# 세미나 및 콜로퀴엄

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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.

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 $\pi/4$.

M. Krasner introduced a notion of valued hyperfield analogous to a valued field with a multivalued addition operation, and used it to do a theory of limits of local fields. P. Deligne did the theory of limits of local fields in a different way by defining a notion of triple, which consists of truncated discrete valuation rings and some additional data. Typical examples of a valued hyperfields and truncated discrete valuation rings are the $n$-th valued hyper field, which is quotient of a valued field by a multiplicative subgroup of the form +m^n$, where $m$ is the maximal ideal of a valuation ring, and the $n$-th residue ring, which is a quotient of a valuation ring by the $n$-th power of the maximal ideal.J. Tolliver showed that discrete valued hyperfields and triples are essentially same, stated by P. Deligne without a proof. W. Lee and the author showed that given complete discrete valued fields of mixed characteristic with perfect residue fields, any homomorphism between the $n$-th residue rings of the valued fields is lifted to a homomorphism between the valued fields for large enough $n$. This lifting process is functorial.

Motivated by above results, we show that given complete discrete valued fields of mixed characteristic with perfect residue fields, any homomorphism between the $n$-th valued hyperfields of the valued fields can be lifted to a homomorphism between the valued fields for large enough $n$, which is functorial. We also compute an upper bound of such a minimal $n$ effectively depending only on the ramification index. Most of all, any homomorphism between the first valued hyperfields of valued fields is uniquely lifted to a homomorphism between the valued fields in the case of tamely ramified valued fields. From this lifting result, we prove a relative completeness AKE-theorem via valued hyperfields for finitely ramified valued fields with perfect residue fields.

Any structure whose language is finite has a model of graph theory which is bi-interpretable with it. From this idea, Mekler further developed a way of interpreting a model into a group. This Mekler's construction preserves various model-theoretic properties such as stability, simplicity, and NTP2, thus helps us find new group examples in model theory. In this talk, I will introduce to you what Mekler's construction is and briefly show that this preserves NTP1.

Spatial sampling is particularly important for environmental statistics. A simple reasoning developed in Grafström and Tillé (2013) shows that under a self-correlated linear model, it is more efficient to select a well-spread sample in space. If we select two neighbouring units in a sample, we will tend to collect partially redundant information. Grafström and Lundström (2013) discuss at length the concept of spreading, also known as spatial balancing and its implication on estimation. The Generalized Random Tesselation Sampling GRTS design has been proposed by Stevens Jr. and Olsen (1999, 2004, 2003) to select spread samples. The pivotal method has been proposed by Deville and Tillé (2000). Grafström et al. (2012) proposed to use the pivotal method for spatial sampling. This method, called the local pivotal method, consists, at each step, in comparing two neighboring units. If the probability of one of these two units is increased, the probability of the other is decreased, which induces a repulsion between the units. The natural extension of this idea is to confront a group of units. The local pivot method was generalized by Grafström and Tillé (2013) to provide samples that are both well-balanced in space and balanced on the totals of the auxiliary variables. This method is called the local cube method. We also propose a new method that enables us to select spreader samples that all existing methods and allows the construction of periodic sampling plans when these plans exist.

Symplectic geometry has one of its origins in Hamiltonian dynamics. In the late 60s Arnold made a fundamental conjecture about the minimal number of periodic orbits of Hamiltonian vector fields. This is a far-reaching generalization of Poincaré's last geometric theorem and completely changed the field of symplectic geometry. In the last 30 years symplectic geometry has been tremendously developed due to the theory of holomorphic curves by Gromov and Floer homology theory by Floer. I will give a gentle introduction to the field of symplectic geometry and explain how modern methods give rise to existence results for periodic orbits and discover rigidity phenomena in symplectic geometry.

It has been established now that long range dependence of stationary infinitely processes is strongly related to ergodic-theoretical properties of the shift operatior acting on its L'evy measure. We discuss one case in which these ideas can can be extended to stationary infinitely divisible random flelds.

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.

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E6-1, ROOM 1401
Discrete Math
이준경 (Universität Hamburg, Hamburg, Germany)
The extremal number of subdivisions

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 C_{4}-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.

In part I, we present a short overview of typical data science (DS) and machine learning (ML) projects in Silicon Valley tech companies, using problems in transportation science and online advertising industry as examples.

In part II, we present a few ways statisticians can contribute to the success of such projects, and how they can be more equipped to make positive impacts.

In part III, we present a case study in anomaly detection, a massive-scale learning problem that is central to many applications in many businesses and science. We explore some of the challenges including modeling, scaling, methods for assessing and visualizing performance, probability calibration, and automated monitoring.

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E6-1, ROOM 1401
Discrete Math
백진언 (NIMS, Daejeon)
On the off-diagonal Erdos-Szekeres convex polygon problem

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.

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E6-1, ROOM 3434
Discrete Math
Hong Liu (University of Warwick, Warwick, UK)
Enumerating sets of integers with multiplicative constraints

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.