Abstract: Assuming the Generalized Riemann Hypothesis, it is known that the smallest quadratic non-residue modulo a prime $p$ is less than or equal to $(log p)^2$. In this talk, we will discuss the distribution of quadratic non-residues in even smaller intervals of size $(log p)^A$ with $A>1$, for almost all primes $p$. We will begin with some background on quadratic non-residues and then give a brief outline of the proof. This is joint work with Kunjakanan Nath and Alexandru Zaharescu.

Abstract: The cohomology and homotopy groups of moduli spaces of smooth manifolds are some of the most basic invariants of manifold bundles: the former contain all the characteristic classes and the latter classify smooth bundles over spheres. Complete calculations of these groups are challenging, even for the simplest compact manifolds. It is then desirable to know, at least, some qualitative information, for example whether these groups are (degreewise) finitely generated. In this talk, I will discuss a method to attack this question which leads to the following theorem: if M is a closed smooth manifold of even dimension > 5 with finite fundamental group, then the cohomology and higher homotopy groups of BDiff(M) are finitely generated abelian groups.

Abstract: The space U(n)/N(n), where N(n) is the normalizer of the maximal torus in U(n), arises naturally in many areas of mathematics and can be identified with the unordered flag manifolds. In this talk, we will introduce the concept of the n-fold extended symmetric power of a space X, and describe its cohomology as a Hopf ring. We will also demonstrate homological stability for the spaces {U(n)/N(n)}, and describe the stable cohomology ring of U(n)/N(n). If there is sufficient time, we may also see some of the low-dimensional computations that have been done in the case where n = 3, 4. This is joint work with Lorenzo Guerra.

Abstract: Presence not detected

Abstract: In this talk, I'll compute the equivariant cohomology algebras of locally k-standard T-manifolds. Then, we discuss when the torus equivariant cohomology algebra distinguishes them up to weakly equivariant homeomorphism.

Abstract: In this talk, I'll introduce the category of locally k-standard T-manifolds which includes well-known classes of manifolds such as toric and quasitoric manifolds, good contact toric manifolds and moment-angle manifolds. They are smooth manifolds with well-behaved actions of tori. Then, I'll compute the fundamental groups and equivariant cohomology rings of these new manifolds.

Abstract: One can naturally associate to any pointed topological space a topological monoid of continuous loops that start and end at base point with multiplication given by concatenation of loops. Every element in this topological monoid has an inverse up to homotopy, so it is “almost” a topological group. This passage does not loose any homotopical information and it is often useful to recast the homotopy theory of spaces in terms of topological monoids/groups. In algebraic topology one studies spaces by means of algebraic invariants. For instance, one can replace a topological space by a chain complex with additional structure, more precisely a coalgebra structure - through the classical singular chains construction. In this series of talks, I will describe how to model algebraically the passage from a pointed space to its topological monoid of based loops as a natural construction -called the “cobar functor” and originally due to F. Adams- that takes a differential graded (dg) coalgebra and produces a dg algebra. When applied to the singular chains on a pointed space, the cobar functor produces a model for the chains on the based loop space. A slightly new perspective on this construction will allow us to generalize many existing results in the literature from simply connected spaces to spaces with arbitrary fundamental groups. We will discuss several applications and consequences of this new perspective such as obtaining algebraic models for non-simply connected spaces (extending the work of Sullivan, Quillen, Goerss, Mandell, and others) and models for the free loop space functor suitable for studying string topology in the non-simply connected setting.

Abstract: Classical permutations are symmetries of sets with finitely many points. In the matrix representation, classical permutations are specific square matrices with entries of either 0 or 1. This perspective was generalised by Shuzhou Wang in 1998, and he introduced the notion of quantum permutation group. This talk primarily aims to motivate Wang's pioneering idea and discuss a few properties of the quantum permutation group.

Abstract: The usual definition of Fourier transform on the Heisenberg $ \mathbb{H}^n $ is in terms of the Schr\"odinger representations $ \pi_\lambda $ which makes it operator valued and hence unwieldy and not suitable for studying several problems in harmonic analysis on $ \mathbb{H}^n.$ In this talk, which is based on a recent article, we propose a scalar valued Fourier transform that shares several properties with the Helgason Fourier transform on noncompact rank one Riemannian symmetric spaces.

Abstract: The celebrated Otal-Croke marked length spectrum rigidity theorem recovers the geometry of a closed negatively curved surface from the lengths of closed geodesics on the surface. As an intermediate step of the proof, the dynamics of the (Anosov) geodesic flow is recovered from the periods. We generalize this dynamics rigidity result to the setting of volume preserving 3-dimensional Anosov flows. In turn, it leads to a more general “weighted” marked length spectrum rigidity for negatively curved surfaces. This is joint work with Federico Rodriguez Hertz.

Abstract: I will discuss what cyclic cohomology and various string topology algebraic operations on it are. I will then discuss how these operations are related to counting the number of possibilities of gluing the sides of a 2n-gon to obtain a surface of genus g.

Abstract: I will discuss how the answer to certain integrals on the space of Hermitian matrices is related to counting the number of possibilities of gluing the sides of a 2n-gon to obtain a surface of genus g.

Abstract: Annular knots can be thought of as knots in a solid torus. Recently, the study of annular knots using invariants from annular Khovanov homology has inspired interesting results. I will talk about a different approach to define invariants of annular knots using combinatorial knot Floer homology. I will also discuss it's relationship with band rank and contact topological invariants of Legendrian knots.

Abstract: Lie groupoids are to be seen as a simultaneous generalisation of Lie groups and smooth manifolds. To any (interesting) geometric structure on a manifold, one can associate a Lie groupoid. We begin the talk by recalling the classical notions of Lie group actions, foliations, principal bundles on smooth manifolds and then associate a Lie groupoid for each of these cases. Given a category C, Yoneda lemma says that the next obvious things to study after studying objects of the category C are the (contravariant) functors C——> Set. If the objects in the category C are “geometric”, one can look at a special class of (contravariant) functors C——> Set, which goes by the name of sheaf. A slight generalisation of the notion of sheaf is that of a stack. We start the second half of the talk by focusing on the notion of stack, and towards the end, we will see the more interesting notion of differentiable stack and see how Lie groupoids and differentiable stacks are related to each other.

Abstract: It is a well-known fact that any compact metrizable space is a continuous image of the Cantor space. Furthermore, any group action on a compact metrizable space can be lifted to an action on the Cantor space such that the covering mentioned above is a factor map. Dualising this result gives us an equivariant AF-embeddability result for commutative unital C*-algebras. We expect such a result to hold true for nuclear C*-algebras of finite decomposition rank in general.

Abstract: We shall start with the definitions of limit and colimit, and see a few of their well known incarnations. Then we will look at a few examples of limit and colimit. Now given a tower of sub-complexes, we will address whether limit/colimit and cohomology/homology commute. It turns out that the answer to one of the questions is affirmative, and the other is more interesting.

Abstract: We shall describe the notion of parabolic vector bundles over an integral smooth projective complex curve and the corresponding moduli space of parabolic vector bundles. Next, we shall look at the quotient of this moduli under a certain finite group, and compute the Chen--Ruan cohomology of this quotient.

Abstract: In several fields like genetics, viral dynamics, pharmacokinetics and pharmacodynamics, population studies, and so on, regression models are often given by differential equations that are not analytically solvable. In this talk, Bayesian estimation and uncertainty quantification are addressed in such models. The approach is based on embedding the parametric nonlinear regression model into a nonparametric regression model and extending the definition of the parameter beyond the original model. The nonparametric regression function is expanded in a basis and normal priors are put on coefficients leading to a normal posterior, which then induces a posterior distribution on the model parameters through a projection map. The posterior can be obtained by simple direct sampling. We show that the posterior distribution of the model parameters is approximately normal. The most important consequence of this result is that the frequentist coverage of Bayesian credible regions approximately matches with their credibility levels, implying that the Bayesian and the frequentist measures of uncertainty quantification approximately agree. We consider different choices of the projection map and study their impact on the asymptotic efficiency of the Bayesian estimator. A simulation study and applications to some real data sets show the practical usefulness of the method. Extensions of the results to generalized regression and to higher order differential equation and partial differential equation models will also be discussed.

Abstract: The nonexistence of a solution to a generic algebraic equation of degree 5 (or higher than 5) as a finite combination of its coefficients using radicals and finite field operations is one of the first and the most important impossibility results in mathematics known as the Abel-Ruffini Theorem. The usual proof is not accessible without the machinery of the Galois theory. Lecturing to Moscow High School students in 1963–1964, Arnold provided an elementary proof of this fundamental result. I will attempt to explain his proof.

Abstract: In this minicourse, we will discuss various topological and analytic properties of the Mandelbrot set M, which is the connectedness locus of quadratic complex polynomials (i.e., parameters c for which the polynomial z 2 + c has a connected Julia set). The main features of M that will be touched upon include: (1) connectedness of M, (2) combinatorial rigidity of geometrically finite parameters in M (analogue of Mostow rigidity), (3) topological model of M in terms of external parameter rays, and (4) renormalization and existence of small copies of M within itself.

Abstract: Deep learning profoundly impacts science and society due to its impressive empirical success by applying data-driven artificial intelligence. A key characteristic of deep learning is that accuracy empirically scales with the sizes of the model and the amount of training data. Over the past decade, this property has enabled dramatic improvements in state-of-the-art learning architectures across various fields. However, due to a lack of mathematical/statistical foundation, the developments are limited to specific applications and do not generalize to a broader class of highly confident applications. This lack of foundation is far more evident under limited training sample regimes when applied to statistical estimation and inference. We attempt to develop statistically principled reasoning and theory to validate the application of deep learning, thereby paving the way for interpretable deep learning. Our approach builds on Bayesian statistical theory and methodology and scalable computation. We illustrate the methodology with a wide range of applications.

Abstract: Random fields indexed by amenable groups arise naturally in machine learning algorithms for structured and dependent data. On the other hand, mixing properties of such fields are extremely important tools for investigating asymptotic properties of any method/algorithm in the context of space-time statistical inference. In this work, we find a necessary and sufficient condition for weak mixing of a left-stationary symmetric stable random field indexed by an amenable group in terms of its Rosinski representation. The main challenge is ergodic theoretic - more precisely, the unavailability of an ergodic theorem for nonsingular (but not necessarily measure preserving) actions of amenable groups even along a tempered Følner sequence. We remove this obstacle with the help of a truncation argument along with the seminal work of Lindenstrauss (2001) and Tempelman (2015), and finally applying the Maharam skew-product. This work extends the domain of application of the speaker’s previous paper connecting stable random fields with von Neumann algebras via the group measure space construction of Murray and von Neumann (1936).

Abstract:

Abstract: Complex Dynamics

Abstract: Certain combinatorial problems about configuration of points in Euclidean space lead directly to topological generalizations. A motivating question in this subject is the topological Tverberg's conjecture. Alongside we also mention results proved by Kakutani and generalized by Yang. These are connected to classical results in algebraic topology such as the Borsuk-Ulam theorem. We discuss some of these results and connections to theorems in equivariant algebraic topology.

Abstract: The Conway–Schneeberger Fifteen theorem states that a given positive definite integral quadratic form is universal (i.e., represents every positive integer with integer inputs) if and only if it represents the integers up to 15. This theorem is sometimes known as “Finiteness theorem" as it reduces an infinite check to a finite one. In this talk, I would like to present my recent work along with Ben Kane where I have investigated quadratic forms which are universal when restricted to almost prime inputs and have established finiteness theorems akin to the Conway–Schneeberger Fifteen theorem.

Abstract: We will present several results related to the Harmonic Analysis of functions defined on the infinite-dimensional torus $\mathbb{T}^{\omega}$, which is the topological compact group consisting of the Cartesian product of countably infinite many copies of the one-dimensional torus, with its corresponding Haar measure. Such results include the study of absolutely divergent series, Calder\'on--Zygmund decomposition, and maximal operators associated with various differentiation bases.

Abstract: I shall start this talk with an introduction of approximation approaches for solving general applied problems so that it reaches the common audience. An introduction of singularly perturbed problems will be given and the nature of difficulty to handle these problems will be discussed. It will be shown that the existing approaches may not produce convergent approximations for several parameterized systems. Thereafter a priori and a posteriori based estimates will be explained. These estimates will be used to develop the adaptive mesh generations for systems of boundary layer originated problems. Experimental results will be explained in favor of theory at every stage of discussions. In addition I shall highlight the future research directions from my topics of interest in the computational field.

Abstract: Randomness is a useful tool to solve problems in Combinatorics. It provokes constructing non-trivial examples with unexpected properties. In this series of talks, we discuss some of such solutions. The reference for this talk is the book "The Probabilistic Method" authored by Noga Alon and Joel H. Spencer. Paul Erdos is the pioneer of this technique. He first applied this technique to show the Ramsey number $R(k,k)$ is at least $\sqrt{2}^{k}$, for each integer $k\geq3$.

Abstract: Fundamental group is an important and useful topological invariant of a topological space. In algebraic geometry, the spaces we are interested in are algebraic varieties or schemes or stacks, which are naturally equipped with Zariski topology. Zariski topology being quite coarse compared to Euclidean topology, the notion of homotopy classes of loops does not generalize in a straight-forward way for an algebraic variety. However, there are many analogs of topological fundamental group for an algebraic variety or a scheme, which are mostly group-schemes (schemes together with compatible group structures). Understanding these group-schemes is useful to understand geometry and topology of an algebraic variety or a scheme. In the first part of the talk, I shall briefly discuss the notions of the etale fundamental group, Nori's fundamental group-scheme and the S-fundamental group-scheme of an algebraic variety, and relationships among them. In the second part of the talk, I shall talk about my joint work with Ronnie Sebastian on the S-fundamental group-scheme and Nori's fundamental group-scheme of the Hilbert scheme of n points on a smooth projective surface defined over an algebraically closed field of characteristic p > 3.

Abstract: Classical Fatou theorems relate different notions of differentiation of measures with different notions of convergence of the Poisson integral of the measure to the boundary of the upper half-plane. In the first lecture, we will describe various generalizations of these results for the upper half-space and the Siegel upper half-space. Some of these results have been proved recently. In the second lecture, we will try to sketch the proofs of some of these results.

Abstract: Classical Fatou theorems relate different notions of differentiation of measures with different notions of convergence of the Poisson integral of the measure to the boundary of the upper half-plane. In the first lecture, we will describe various generalizations of these results for the upper half-space and the Siegel upper half-space. Some of these results have been proved recently. In the second lecture, we will try to sketch the proofs of some of these results.

Abstract: Brain signals often show intrinsic oscillations at “gamma” frequency range (30-80 Hz), which can be induced by presenting visual stimuli such as bars, gratings and some colors. Gamma oscillations are also modulated by high-level cognitive processes such as attention and memory and are abnormal in patients suffering from mental disorders such as Autism and Schizophrenia. These oscillations thus provide ways to investigate neural processes in health and disease. In the first part of the talk, I will show that gamma oscillations are severely attenuated in the presence of small discontinuities in the stimulus, which could be explained by a simple rate model with an excitatory and inhibitory population with stimulus-tune recurrent inputs. This suggests that gamma rhythm could be a resonant phenomenon arising from a fine balance of excitation and inhibition. In the second part, I will discuss how stimulus induced gamma oscillations recorded using EEG vary with healthy ageing in a large cohort of elderly subjects (> 50 years) as well as with the onset of Alzheimer’s Disease (AD) and can be used as a biomarker in early diagnosis of AD.

Abstract: This last talk will be about studying the space of almost complex structures on a given manifold, through the specific example of the six sphere S^6; beyond the definition of an almost complex structure, it will be independent of the previous talks. By thinking of S^6 as the unit sphere in the imaginary octonions, one detects a real projective seven-space RP^7 in the space of all almost complex structures on S^6. On the other hand, using the Haefliger-Sullivan rational homotopy theoretic model for the space of sections of a fiber bundle applied to the twistor space construction, one can abstractly calculate that the rational homology of the space of almost complex structures on S^6 agrees with that of RP^7. We show that, even though the inclusion of the octonionic RP^7 into the space of all almost complex structures is not a homotopy equivalence, it is a rational homology equivalence that induces an isomorphism on fundamental groups, and we describe the homotopy fiber.

Abstract: Understanding how adult humans learn to categorize can shed insights into the mechanisms underlying experience-dependent brain plasticity. Drift-diffusion processes are popular in such contexts for their ability to mimic underlying neural mechanisms but require data on both category responses and associated response times for inference. Category response accuracies are, however, often the only measure recorded by behavioural scientists to describe human learning. Building carefully on drift-diffusion models with latent response times, we derive a biologically interpretable inverse-probit categorical probability model for such data. The model, however, presents significant identifiability and inference challenges. We address these challenges via a novel projection-based approach with a symmetry preserving identifiability constraint that allows us to work with conjugate priors in an unconstrained space. We adapt the model for group and individual level inference in longitudinal settings. Building again on the model's latent variable representation, we design an efficient Markov chain Monte Carlo algorithm for posterior computation. The method's practical efficacy is illustrated in applications to longitudinal tone learning studies.

Abstract: We continue with the characterization of simply connected rational homotopy types realized by closed almost complex manifolds, invoking techniques from surgery theory to show that the necessary conditions for realization that we began and will continue to list are in fact sufficient. We will then go through some explicit examples of obtaining closed almost complex manifolds with perhaps surprising rational homotopy types.

Abstract: Classical distance based classifiers like the nearest neighbor and average distance are quite effective methods for discrimination. However, these methods suffer from severe problems in high dimension low sample size (HDLSS) settings, and yield poor performance if the differences in their locations get masked by the scale differences. In the literature, several methods have been developed that are quite effective when the competing populations differ either in locations, scales or both. In this talk, we shall propose some transformations of these classical methods that enable us to discriminate between populations which possess differences under more general conditions (e.g., the underlying populations have the same location and scale). Further, we propose a generalization of these classifiers based on the idea of grouping of variables. We shall then analyze the performance of the proposed classifiers using some numerical examples, and state their theoretical properties using ideas from HDLSS asymptotics.

Abstract: We will obtain large values of the argument of the Riemann zeta function using the resonance method. We will also apply the method to the iterated arguments. This is a joint work with A. Chirre.

Abstract: This talk is on understanding how the rare events of deviation of sample mean from population mean cluster. Consider data coming from a stationary process whose marginal distribution has all exponential moments finite. Conditional on the event that the average of the first n observations exceeds the population mean by a prefixed threshold, we study the asymptotic distribution of the length of time for which this deviation persists, as n goes to infinity. It turns out that the answer depends on the memory of the stationary process. In the short memory regime, the asymptotic conditional distribution of the persistence time is a law on the set of natural numbers, whereas in the long memory regime, the persistence time goes almost surely to infinity. In the latter regime, the asymptotic conditional law under an appropriate scaling is studied. This is a joint work with Gennady Samorodnitsky.

Abstract: In this second talk, we will consider the topological obstructions to admitting an almost complex structure on a compact manifold. These will be formulated in terms of characteristic classes, namely the Stiefel-Whitney, Chern, and Pontryagin classes. We will discuss the relations among them, and how certain rational polynomials in the Chern classes have to evaluate to integers on almost complex manifolds. As applications, we will see e.g. that the four-sphere does not admit an almost complex structure, as well as the slightly more involved example where hints of number-theoretic problems begin to appear, the quaternionic projective plane.

Abstract: A tolerance band for a functional response provides a region that is expected to contain a given fraction of observations from the sampled population at each point in the domain. This band is a functional analog of tolerance interval for univariate response. Although the problem of constructing functional tolerance bands has been considered for a Gaussian response, it has not been considered for non-Gaussian responses, which are common in biomedical applications. In this talk, a methodology is described for constructing tolerance bands for two non-Gaussian members of the exponential family: binomial and Poisson. The approach is to first model the data using the framework of generalized functional principal components analysis. Then, a parameter is identified in which the marginal distribution of the response is stochastically monotone. It is shown that the tolerance limits can be readily obtained from confidence limits of this parameter, which in turn can be computed using large-sample theory and bootstrapping. The proposed methodology works for both dense and sparse functional data. Simulation studies are conducted to evaluate its performance and get recommendations for practical applications. The methodology is illustrated by analyzing two real biomedical datasets.

Abstract: We consider pairs of Banach spaces E and E_0 which are defined, respectively, by a “big O”-condition and the corresponding “little o”-condition. Prototypes are given by \ell^\infty and c_0; the Bloch space and the little Bloch space; and the space of functions of bounded mean oscillation (BMO), and the corresponding space of vanishing mean oscillation (VMO). In all these examples, the bidual of E_0 is equal to E, E_0^{**} = E. For example, VMO* = H^1 (Hardy space with p = 1), and VMO** = (H^1)* = BMO. I will describe a construction of pairs (E, E_0) which is quite general for pairs where the space E_0 is separable. Some features of this construction are: i) the second dual of E_0 isometrically coincides with E, ii) the middle space E_0^* is always given by a kind of atomic decomposition, and iii) E_0 is an M-ideal in E. The notion of M-ideals in Banach spaces generalizes that of two-sided ideals in C*-algebras. Through M-structure theory, property iii) provides a lot of metric and geometric information for the spaces E_0 and E. As an application, we may give a useful characterization of all weakly compact operators on M_0. The results described apply in many classical situations, but also in the context of modern function spaces such as the Bourgain—Brezis—Mironescu space.

Abstract: In recent years there has been a resurgence of interest in almost complex geometry and topology, where one would like to see which phenomena from complex geometry carry over to the context of a not-necessarily-integrable complex structure: Dolbeault cohomology and its relation to harmonic theory, the Frölicher spectral sequence, Kodaira dimension, etc. My work in the subject has mostly been focused on the topological aspects of almost complex manifolds, along with the space of all such structures on a given manifold; in this series I will discuss these results, together with the necessary background. As needed I will give an overview of characteristic classes, index theory, and tools from classical and rational homotopy theory, in a user-friendly way, with explicit computations. In this first talk I will discuss the contrast in what is known about the topology of compact almost complex (equivalently, almost symplectic), complex, symplectic, and Kähler manifolds, including explicit examples, in a non-technical way. In the subsequent talks I will give a characterization of the rational homotopy types of simply connected compact almost complex manifolds, together with their rational Chern classes. The proof of this characterization relies on surgery theory, while explicit computations for obtaining almost complex manifolds with a given rational homotopy type often come down to number-theoretic problems, as we will see. After this, we will have a good stock of almost complex manifolds in hand, and one can turn to the question of what the space of all almost complex structures on a manifold looks like: an illustrative case we will consider is that of the six-dimensional sphere, where one obtains many almost complex structures from thinking of it as the sphere of unit vectors in the imaginary octonions.

Abstract: We consider Bayesian empirical likelihood estimation and develop an efficient Hamiltonian Monte Carlo method for sampling from the posterior distribution of the parameters of interest. The proposed method uses hitherto unknown properties of the gradient of the underlying log-empirical likelihood function. It is seen that these properties hold under minimal assumptions on the parameter space, prior density and the functions used in the estimating equations determining the empirical likelihood. We overcome major challenges posed by complex, non-convex boundaries of the support routinely observed for empirical likelihood which prevents efficient implementation of traditional Markov chain Monte Carlo methods like random walk Metropolis-Hastings etc. with or without parallel tempering. Our method employs a finite number of estimating equations and observations but produces valid semi-parametric inference for a large class of statistical models including mixed effects models, generalised linear models, hierarchical Bayes models etc. A simulation study confirms that our proposed method converges quickly and draws samples from the posterior support efficiently. We further illustrate its utility through an analysis of a discrete data-set in small area estimation.

Abstract: We are concerned here with some statistical problems (in particular, related to classification) in which the available data are not elements of the usual finite-dimensional Euclidean space. Such “general” cases include, for example, situations where the data are functions (not necessarily living in a Hilbert space), images, point patterns, graphs or shapes. In these situations, the choice of a suitable distance between data turns out to be crucial. Three particular examples will be briefly discussed: a functional version of the Mahalanobis distance (suitable for trajectories of L2-processes), a “visual metric” suitable for spectrometric curves, and a “probabilistic distance” adapted to problems of shape discrimination. The contents of this talk are a summary of several recent joint works with different co-authors: J. R. Berrendero, A. Cholaquidis, B. Bueno, R. Fraiman and B. Pateiro.

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.