Abstract: The notion of a holomorphic function (in open subsets of) $\mathbb{C}$ has a natural generalization to $\mathbb{C}^n$, $n>1$. In this talk we explore generalizing holomorphicity to functions defined on submanifolds of $\mathbb{C}^n$. This leads to the notion of CR manifolds and CR functions. We will spend much of the talk discussing various properties of these objects. A particular point of focus would be the holomorphic extendibility of CR functions. Concrete examples will be extensively used to highlight concepts, phenomena, and results. No prior knowledge of complex analysis in higher dimensions will be assumed.

Abstract: Optimal transport is now a popular tool in statistics, machine learning, and data science. A major challenge in applying optimal transport to large-scale problems is its excessive computational cost. We propose a simple resampling scheme for fast randomised approximate computation of optimal transport distances on finite spaces. This scheme operates on a random subset of the full data and can use any exact algorithm as a black-box back-end, including state-of-the-art solvers and entropically penalized versions. We give non-asymptotic bounds for the expected approximation error. Remarkably, in many important instances such as images (2D-histograms), the bounds are independent of the size of the full problem. Our resampling scheme can also be employed for the barycentre problem, namely computing Fréchet means with respect to the optimal transport metric. We present numerical experiments demonstrating very good approximations can be obtained while decreasing the computation time by several orders of magnitude.

Abstract: Calder\'on-Zygmund operators play an important role in partial differential equations and complex analysis. Some problems in analysis benefit from an understanding of the commutation between certain operators or the factorization of functions from natural function spaces. These topics all interact when studying the commutators of Calder\'on-Zygmund operators and multiplication operators. In this talk, we will discuss some recent results about commutators of certain Calderon-Zygmund operators and BMO spaces and how these generate bounded operators on Lebesgue spaces. Motivations and connections to operator theory and partial differential equations will be provided. Versions of these results on the Heisenberg group, pseudoconvex domains with $C^2$ boundary, and other examples will be explained to show how the general theory carries over to many other settings. This talk is based on joint collaborative work.

Abstract: Unlike in regular statistical models, the posterior distribution over neural network weights is not asymptotically Gaussian. As established in singular learning theory, the posterior distribution over the parameters of a singular model is, asymptotically, a mixture of standard forms. Loosely, this means the parameter space can be partitioned such that in each local parameter set, the average log likelihood ratio can be made "normal crossing" via an algebraic-geometrical transform known as a resolution map. We leverage this under-appreciated result to propose a new variational family for Bayesian deep learning. Affine coupling layers are employed to learn the unknown resolution map, effectively rendering the proposed methodology a normalizing flow with the generalized gamma as the source distribution, rather than the multivariate Gaussian typically employed.

Abstract: Let X be your favorite Banach space of continuous functions on Rn. Given a real-valued function f defined on some (possibly awful) set E in Rn, how can we decide whether f extends to a function F in X? If such an F exists, then how small can we take its norm? Can we make F depend linearly on f? What can we say about the derivatives of F at or near points of E (assuming X consists of differentiable functions)? Suppose E is finite. Can we compute a nearly optimal F? How many computer operations does it take? What if we demand merely that F agree approximately with f? Suppose we allow ourselves to discard a few data points as “outliers”. Which points should we discard?

Abstract: We consider the problem of testing for association between a functional variable belonging to a Hilbert space and a scalar variable. Particularly, we propose a distribution-free test statistic based on Kendall's Tau, which is a popular method for determining the association between two random variables. The distribution of the test statistic under the null hypothesis of independence is established using the theory of U-statistics taking values in a Hilbert space. We also consider the case where the functional data is sparsely observed, a situation that arises in many applications. Simulations show that the proposed method outperforms the alternatives under multiple settings, demonstrating the effectiveness and robustness of our approach. We provide data applications that further showcase the utility of our method.

Abstract: Random matrices and random graphs have evolved into areas of great interest for scientists of different backgrounds. In this talk, I will try to make a survey on some aspects of the spectrum of random graphs and some of the known results and open conjectures. It is not possible to cover different kinds of random graphs in a short time, so I will speak mainly about Erdos-Renyi random graphs.

Abstract: The idea of analytic continuation of representations of real simple (linear) Lie groups $G$ to representations of their complexification $G_\mathbb C$ was proposed by Gelfand and Gindikin. In view of this old idea and some of its modern manifestations, in this talk, we will explain what each of the terms in the title mean and how they are related to each other!

Abstract: We consider the problem of estimating the number of principal components in the high-dimensional asymptotic regime where $p$, the number of variables, grows at the same rate as $n$, the number of observations, i.e. $p/n \rightarrow c \in (0, \infty)$. Under the spiked covariance model of Johnstone (2001), the Akaike Information Criterion (AIC) is known to be strongly consistent [Bai et al. (2018)], although under a certain "gap condition" which requires the dominant population eigenvalues to be above a threshold depending on $c$, which is strictly larger than the BBP threshold $1 + \sqrt{c}$, below which a spiked covariance structure becomes indistinguishable from one with no spikes [Baik et al. (2005)]. We show how to modify the penalty term of AIC to yield a strongly consistent estimator under an arbitrarily small "gap", i.e. when the dominant population eigenvalues exceed the BBP threshold by an arbitrarily small amount $\delta > 0$. We also propose another intuitive alteration of the penalty which results in a weakly consistent estimator under exactly zero gap, i.e., above the BBP threshold. We empirically compare the proposed estimators with other existing estimators in the literature.

Abstract: We consider the problem of testing linear hypotheses associated with a high-dimensional multivariate linear regression model. The classical test for this type of hypothesis based on the likelihood ratio statistic suffers from substantial loss of power when the dimensionality of the observation is comparable to the sample size. To mitigate this problem, we propose a class of regularized test procedures that rely on a nonlinear shrinkage of the eigenvalues and eigenprojections of the sample noise covariance matrix, under the assumption that the population noise covariance matrix has a spiked covariance structure. We solve the problem of finding the optimal regularization parameter through a probabilistic formulation of the alternatives and making use of decision-theoretic principles. We illustrate the performance of the proposed test through simulation studies. We also apply the method to detect possible associations among some human behavioral measurements and volumetric measurements on various brain regions.

Abstract: The Laplacian is a ubiquitous object appearing in both analysis and geometry. In this talk, we will give an overview of the Laplace operator and its geometric properties.

Abstract: Covariance estimation is ubiquitous in functional data analysis. Yet, the case of functional observations over multidimensional domains introduces computational and statistical challenges, rendering the standard methods effectively inapplicable. To address this problem, we introduce Covariance Networks (CovNet) as a modeling and estimation tool. The CovNet model is universal – it can be used to approximate any covariance up to desired precision. Moreover, the model can be fitted efficiently to the data and its neural network architecture allows us to employ modern computational tools in the implementation. The CovNet model also admits a closed-form eigen-decomposition, which can be computed efficiently, without constructing the covariance itself. This facilitates easy storage and subsequent manipulation of the estimator. Moreover, the proposed estimator comes with theoretical guarantees in the form of consistency and rate of convergence. We demonstrate the usefulness of the proposed method using several numerical examples. Based on joint work with Victor M. Panaretos.

Abstract: Bayesian multichannel change-point detection problem is studied in the following general setting. A multidimensional stochastic process is observed; some or all of its components may experience changes in distribution, simultaneously or not, according to the joint prior distribution of change-points. The loss function penalizes for false alarms and detection delays, and the penalty increases with each missed change-point. In this talk, we review several approaches to Bayesian detection and introduce asymptotically pointwise optimal (APO) rules, translating the classical concept of Bickel and Yahav to the sequential change-point detection. Extending the method to multiple channels, we derive APO stopping rules for wide classes of stochastic processes, with or without nuisance parameters and practically any prior distribution of change-points. These APO solutions are attractive because of their simple analytic form, straightforward computation, and numerous potential applications. The case of a multidimensional autoregressive time series is considered in detail and applied to a modern problem of disaggregation of energy consumption.

Abstract: In this talk, we discuss about the boundary controllability of some coupled parabolic systems with a single control and with Robin or Kirchhoff-type conditions. Showing the controllability for Robin cases, we play with either the classical moments technique (by Fattorini & Russell, 1971) or with the very recent block moments technique (by Benabdallah, Boyer & Morancey, 2020) depending on the choices of the Robin parameters. For the coupled systems with Kirchhoff-type of boundary conditions at one end and a Dirichlet control at the other end, we show the changes of the controllability phenomenon depending on which component we exert a control. In fact, when the control is acting on the first component, there are some unobservable eigenmode and thus, even the approximate controllability fails by Fattorini-Hautus criterion. Finally, for some 3*3 Kirchhoff-system with space dependent coupling coefficient and with a single boundary control, we show that a minimal time for null-controllability would appear subject to the behavior of the coupling coefficient.

Abstract: This two-part minicourse will revolve around topics in ergodic theory, and the various roles it plays in the study of discrete dynamical systems. In the first part, we will give an overview of basic notions from ergodic theory, which will be followed by an excursion into two classical results in the classification theory of measurable dynamical systems; namely, the classification of discrete dynamical systems with pure point spectrum up to measure theoretic isomorphism using the notion of Koopman operators (von Neumann, 1932), and the classification of Bernoulli automorphisms up to measure theoretic isomorphism using the notion of Kolmogorov-Sinai entropy (Ornstein, 1970). The second part of the mini-course will concern measure-theoretic aspects of dynamics of rational maps on the Riemann sphere. After reviewing some basic results from rational dynamics, we will discuss the existence of the unique measure of maximal entropy on the Julia set of a rational map (Lyubich, 1983). Finally, we will illustrate the dynamical importance of this measure by studying its ergodic and equidistribution properties.

Abstract: Given a finite index $II_1$ subfactor $N \subset M$, we formulate a notion and prove the existence of a universal Hopf *-algebra (of discrete type) which can act on $M$ s.t. $N$ is fixed by the action. Such a Hopf algebra (quantum group) can naturally be thought of as a quantum generalization of some kind of `Galois' group of the subfactor. We compute this quantum group for several interesting cases, including a generic depth 2 subfactor

Abstract: Originating in work of Radchenko and Viazovska, a new kind of Fourier analytic duality, known as Fourier interpolation, has recently been developed. I will discuss this general duality principle and present a new construction associated with the nontrivial zeros of the Riemann zeta function, obtained in joint work Andriy Bondarenko and Danylo Radchenko. I will emphasize how the latter construction fits into the theory of the Riemann zeta function.

Abstract: We say a number x in [0,1] is normal if for any positive integer D, all finite words of same length with letters from the alphabet {0, 1, ... , D-1} occurs with the same asymptotic frequency in the representation of x in base D, or in simple words, its digital expansion is uniformly random in any base. The famous Normal number theorem of E. Borel says that almost every number possesses this phenomenon. It is generally believed that some naturally defined subsets of $\mathbb{R}$ also inherit the above property unless the set under consideration displays an obvious obstruction. This talk is about the study of Borel's theorem on fractals; Cantor type sets for instance. We show that for certain fractals how the property of being normal can be related to the behaviour of trajectories under some random walk on tori, and consequently can be settled studying measures which are `stationary' with respect to the random walk.

Abstract: Pick's theorem is a century-old theorem in complex analysis about interpolation with bounded analytic functions. This classical result inspired the study of a class of Hilbert function spaces including the Dirichlet space and the Drury-Arveson space on the ball. I will talk about the basics and about some of the recent developments in this area. In particular, I will explain how the column-row property has emerged and why it holds automatically.

Abstract: Among all shade functions in Euclidean space, only the black on white regions delimited by a single polynomial inequality can be identified by finitely many power moments. This observation is a consequence of a foundational result of A. Markov. Superresolution is a continuity phenomenon, in some specific functional norm, of a shade function belonging to a neighborhood of such a special uniquely determined domain. We will discuss some recent sharp bounds of superresolution type, and apply them to two dimensions, where a fortunate generalization of Markov theory was developed.

Abstract: In 1971, Graham and Pollak showed that if $D_T$ is the distance matrix of a tree $T$ on $n$ nodes, then $\det(D_T)$ depends only on $n$, not $T$. This independence from the tree structure has been verified for many different variants of weighted bi-directed trees. In my talk: 1. I will present a general setting which strictly subsumes every known variant, and where we show that $\det(D_T)$ - as well as another graph invariant, the cofactor-sum - depends only on the edge-data, not the tree-structure. 2. More generally - even in the original unweighted setting - we strengthen the state-of-the-art, by computing the minors of $D_T$ where one removes rows and columns indexed by equal-sized sets of pendant nodes. (In fact, we go beyond pendant nodes.) 3. We explain why our result is the "most general possible", in that allowing greater freedom in the parameters leads to dependence on the tree-structure. 4. Our results hold over an arbitrary unital commutative ring. This uses Zariski density, which seems to be new in the field, yet is richly rewarding. We then discuss related results for arbitrary strongly connected graphs, including a third, novel invariant. If time permits, a formula for $D_T^{-1}$ will be presented for trees $T$, whose special case answers an open problem of Bapat-Lal-Pati (Linear Alg. Appl. 2006), and which extends to our general setting a result of Graham-Lovasz (Advances in Math. 1978). (Joint with Apoorva Khare.)

Abstract: Cooking forms the core of our cultural identity other than being the basis of nutrition and health. The increasing availability of culinary data and the advent of computational methods for their scrutiny is dramatically changing the artistic outlook towards gastronomy. Starting with a seemingly simple question, ‘Why do we eat what we eat?’ data-driven research conducted in our lab has led to interesting explorations of traditional recipes, their flavor composition, and health associations. Our investigations have revealed ‘culinary fingerprints’ of regional cuisines across the world, starting with the case study of Indian cuisine. Application of data-driven strategies for investigating the gastronomic data has opened up exciting avenues giving rise to an all-new field of ‘Computational Gastronomy’. This emerging interdisciplinary science asks questions of culinary origin to seek their answers via the compilation of culinary data and their analysis using methods of statistics, computer science, and artificial intelligence. Along with complementary experimental studies, these endeavors have the potential to transform the food landscape by effectively leveraging data-driven food innovations for better health and nutrition.

Abstract: This talk is broadly divided into three parts. We first discuss on hole probabilities of the finite and infinite Ginibre ensembles, Beta ensembles and the Mittag-Leffler determinantal point processes in the complex plane. The hole probability means the probability that there is no points in a given region for a given point process. We study the asymptotic of the hole probabilities as the size of the region increases. The equilibrium measure plays a crucial role in calculating the hole probabilities. The equilibrium measure and the minimum energy related results will be discussed. Second, we discuss on random matrix related results. We show that the eigenvalues of product of independent Ginibre matrices form a determinantal point process in the complex plane. A matrix with iid standard complex normal entries is known as Ginibre matrix. Then we introduce basic notion of free probability and Brown measure. We show that the limiting spectral distribution of the product of elliptic matrices is same as the Brown measure of its limiting element (*-distribution sense). We calculate the limiting spectral distribution of the product of truncated unitary matrices using free probability and Brown measure techniques. Finally, we discuss on the fluctuations of the linear statistics of the eigenvalues of circulant, reverse circulant, symmetric circulant, Hankel, and variance profile random matrices. We re-establish some existing results on fluctuations of linear statistics of the eigenvalues by choosing appropriate variance profiles.

Abstract: Thurston's geometrization conjecture and its subsequent proof for Haken manifolds distinguish knots in S^3 by the geometries in the complement of the knots. While the definition of alternating knots make use of nice knot diagrams, Knot Floer homology, a knot invariant toolbox, defined by Ozsvath-Szabo and Rasumussen, generalizes the definition of alternating knots in the context of knot Floer homology and defines family of quasi-alternating knots which contains all alternating knots. Using Lipshitz-Ozsvath-Thurston's bordered Floer homology, we prove a partial affirmation of a folklore conjecture in knot Floer theory, which bridges these two viewpoints of looking at knots.

Abstract: Genetic predisposition for complex traits is often manifested through multiple tissues of interest at different time points during their development. For example, the genetic predisposition for obesity could be manifested either through inherited variants that control metabolism through regulation of genes expressed in the brain, or through the control of fat storage by dysregulation of genes expressed in adipose tissue, or both. Here we describe a statistical approach that leverages tissue-specific expression quantitative trait loci (eQTLs) to prioritize the tissue of interest underlying the genetic predisposition of a given individual for a complex trait. Unlike existing approaches that prioritize tissues of interest for the trait in the population, our approach probabilistically quantifies the tissue-specific genetic contribution to the trait for a given individual. We implement a variant of finite mixture of regression models based on a maximum a posteriori (MAP) expectation-maximization (EM) algorithm. Through simulations using the UK Biobank genotype data, we show that our approach can predict the relevant tissue of interest accurately and can cluster individuals according to their tissue-specific genetic architecture. We analyze body mass index (BMI) in the UK Biobank to identify individuals who have their genetic contribution manifested through their brain versus adipose tissue. Notably, we find that the individuals with a particular tissue of interest have specific phenotypic features beyond BMI that distinguish them from random individuals in the data, demonstrating the role of tissue-specific genetic contribution for these traits.

Abstract: A reflection group is a subgroup of orthogonal or unitary group generated by (possibly complex) reflections. In the first part of the talk we shall describe nice Coxeter-type generators and relations for many interesting reflection groups. In the second part we shall focus on one particular reflection group R in U(13,1). This group R naturally acts on the the unit ball B in the complex 13 dimensional vector space. Let B_reg be the subset of B on which R acts freely. We shall describe generators and relations for the fundamental group G of (B_reg/R). The generators and relations for R and G are similar to generators and relations known for a group closely related to the monster simple group. We shall discuss a precise conjecture relating G and the monster due to Daniel Allcock. We will not assume familiarity with the theory of complex or hyperbolic reflection groups or the monster.

Abstract: Contrary to the two dimensional situation where blow-up occurs only on a finite set, in an open Euclidean domain of dimension four or higher it is possible to have blow-up on larger sets. It can be written as a union of a finite set and the zero set of a poly-harmonic function. I will talk about the role of the zero set in quantization of energy.

Abstract: Warped cone is a geometric object associated with a measure preserving isometric action of a finitely generated group on a compact manifold. It encodes the geometry of the manifold, geometry of the group (Cayley graph) and the dynamics of the group. This geometric object has been introduced by J. Roe in the context of Coarse Baum-Connes conjecture (CBC conjecture). Warped cones associated with the action of amenable groups give examples of CBC conjecture and some expander graphs can be constructed from the warped cones associated with the action of Property (T) group. On the other hand, Measured Equivalence (ME) is an equivalence relation between two countable groups introduced by M. Gromov as a measure-theoretic analogue of quasi-isometry. If the `cocyles' associated with a measured equivalence relation are bounded, the relation is called Uniform Measured Equivalence. In this lecture, we prove that if two warped cones are quasi-isometric, then the associated groups are Uniform Measured Equivalent. As an application, we will talk about different ME-invariants which distinguish two warped cones up to quasi-isometry. This is a work in progress.

Abstract: Estimation of animal abundance and distribution over large regions remains a central challenge in statistical ecology. In our first study, we use a Bayesian smoothing technique based on a conditionally autoregressive (CAR) prior distribution and Bayesian regression to integrate data from reliable but expensive surveys conducted at smaller scales with cost-effective but less reliable data generated from surveys at wider scales to address this problem. We also investigate whether the random effects which represent the spatial association due to the CAR structure have any confounding effect on the fixed effects of the regression coefficients. Next, we develop a novel Bayesian spatially explicit capture-recapture (SECR) model that disentangles the latent ecological process of animal arrival within a detection region from the process of recording this arrival by a given set of detectors. We integrate this into an advanced version of a recent SECR model by Royle (2015) involving partially identified individuals. This is a joint work with Prof. Mohan Delampady.

Abstract: In this talk, I would like to present some recent results regarding the behaviour of functions which are uniformly bounded under the action of a certain class of non-convex non-local functionals. In the literature, this class of functionals happens to be a very good substitute of the first order Sobolev spaces. As a consequence various improvements of the classical Poincaré’s inequality, Sobolev’s inequality and Rellich-Kondrachov’s compactness criterion were established. This talk will be focused on addressing the gap between a certain exponential integrability and the boundedness for functions which are finite under the action of these class of non-convex functionals.

Abstract: The analysis for Yang-Mills functional and in general, problems related to higher dimensional gauge theory, often requires one to work with notions of Sobolev principal bundles and Sobolev connections on them. The bundle transition functions for a Sobolev principal $G$-bundle are not continuous in the critical dimension and thus the usual notion of topology does not make sense. In this talk, we shall see that if a bundle $P$ is equipped with a Sobolev connection $A$, then one can associate a topological isomorphism class to the pair $\left( P, A\right)$, which is invariant under Sobolev gauge changes. In stark contrast to classical notions, this notion of `bundle topology' is \emph{not} independent of the connection. However, for more regular bundles and connections, this coincides with the usual notion. On the other hand, we shall see that this notion behaves well with respect to passage to the limit of sequences with control on $n/2$-Yang-Mills energies and is thus more suitable to capture the change of topology in the limit due to concentration of curvatures.

Abstract: The problem of constructing flat representations of spherical surfaces arises naturally in geography and astronomy while making maps. We look at a mathematical formulation of this problem using the notion of conformal mapping, and discuss its relation with complex analysis. After reviewing the contributions of Gauss, Riemann, and Poincaré to this problem, we end with some glimpses of 20th century developments. This will be an expository talk accessible to undergraduate and postgraduate students.

Abstract: In this talk we shall discuss the theory of braided compact quantum groups in the C*-algebraic framework. We shall begin with the motivation behind the definition, then continue with some results related to the Haar state and finally, if time permits, we shall conclude with an example. This talk is based on a joint work in progress with Thomas Timmermann.

Abstract: Let $\Gamma$ be a hyperbolic group acting geometrically on a proper, geodesic, hyperbolic metric space $X$. We prove that the $\Gamma$ action on the visual boundary of $X$ has the large intersection property of Falconer. We deduce some geometric and arithmetic consequences. This is joint work with Anish Ghosh.

Abstract: We shall talk about some basic results on additions of subsets in groups. These are fundamental questions in Additive Combinatorics. Then we shall discuss some related zero-sum theorems. If there is time, we may tell about weighted generalizations.