more shape-sensitive objects such as Macbeath regions (Arya-da Fonseca-Mount, 2017; Abdelkader-Mount, 2018, 2024). A crucial ingredient in the analysis of these data structures is the Hilbert metric of K, which is a complete, projectively-invariant metric on K, for which straight lines give the shortest paths.
In this talk, we discuss a more refined version of the polytope membership problem. Given a convex body K⊂ℝd, the depth of a query point q∈ℝd is the maximum value δ∈[0,1) such that q lies outside all those half-spaces that contain δ proportion of the volume of K. We will discuss how the problem of determining the approximate depth of a query point can be converted into an approximate membership problem using the notion of convex floating bodies from convex geometry. We will then discuss the role of the Hilbert metric in the construction of a data structure to answer such queries. This is joint work with A. Narayanan.
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Title : Protecting quantum states against environmental perturbations by dynamical decoupling using inverting pulses
Speaker : Deepak Dhar, ICTS Bangalore
Abstract : One important difficulty in developing powerful quantum computers is the fact that information stored in qubits is susceptible to small changes in environment, and suffers degradation and loss of fidelity with time. One way to protect quantum states against this to subject the quantum system to a series of magnetic field pulses. In this, a particularly promising candidate are “inverting pulses”, which are very short duration magnetic field pulses, approximated as delta functions in time, that are designed so that one subspace of the Hilbert space picks up a phase π compared to perpendicular subspace.
I will discuss the problem of designing the optimal sequence of interval between pulses to to supress the transition rate between these two subspaces to be of Ο(λ
2N+2), where λ is the strength of pertubation. It was shown by Uhrig that one can achieve this by only using N pulses. I will discuss the problem in a general setting, and show that optimization problem involves solution of order 2N coupled polynomial equations in M variables. One might expect that M will have to be
of order 2N. Miraculously, it turns out there is consistent solution exists with only N variables. I note that a transparent argument for this miracle is not yet available, and remains a challenging unsolved problem.
The approach uses very elementary mathematics involving only linear algebra of matrices, and should be accessible to all undergraduate students.
Reference : Preserving quantum states: a super-zeno effect, D.Dhar, L.K. Grover and S. M. Roy, Phys. Rev. Lett., 96, 100405 (2006).
Title: Inequivalence of the Ball and Polydisc in ℂ²
Speaker: Sivaguru R, TIFR CAM , Bangalore
Abstract: Riemann mapping theorem tells us that all simply connected domains in ℂ (except ℂ itself) are biholomorphic to the unit disc 𝔻. This is no longer true in ℂ² (and in higher dimensions) – for instance, the unit ball in ℂ² is not biholomorphic to 𝔻 × 𝔻. We will prove this result by studying the Bergman kernel of these domains. The Bergman kernel of a domain is the reproducing kernel for the Hilbert space of square integrable holomorphic functions on that domain.
Title: Egorychev method for computation of combinatorial sums
Speaker : Venky Krishnan, TIFR CAM, Bangalore
Abstract : In this talk, we will explore the method of Egorychev based on contour integration in the complex plane to analyze and simplify combinatorial terms. Several examples will be presented. The talk will be accessible to a wide audience. Basic knowledge of complex analysis is all that is required
Title: Momentum ray transforms and some applications
Speaker : Venky Krishnan, TIFR CAM, Bangalore
Abstract : Momentum ray transforms are certain weighted transforms that integrates symmetric tensor fields over lines in Eucildean space with weights that are powers of the integration parameter. This is a generalization of the standard longitudinal ray transforms which has attracted significant attention due to its several tomographic applications. We present an inversion algorithm recovering the full symmetric tensor field of rank m from its first m + 1 momentum ray transforms. Some applications will be presented.
Title: Sorting a poset using the Brunn-Minkowski inequality
Speaker : Jaikumar Radhakrishnan
Abstract : We consider the problem of sorting under partial information. We are given a partially ordered set X= {x_1,x_2, ... x_n}. We are told that {x_1, x_2, ..., x_n} have an underlying total order Q that is consistent with X. We wish to determine Q using the fewest comparisons between elements of X. The number of comparisons needed has a lower bound log_2 e(X), where e(X) is the number of linear extensions of X. We will present an argument based on the Brunn-Minkowski inequality due to Jeff Kahn and Nati Linial that shows that O(log_2 e(X)) comparisons suffice.