8-12 January, 2018 at the Indian Institute of Science, Bangalore

Sponsored by the National board for higher mathematics

We will consider several classic models of stochastic geometry. After that we will talk about some not widely known connections between probability theory and convex geometry. A number of open problems will be formulated and discussed.

Abstract for Parthanil's lectures here

We generalize Kesten's IIC in two ways. Firstly we establish multiple-arm IIC measure in planar lattices as certain local limits. Secondly, we prove the existence of IIC measure on slabs and then establish it as a certain limit. We also prove certain tools on slabs, namely quasi-multiplicativity and RSW theorem, (which were previously known to be true for only the planar lattices) which function as key components of the proof.

Joint work with Artem Sapozhnikov.

Joint work with Artem Sapozhnikov.

For a centred stationary Gaussian process on the real line whose spectral measure has finite second moment, the mean number of zeroes is given by the well-known Kac-Rice formula. We show that under certain additional regularity assumptions, the number of zeroes in an interval is concentrated around its mean with failure probability that is exponentially small in the length of the interval.

Joint work with Amir Dembo, Naomi Feldheim and Ofer Zeitouni.

Joint work with Amir Dembo, Naomi Feldheim and Ofer Zeitouni.

A connection between stochastic differential equations with piecewise linear coefficients and Riemann—Hilbert boundary value problems for analytic functions (RBVP) is going to be discussed. We present an approach that makes possible to derive probabilistic characteristics of some piecewise linear Ito diffusions as well as of some integral functionals of them. This approach has certain advantages over the standard one based on the Fokker—Planck—Kolmogorov or Feynman—Kac equations. In order to succeed, we investigate an evolution of the characteristic function of the diffusion using the so-called Pugachev equation, which is an analog of the Kolmogorov forward equation. In a piecewise linear case this equation comes down to the singular integral differential equation of the convolution type. The latter is called the Pugachev—Sveshnikovequation and can be reduced to the Riemann—Hilbert linear conjugation problem for analytic functions. Some explicit results are going to be shown, and the approach used is going to be illustrated.

Abstract for Anup Biswas's lecture

Consider a Gaussian Analytic Function on the disk.
In joint work with Yanqi Qiu and Alexander Shamov,
we show that, almost surely, there does not exist a nonzero square-integrable holomorphic function with these zeros.
By the Peres-Virag Theorem, zeros of a Gaussian Analytic Function on the disk are a
determinantal point process governed by the Bergman kernel, and we prove, for general determinantal point processes,
the conjecture of Russell Lyons and Yuval Peres that reproducing kernels sampled along a trajectory form a complete system in the ambient Hilbert space.
The key step in our argument is that the determinantal property is preserved under conditioning.
The problem, posed by Russell Lyons, of describing these conditional measures explicitly remains open even
for the sine-process, and I will report on partial progress: the analogue of the Gibbs property for one-dimensional determinantal processes governed by integrable kernels.
The talk is based on the preprint arXiv:1605.01400
as well as on the preprint arXiv:1612.06751 joint with Yanqi Qiu and Alexander Shamov.

Abstract for Probal Chaudhuri's lecture

Statistical estimates of the Shannon entropy constructed by means of observations $X_1,\ldots,X_N$ having the same law as $X$ are very important.
They permit to estimate the mutual information and other related characteristics of a random vector $X$. Such estimates are widely used in machine learning, they are essential for tests concerning independence hypothesis for collections of random variables and they are employed in feature selection theory and various applications.
The behavior of the Kozachenko - Leonenko (\cite{Kozachenko}) estimates for the (differential) Shannon entropy, when the number of i.i.d. vector-valued observations tends to infinity, was studied by different authors. D.P\'al et al. (\cite{Pal}) indicated the defects in the previous existing proofs of
the asymptotic unbiasedness and $L^2$-consistency of these estimates. In a quite recent paper \cite{Bulinski} we also turn to the mentioned results and establish them under wide conditions.
To this end the analogues of the Hardy - Littlewood maximal function are proposed and employed. It is shown that our approach applies, in particular, to the entropy estimation of any nondegenerate Gaussian distribution. Moreover, we provide conditions to
guarantee the validity of these new results for distributions mixture.

Abstract for Mathew Joseph's lecture

Abstract for Apoorva Khare's lecture

Abstract for Alexey Naumov's lecture

Abstract for Iuliia Petrova's lecture

Several recent breakthrough results in additive combinatorics, first of allCroot, Lev, Pach theorem and method, relies both on polynomial and probabilistic arguments. I want to survey this intensively developing area for a probabilistic audience.

We consider a continuous-time branching random walk on $\mathbb{Z}^d$ where the particles
are born and die at a periodic set of points (the sources of branching).
Spectral properties of the evolution
operator of the mean number of particles are studied.
In particular we prove that this operator has a positive spectrum which leads to an
exponential asymptotic behavior of the mean number of particles when $t \to \infty$.
Based on join work with K. Ryadovkin.

Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $\Phi$. Let $\{G_{n}\}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. Let $\mu_{n}$ be the Gibbs measure on $G_{n}$ constructed via pull-back from $\Phi$. Assume $\{\mu_{n}\}$ converges to a Gibbs measure $\mu$ on $T_{d}$ in the local weak$^{*}$ sense, and let $\Phi$ exhibit strong spatial mixing. We show that the limit of the specific entropies $|V_n|^{-1}H(\mu_n)$ is equal to the \emph{percolative entropy} $H_{perc}(\mu)$. We emphasize here that the percolative entropy is a different quantity from the much more commonly studied Bethe-Ansatz limit.

Abstract for Rishideep Roy's lecture

Starting from the paper of Hugo Steinhaus (1948), where he considered a problem of fair division of a cake with different toppings among a group of friends having different tastes, economists became interested in fair division mechanisms.
In practice we usually face a problem of distributing a family of indivisible goods (e.g., when a divorcing couple divides common assets) rather than cake-cutting problems. The setting is as follows. A finite set of goods $G$ is to be allocated among a finite set $N$ of agents. An agent $i$ has a value $u_{ig}$ for a good $g\in G$. An allocation $z=(z_{ig})_{i\in N, g\in G}$ is a $N\times G$ bi-stochastic matrix, where element $z_{ig}$ is the probability that agent $i$ receives a good $g$; $U_i=U_i(z)$ is the expected value received by an agent.
What allocations are considered as appropriate? There are two requirements usually imposed in the economic literature: Efficiency and Envy-freeness.
An allocation $z$ is called \emph{Efficient} if there is no other allocation $z'$ giving higher utilities to all agents. An allocation is \emph{Envy-free} if $U_i(z)\geq U_i(\pi_{i,j}(z))$, where $\pi_{i,j}$ is a permutation of rows, corresponding to agents $i$ and $j$.
Efficient envy-free allocations always exist. Varian~(1974) suggested the Competitive division rule having both properties and based on the ideas from economic General Equilibrium theory. A version of this rule is implemented on~\url{spliddit.org}.
In this talk we will discuss
\begin{itemize}
\item mathematical side of mentioned economic ideas
\item typical properties of fair division mechanisms, when the matrix $u$ is random and $G$ is large
\item {\bf probabilistic, algorithmic and combinatorial} open problems corresponding to deterministic fair division mechanisms (i.e., $z_{ig}\in\{0,1\}$)
\end{itemize}

Abstract for Anish Sarkar's lecture

Abstract for Himanshu Tyagi's lecture

We construct integral and supremum type goodness-of-fit tests for the uniform law based on Ahsanullah's characterization of the uniform law. We discuss limiting
distributions of new tests and describe the logarithmic large
deviation asymptotics of test statistics under the null-hypothesis. This
enables us to calculate their local Bahadur efficiency under some parametric
alternatives. Conditions of the local optimality of new statistics are
given. Joint work with M. S. Karakulov and Ya. Yu. Nikitin.

Abstract for Vladislav Vysotsky's lecture

Manjunath Krishnapur (IISc, Bangalore) Krishanu Maulik (ISI, Kolkata)

Govindan Rangarajan (IISc, Bangalore) Bimal Roy (ISI, Kolkata)