We consider Lipschitz percolation in d + 1 dimensions above planes tilted by an angle $gamma$ along one or several coordinate axes. In particular, we discuss the asymptotics of the critical probability as d goes to infinity as well as $gamma$ going to $pi/4$.

We consider the problem of community detection or clustering in the labeled Stochastic Block Model (LSBM) with a finite number K of clusters of sizes linearly growing with the global population of items n. Every pair of items is labeled independently at random, and label ℓ appears with probability p(i,j,ℓ) between two items in clusters indexed by i and j, respectively. The objective is to reconstruct the clusters from the observation of these random labels. Clustering under the SBM and their extensions has attracted much attention recently. Most existing work aimed at characterizing the set of parameters such that it is possible to infer clusters either positively correlated with the true clusters, or with a vanishing proportion of misclassified items, or exactly matching the true clusters. We find the set of parameters such that there exists a clustering algorithm with at most smisclassified items in average under the general LSBM and for any s=o(n), which solves one open problem raised in Abbe et al 2015. We further develop an algorithm, based on simple spectral methods, that achieves this fundamental performance limit within O(n polylog(n)) computations and without the a-priori knowledge of the model parameters.

In this paper we study the fast computation of the lower and upper bounds on the value function for utility maximization under the Heston stochastic volatility model with general utility functions. It is well known there is a closed form solution of the HJB equation for power utility due to its homothetic property. It is not possible to get closed form solution for general utilities and there is little literature on the numerical scheme to solve the HJB equation for the Heston model. In this paper we propose an efficient dual control Monte Carlo method for computing tight lower and upper bounds of the value function. We identify a particular form of the dual control which leads to the closed form upper bound for a class of utility functions, including power, non-HARA and Yarri utilities. Finally, we perform some numerical tests to see the efficiency, accuracy, and robustness of the method. The numerical results support strongly our proposed scheme. (Joint work with W.Y. Li and J.T. Ma)

Shimura varieties, which are generalisations of modular curves, have been intensely studied in recent decades with strong motivations from the Langlands programme. For applications in number theory, it is very important to study the mod p reductions of Shimura varieties for primes p. One way to study is via stratifying mod p Shimura varieties and analysing them stratum by stratum, and among them the Newton stratification has been studied very exhaustively by Frans Oort, Elena Mantovan, etc. In this talk, I will introduce my recent work on the structure of Newton strata of a certain general class of Shimura varieties (some of which is joint work with Paul Hamacher), generalising the earlier work of Oort and Mantovan. We will start by reviewing the Newton stratification in the simplest case of modular curves and Siegel modular varieties. If time permits, I’d like to explain the link between the geometry of Newton strata and the Langlands programme.

The celebrated conjecture of Birch and Swinnerton-Dyer (or the BSD conjecture) claims that the arithmetic of an elliptic curve over the field of rational numbers is encoded in the associated L-function in some prescribed way. Although there have been many important progresses in some special cases, the conjecture remains wide open in general. 

The BSD conjecture has a “geometric analogue”; namely, for elliptic curves over the field of rational functions in one variable over a finite field (and finite extensions thereof). The geometric analogue of BSD conjecture was formulated by John Tate, perhaps in an attempt to provide more theoretical evidence to the BSD conjecture over number fields. Also, there are more techniques (from geometry) to tackle the geometric analogue of BSD conjecture.

In this colloquium talk, I will outline the statement and the ‘early history' of the BSD conjecture and its geometric analogue, and conclude the talk with some recent developments in the geometric BSD conjecture and its equivariant refinement, which is joint work with David Burns and Mahesh Kakde.

The present work is devoted to construction of optimal quadrature formulas in the sense of Sard
and optimal interpolation formulas, and calculation of estimations of their errors in Hilbert spaces.
Here the main aim is to get explicit forms for coefficients of optimal formulas
using discrete analogs of differential operators.

One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if H is a bipartite graph with maximum degree r on one side, then there is a constant C such that every graph with n vertices and C n^{2 – 1/r} edges contains a copy of H. This result is tight up to the constant when H contains a copy of K_r,s with s sufficiently large in terms of r. We conjecture that this is essentially the only situation in which Füredi’s result can be tight and prove this conjecture for r = 2. More precisely, we show that if H is a C₄-free bipartite graph with maximum degree 2 on one side, then there are positive constants C and δ such that every graph with n vertices and C n^{3/2 – δ} edges contains a copy of H. This answers a question by Erdős from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest. This is joint work with David Conlon. 

The infamous Erdős-Szekeres conjecture, posed in 1935, states that the minimum number ES(n) of points on a plane in general position (that is, no three colinear points) that guarantees a subset of n points in convex position is equal to 2^(n-2) + 1. Despite many years of effort, the upper bound of ES(n) had not been better than O(4^{n – o(n)}) until Suk proved the groundbreaking result ES(n)≤2^n+o(n) in 2016.

Counting problems on sets of integers with additive constraints have been extensively studied. In contrast, the counting problems for sets with multiplicative constraints remain largely unexplored. In this talk, we will discuss two such recent results, one on primitive sets and the other on multiplicative Sidon sets. Based on joint work with Peter Pach, and with Peter Pach and Richard Palincza.

The branched virtual fibering theorem by Sakuma states that every closed orientable 3-manifold with a Heegaard surface of genus g has a branched double cover which is a genus g surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy of pseudo-Anosov monodromies of all hyperbolic, genus g surface bundles as branched double covers of the 3-sphere behaves like 1/g. We also give an alternative construction of surface bundles over the circle in Sakuma's theorem when closed 3-manifolds are branched double covers of the 3-sphere branched over links. A feature of surface bundles coming from our construction is that their monodromies can be read off the braids obtained from the links as the branched set.

The branched virtual fibering theorem by Sakuma states that every closed orientable 3-manifold with a Heegaard surface of genus g has a branched double cover which is a genus g surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy of pseudo-Anosov monodromies of all hyperbolic, genus g surface bundles as branched double covers of the 3-sphere behaves like 1/g. We also give an alternative construction of surface bundles over the circle in Sakuma's theorem when closed 3-manifolds are branched double covers of the 3-sphere branched over links. A feature of surface bundles coming from our construction is that their monodromies can be read off the braids obtained from the links as the branched set.

Thanks to Gromov's link condition, it is easy to construct many CAT(0) cube complexes. On the contrary, constructing hyperbolic cube complexes is often a delicate matter. In this talk I will briefly explain the standard technique that is used to show that some Right Angled Coxeter Groups are hyperbolic and I will then introduce a new technique which applies to a larger class of cube complexes (cube complexes with coupled links).

This is a collaboration with Harry Baik and Farbod Shokrieh. We generalized a classical theorem by Kazhdan on the convergence of canonical metric under a sequence of regular covers to the case of metrizable complexes as well as Riemannian manifolds.

In many applications, the dynamics of gas and plasma can be accurately modeled using kinetic Boltzmann equations. These equations are integro-differential systems posed in a high-dimensional phase space, which is typically comprised of the spatial coordinates and the velocity coordinates. If the system is sufficiently collisional the kinetic equations may be replaced by a fluid approximation that is posed in physical space (i.e., a lower dimensional space than the full phase space). The precise form of the fluid approximation depends on the choice of the moment-closure. In general, finding a suitable robust moment-closure is still an open scientific problem.

In this work we consider two specific closure methods: (1) a regularized quadrature-based closure (QMOM) and (2) a nonextensible entropy-based closure (QEXP).

In QMOM, the distribution function is approximated by Dirac deltas with variable weights and abscissas. The resulting fluid approximations have differing properties depending on the detailed construction of the Dirac deltas. We develop a high-order discontinuous Galerkin scheme to numerically solve resulting fluid equations. We also develop limiters that guarantee that the inversion problem between moments of the distribution function and the weights and abscissas of the Dirac deltas is well-posed.

In QEXP, the true distribution is replaced by a Maxwellian distribution multiplied by a quasi-exponential function. We develop a high-order discontinuous Galerkin scheme to numerically solve resulting fluid equations. We break the numerical update into two parts: (1) an update for the background Maxwellian distribution, and (2) an update for the non-Maxwellian corrections. We again develop limiters to keep the moment-inversion problem well-posed.