Tuesday, October 1, 2024

2024-10-07 / 16:30 ~ 18:00

by 최도영(KAIST)

We present a formula for the degree of the 3-secant variety of a nonsingular projective variety embedded by a 5-very ample line bundle. The formula is provided in terms of Segre classes of the tangent bundle of a given variety. We use the generalized version of double point formula to reduce the calculation into the case of the 2-secant variety. Due to the singularity of the 2-secant variety, we use secant bundle as a nonsingular birational model and compute multiplications of desired algebraic cycles. Additionally, we compute the Hilbert-Samuel multiplicity of 2-secant variety along given variety.

2024-10-02 / 16:30 ~ 18:00

학과 세미나/콜로퀴엄 - 대수기하학:

by 정재우()

The study of monomial ideals is central to many areas of commutative algebra and algebraic geometry, with Stanley-Reisner theory providing a crucial bridge between algebraic invariants and combinatorial structures. We explore how the syzygies and Betti diagrams of Stanley-Reisner ideals can be understood through combinatorial operations on simplicial complexes. In this talk, we focus on the regularity of Stanley-Reisner ideals. We introduce a graph decomposition that bounds the regularity and a decomposition of simplicial complexes with respect to facets. In addition, we introduce secant complexes corresponding to the joins of varieties defined by Stanley-Reisner ideals and investigate the secant variety of minimal degree defined by the Stanley-Reisner ideals. This talk includes multiple collaborative works with G. Blekherman, J. Choe, J. Kim, M. Kim, and Y. Kim.

2024-10-04 / 10:30 ~ 12:00

학과 세미나/콜로퀴엄 - 대수기하학:

by 최준호(고등과학원)

In this talk we present a construction of quadratic equations and their weight one syzygies of tangent varieties using 4-way tensors of linear forms. This is in line with the 2-minor technique for quadratic equations of projective varieties and with the Oeding-Raicu theorem on equations of tangent varieties to Segre-Veronese varieties. We also discuss generalizations of the method if time permits. This is an early stage research.

2024-10-04 / 14:00 ~ 16:00

IBS-KAIST 세미나 - 수리생물학:

by 임동주(IBS 의생명수학그룹)

In this talk we discuss the paper “Analysis of a detailed multi-stage model of stochastic gene expression using queueing theory and model reduction” by Muhan Ma, et.al., Mathematical Biosciences, 2024.

2024-10-08 / 16:00 ~ 17:00

SAARC 세미나 - SAARC 세미나:

by 손영탁()

Modern machine learning methods such as multi-layer neural networks often have millions of parameters achieving near-zero training errors. Nevertheless, they maintain strong generalization capabilities, challenging traditional statistical theories based on the uniform law of large numbers. Motivated by this phenomenon, we consider high-dimensional binary classification with linearly separable data. For Gaussian covariates, we characterize linear classification problems for which the minimum norm interpolating prediction rule, namely the max-margin classification, has near-optimal generalization error. In the second part of the talk, we consider max-margin classification with non-Gaussian covariates. In particular, we leverage universality arguments to characterize the generalization error of non-linear random features model, a two-layer neural network with random first layer weights. In the wide-network limit, where the number of neurons tends to infinity, we show how non-linear max-margin classification with random features collapse to a linear classifier with a soft-margin objective.

2024-10-02 / 16:00 ~ 17:00

SAARC 세미나 - SAARC 세미나:

by 류한백()

We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margin). This problem has rich connections to relative entropy minimization, Schrodinger bridge, the enumeration of contingency tables, and random graphs with given degree sequences. Such margin-constrained random matrix turns out to be sharply concentrated around a certain deterministic matrix, which we call the "typical table". Typical tables have dual characterizations: (1) the expectation of the random matrix ensemble with minimum relative entropy from the base model constrained to have the expected target margin, and (2) the expectation of the maximum likelihood model obtained by rank-one exponential tilting of the base model. The structure of the typical table is dictated by two dual variables, which give the maximum likelihood estimates of the tilting parameters. Based on these results, for a sequence of "tame" margins that converges in $L^{1}$ to a limiting continuum margin as the size of the matrix diverges, we show that the sequence of margin-constrained random matrices converges in cut norm to a limiting kernel, which is the $L^{2}$-limit of the corresponding rescaled typical tables. The rate of convergence is controlled by how fast the margins converge in $L^{1}$. We also propose a Sinkhorn-type alternating minimization algorithm for computing typical tables, which speicalizes to the classical Sinkhorn algorithm for the Poisson base measure. We derive several new results for random contingency tables from our general framework. This talk is based on a Joint work with Sumit Mukherjee (Columbia).

2024-10-04 / 13:15 ~ 14:30

학과 세미나/콜로퀴엄 - Topology, Geometry, and Data Analysis: Introduction to Graph Neural Networks (Part 4)

by 서동엽(KAIST)

2024-10-02 / 13:30 ~ 14:30

학과 세미나/콜로퀴엄 - Topology, Geometry, and Data Analysis: Introduction to Graph Neural Networks (Part 3)

by 서동엽(KAIST)

2024-10-02 / 16:00 ~ 17:00

IBS-KAIST 세미나 - IBS-KAIST 세미나:

by ()

The study of sleep and circadian rhythms at scale requires novel technologies and approaches that are valid, cost effective and do not pose much of a burden to the participant. Here we will present our recent studies in which we have evaluated several classes of technologies and approaches including wearables, nearables, blood based biomarkers and combinations of data with mathematical models.

2024-10-01 / 10:30 ~ 11:30

학과 세미나/콜로퀴엄 - 대수기하학:

by ()

The lecture series gives a view on computational methods and their some applications to existence and classfication problems. In the first lectures I will introduce Groebner basis and their basic applications in commutative algebra such as computing kernel and images of morphism between finitely presented modules over polynomial rings. As a theoretical application of Groebner basis I will give Petri's analysis of the equations of a canonical curve. The second topic will be Computer aided existence and unirationality proofs of algebraic varieties and their moduli spaces. In case of curves liaison theory is needed, which will be developed. For existence proofs random searches over finite fields is a technique that has not been exploited very much. I will illustrate this technique in a number of examples, in particular for the construction of certain surfaces. Classification of non-minimal surfaces uses adjunction theory. We will discuss this from a computational point of view.

2024-10-08 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Canonical colourings in random graphs

by Mathias Schacht(University of Hamburg)

Rödl and Ruciński established Ramsey’s theorem for random graphs. In particular, for fixed integers $r$, $\ell\geq 2$ they showed that $n^{-\frac{2}{\ell+1}}$ is a threshold for the Ramsey property that every $r$-colouring of the edges of the binomial random graph $G(n,p)$ yields a monochromatic copy of $K_\ell$. We investigate how this result extends to arbitrary colourings of $G(n,p)$ with an unbounded number of colours. In this situation Erdős and Rado showed that canonically coloured copies of $K_\ell$ can be ensured in the deterministic setting. We transfer the Erdős-Rado theorem to the random environment and show that for $\ell\geq 4$ both thresholds coincide. As a consequence the proof yields $K_{\ell+1}$-free graphs $G$ for which every edge colouring yields a canonically coloured $K_\ell$. This is joint work with Nina Kamčev.

Events for the 취소된 행사 포함