Let X denote the number of triangles in the random graph G(n,p). The problem of determining the asymptotic of the rate of the upper tail of X - that is, the function f_{n,p}(c) = log Pr(X > (1+c)E[X]) - has attracted considerable attention from both the combinatorics and probability communities. We will present a proof that, whenever log(n)/n << p << 1, the function f_{n,p}(c) = - [r(c) + o(1)] n^2 p ^2 log(1/p), for an explicit function r(c). This will demonstrate an approach for the study of the upper tail of behavior of "highly structured" nonlinear polynomials of Bernoulli random variables, where we expect the large deviation to be dominated by the appearance of small, dense structures. This is joint work with Frank Mousset and Wojciech Samotij.

I will introduce the notion of large fields and move toward the proof that large stable fields are separably closed.

The theory of graph quasirandomness studies several asymptotic properties of the random graph that are equivalent when stated as properties of a deterministic graph sequence and was one of the main motivations for the theory of dense graph limits, also known as theory of graphons. Since the theory of graphons can itself be used to study graph quasirandomness and can be generalized to a theory of dense limits of models of a universal first-order theory, a natural question is whether a general theory of quasirandomness is possible. In this talk, I will briefly introduce the general theory of dense limits of combinatorial objects (often associated with the name continuous combinatorics) and talk about the notion of natural quasirandomness, a generalization of quasirandomness to the same general setting of universal first-order theories. The main concept explored by our quasirandomness properties is that of unique coupleability that roughly means that any alignment of two limit objects on the same ground set "looks random". This talk is based on joint work with Alexander A. Razborov.

With the rapid growth of modern technology, many large-scale imaging studies have been or are being conducted to collect massive datasets with large volumes of imaging data, thus boosting the investigation of "next-generation functional data." These enormous collections of imaging data contain interesting information and valuable knowledge, whichhas raised the demand for further advancement in functional data analysis. In this talk, we mainly focus on modeling and inference of the next-generation functional data. We propose using flexible multivariate splines over triangulation or tetrahedral partitions to handle irregular domain of the images that are common in brain imaging studies and in other biomedical imaging applications. The proposed spline estimators are shown to be consistent and asymptotically normal under some regularity conditions. We also provide a computationally efficient estimator of the covariance function and derive its uniform consistency. Finally, we discuss the inferential capabilities of the proposed method. To be more specific, we develop simultaneous confidence corridors for the mean of the next-generation functional data. The procedure is also extended to the two-sample case in which we focus on comparing the mean functions of random samples drawn from two populations. The proposed method is applied to analyze brain Positron Emission Tomography (PET) data of Alzheimer's Disease.

The Bernstein-Sato polynomial of a holomorphic function is an invariant that originated in complex analysis, and with now strong applications to birational geometry and singularity theory over the complex numbers. For example, it detects the log-canonical threshold as well as the eigenvalues of the monodromy action on the cohomology of the Milnor fibre. In this talk I will present an analogue of this invariant for polynomials with Z/p^m coefficients and explain some connections to the characteristic-0 theory. This is joint work with T. Bitoun.

This talk will discuss our search for a quantum algorithm for Khovanov Homology. The project began a long time ago with the speaker and more recently with Nadia Shirakova and Sam Lomonaco. It remains in an unfinished state. The problem is how could a quantum black box efficiently produce the chain complex for Khovanov homology from the combinatorial input data of a knot or quantum knot? Once we have such data, it is possible to apply combinatorial Hodge Theory and the phase estimation methods of Lloyd and Zanardi to compute ranks of homology groups. But how can a quantum black box know the chain complex?

I'll present a new result with A. Mellit, which gives a combinatorial formula for a remarkable diagonalizing operator for the modified Macdonald polynomials, known as the nabla operator. This formula was discovered by finding a Schubert-type basis of a certain explicit module from Haiman's polygraph theory, which is conjecturally identified with the equivariant homology of the unramified affine Springer fiber studied by Goresky, Kottwitz, and Macpherson.

The hard-core/independent set model describes a system of non-overlapping identical hard spheres in a space or on a lattice. One of the outstanding open problems of statistical mechanics is: do hard disks in a plane exhibit a phase transition? It seems natural to approach this question by possible discrete approximations where disks must have the centers at sites of a lattice. Unlike most models in statistical physics, we find non-universality and connections to algebraic number theory, with different new phenomena arising in triangular, square lattices and in a hexagonal tiling. We analyze the structure of Gibbs measures for large fugacities (i.e., high densities), intrinsically related to the disk-packing problem. This is joint work with A. Mazel and Y. Suhov.

In joint work with Segal we use the fact that for Chevalley groups $G(R)$ of rank at least $2$ over a ring $R$ the root subgroups are (nearly always) the double centralizer of a corresponding root element to show under mild restrictions on the ring $R$ that $R$ and $G(R)$ are bi-interpretable. (This holds in particular for any field $k$.) For such groups it then follows that the group $G(R)$ is finitely axiomatizable in the appropriate class of groups provided $R$ is finitely axiomatizable in the corresponding class of rings.

We present a novel spatial model that predicts scalar responses based on functional predictors observed at spatial locations. We incorporate two spatial components in the modeling, (i) spatial correlation between infinite-dimensional functional predictors and (ii) spatially heterogeneous associations between responses and functional covariates at different locations, by introducing a spatially varying functional coefficient model. It allows the functional coefficients to vary with location. To preserve spatial continuity on the low dimensional representation of functional predictors, we employ nonparametric data-adaptive functions for basis expansion under a Bayesian framework and place spatial priors on projection coefficients. We further propose the spatial variable selection, which allows spatially heterogeneous sets of non-null coefficients over locations by borrowing information across neighbors. The basis function estimation, model parameter estimation, and model selection can be jointly performed through Bayesian hierarchical modeling. For the prediction on new observations, we propose the unified approach which enables the estimation of nonparametric basis functions adaptive to new functional predictors and simultaneously draws predictive values from posterior prediction distribution in MCMC implementation. The model performance is demonstrated in simulation studies and an application to a crop yield prediction.

Hyperelastic constitutive models describe a wide range of materials. Examples include biomechanical soft tissues like muscle, tendon, skin etc. Elastoplastic materials consisting of a hyperelastic constitutive model combined with a notion of stress constraint (or feasible stress region) describe an even wider range of materials. A very interesting class of these models arises from frictional contact considerations. Examples include granular materials like sand and snow. I will present recent models developed for novel applications including frictional contact for membrane and thin shell simulation, ductile fracture and baking of breads and cookies. I will also present novel Material Point Methods (MPM) used for approximating the governing equations.

Slender structures are subject to various localised instabilities: necking of bars under traction, bulging of cylindrical party balloons, beading in cylinders made up of soft gels, or folding of tape-springs. In all these examples, distinct states of deformation may coexist and classical one-dimensional (1D) models predict singular solutions. In particular, classical 1D models fail to describe interfaces or finite size effects. The most common remedy is to use full structural models based on 3D finite elasticity or nonlinear shell/membrane equations. However, this is computationally costly and often impracticable: simpler 1D regularised models depending on the strain and the strain gradient are therefore attractive. There is a recent effort to rigorously establish 1D higher-order models for the analysis of localisation in slender structures. I will introduce a systematic method to derive such models by a formal expansion, starting from a variety of full structural models for slender elastic structures. The expansion is performed near a finitely pre-strained state and therefore retains all sources of nonlinearity, coming from the geometry and the constitutive law. I will illustrate the method in the case of bulging and beading and demonstrate its accuracy by comparing solutions of the 1D gradient model with solutions of the original structural model.

We introduce a Sufficient Graphical Model by applying the recently developed nonlinear sufficient dimension reduction techniques to the evaluation of conditional independence. The graphical model is nonparametric in nature, as it does not make distributional assumptions such as the Gaussian or copula Gaussian assumptions. However, unlike a fully nonparametric graphical model, which relies on the high-dimensional kernel to characterize conditional independence, our graphical model is based on conditional independence given a set of sufficient predictors with a substantially reduced dimension. In this way we avoid the curse of dimensionality that comes with a high-dimensional kernel. We develop the population-level properties, convergence rate, and variable selection consistency of our estimate. By simulation comparisons and an analysis of the DREAM 4 Challenge data set, we demonstrate that our method outperforms the existing methods when the Gaussian or copula Gaussian assumptions are violated, and its performance remains excellent in the high-dimensional setting.

TBA

TBA

Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand maps of C to projective space of dimension r of degree d. When the curve C is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. However, in nature, curves C are often encountered already equipped with a map to some projective space, which may force them to be special in moduli. The simplest case is when C is general among curves of fixed gonality. Despite much study over the past three decades, a similarly complete picture has proved elusive in this case. In this talk, I will discuss joint work with Eric Larson and Isabel Vogt that completes such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting.

Weighted nearest neighbor (WNN) classifiers are fundamental non-parametric classifiers for classification. They have become the methods of choice in many applications where limited knowledge of the data generation process is available a priori. There exists a vast room of flexibility in the choice of weights for the neighbors in a WNN classifier. In this talk, I will introduce a new locally weighted nearest neighbor (LWNN) classifier, which adaptively assigns weights for different test data points. Given a training data set and a test data point x0, the weights for classifying x0 in LWNN is obtained by minimizing an upper bound of the conditional expected estimation error of the regression function at x0. The resultant weights have a neat closed-form expression, and therefore the computation of LWNN is more efficient than some existing adaptive WNN classifiers that require estimating the marginal feature density. Like most other WNN classifiers, LWNN assigns larger weights for closer neighbors. However, in addition to the ranks of neighbors' distances, the weights in LWNN also depend on the raw values of the distances. Our theoretical study shows that LWNN achieves the minimax rate of convergence of the excess risk, when the marginal feature density is bounded away from zero. In the general case with an additional tail assumption on the marginal feature density, the upper bound of the excess risk of LWNN matches the minimax lower bound up to a logarithmic term.

TBA

TBA

TBA

TBA

TBA