Saturday, October 12, 2024

2024-10-16 / 17:00 ~ 18:00

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A vector bundle on projective space is called "Steiner" if it can be recognized simply as the cokernel of a map given by a matrix of linear forms. Such maps arise from various geometric setups and one can ask: from the Steiner bundle, can we recover the geometric data used to construct it? In this talk, we will mention an interesting Torelli-type result of Dolgachev and Kapranov from 1993 that serves as an origin of this story, as well as other work that this inspired. We'll then indicate our contribution which amounts to analogous Torelli-type statements for certain tautological bundles on the very ample linear series of a polarized smooth projective variety. This is joint work with R. Lazarsfeld.

2024-10-16 / 15:30 ~ 16:30

학과 세미나/콜로퀴엄 - 대수기하학:

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A fundamental problem at the confluence of algebraic geometry, commutative algebra and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on flag varieties. Over fields of characteristic zero, this is the content of the Borel-Weil-Bott theorem and is well-understood, but in positive characteristic it remains wide open, despite important progress over the years. By embedding smaller flag varieties as Schubert subvarieties in larger ones, one can compare cohomology groups on different spaces and study their eventual asymptotic behavior. In this context I will describe a sharp stabilization result, and discuss some consequences and illustrative examples. Joint work with Keller VandeBogert.

2024-10-16 / 14:00 ~ 15:00

학과 세미나/콜로퀴엄 - 대수기하학:

by 문현석(고등과학원)

Let L be a ample line bundle on a projective scheme X. We say that (X,L) satisfies property QR(k) if the homogeneous ideal can be generated by quadrics of rank less than or equal to k. In the previous paper, we show that the Veronese embedding satisfies property QR(3). Let (X,L) be a Segre-Veronese embedding where X is a product of P^{a_i} with i=1,...,l and L is a very ample lines bundle O_X(d_1,d_2,...,d_l). In the paper [Linear determinantal equations for all projective schemes, SS2011], they prove that (X,L) satisfies QR(4) and it is determinantally presented if at least l-2 entries of d_1,...,d_l are at least 2. in this talk, we prove that (X,L) satisfies Qr(3) if and only if all the entries of d_1,...,d_l are at least 2. For one direction, we compute the radical ideal of 4 by 4 minors of a big matrix with linear forms, and for the other direction, we use the inducution on the sum of entries of (d_1,...,d_l).

2024-10-16 / 11:00 ~ 12:00

학과 세미나/콜로퀴엄 - 대수기하학:

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Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. Starting with the pioneering work of Mark Green on curves, numerous attempts have been made to extend these results to higher dimensions. Ein and Lazarsfeld proved that if A is a very ample line bundle, then K_X + mA satisfies property N_p for any m>=n+1+p. It has ever since been an open question if the same holds true for A ample and basepoint free. In recent joint work with Purnaprajna Bangere we give a positive answer to this question.

2024-10-18 / 10:30 ~ 12:00

학과 세미나/콜로퀴엄 - 대수기하학:

by 조용화(경상대)

In the 19th century, Kummer extensively studied quartic surfaces in the complex projective 3-space containing 16 nodes(=ordinary double points). One of his notable results states that a quartic surface cannot contain more than 16 nodes. This leads to a classic question: how many nodes may a surface of degree d contain? The answer to this question is known only for a very low degrees, namely, degrees 5 and 6. To find the optimal answer(31) for quintics, Beauville introduced the concept of "even sets of nodes," which turned out to be highly influential in the study of nodal surfaces. Based on the structure theorem of even sets by Casnati and Catanese, we will discuss some structure theorems of nodal quintics and sextics with maximal number of nodes. This talk is based on joint works with Fabrizio Catanese, Stephen Coughlan, Davide Frapporti, Michael Kiermaier, and Sascha Kurz.

2024-10-14 / 16:00 ~ 17:00

편미분방정식 통합연구실 세미나 - 편미분방정식: Null shell solutions - stability and instability

by (고등과학원)

In this talk, we study initial value problem for the Einstein equation with null matter fields, motivated by null shell solutions of Einstein equation. In particular, we show that null shell solutions can be constructed as limits of spacetimes with null matter fields. We also study the stability of these solutions in Sobolev space: we prove that solutions with one family of null matter field are stable, while the interaction of two families of null matter fields can give rise to an instability.

2024-10-15 / 16:00 ~ 17:00

SAARC 세미나 - SAARC 세미나:

by 라준현(KIAS)

Wave turbulence refers to the statistical theory of weakly nonlinear dispersive waves. In the weakly turbulent regime of a system of dispersive waves, its statistics can be described via a coarse-grained dynamics, governed by the kinetic wave equation. Remarkably, kinetic wave equations admit exact power-law solutions, called Kolmogorov-Zakharov spectra, which resemble Kolmogorov spectrum of hydrodynamic turbulence, and is often interpreted as a transient equilibrium between excitation and dissipation. In this talk, we will outline a local well-posedness result for kinetic wave equation for a toy model for wave turbulence. The result includes well-posedness near K-Z spectra, and demonstrates a surprising smoothing effect of the kinetic wave equation. The talk is based on the joint work with Pierre Germain (ICL) and Katherine Zhiyuan Zhang (Northeastern).

2024-10-17 / 16:15 ~ 17:15

학과 세미나/콜로퀴엄 - 콜로퀴엄:

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2-linear varieties are a rich topic. Sijong Kwak initiated the study of 3-regular varieties. In this talk I report on joined work Haoang Le Truong on the classification of smooth 3-regular varieties of small codimension 3. Some of these varieties are analogously to the 2-regular case determinantal. This first non-determinantal cases occurs in codimension 3. In this talk I report on the classification of varieties with Betti table $$ \begin{matrix} & 0 & 1 & 2 & 3\\ \hline 0: & 1 & . & . & .\\ 1: & . & . & . & .\\ 2: & . & 10 & 15 & 6 \end{matrix} $$ Our approach consist of studying extension starting from curves. Let $X \subset \mathbb P^n$ be a variety. An e-extension $Y \subset \mathbb P^{n+e}$ of $X$ is a variety, which is not a cone, such that there exists a regular sequence $y_1,\ldots,y_e$ of linear forms for the homogeneous coordinate ring $S_Y$ of $Y$ such that $S_Y/(y_1,\ldots,y_e) = S_X$ is the coordinate ring of $X$. Using a computationally easy deformation theoretic method to compute extensions, we classify the extensions of 3-regular curves in $\mathbb P^4$ to surfaces in $\mathhbb P^5$ completely.

2024-10-18 / 11:00 ~ 12:00

IBS-KAIST 세미나 - IBS-KAIST 세미나:

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There has been an increased use of scoring systems in clinical settings for the purpose of assessing risks in a convenient manner that provides important evidence for decision making. Machine learning-based methods may be useful for identifying important predictors and building models; however, their ‘black box’ nature limits their interpretability as well as clinical acceptability. This talk aims to introduce and demonstrate how interpretable machine learning can be used to create scoring systems for clinical decision making.

2024-10-17 / 10:30 ~ 11:30

학과 세미나/콜로퀴엄 - 대수기하학:

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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.

2024-10-15 / 10:30 ~ 11:30

학과 세미나/콜로퀴엄 - 대수기하학:

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2024-10-15 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Random walks on percolation

by Kyeongsik Nam(KAIST)

In general, random walks on fractal graphs are expected to exhibit anomalous behaviors, for example heat kernel is significantly different from that in the case of lattices. Alexander and Orbach in 1982 conjectured that random walks on critical percolation, a prominent example of fractal graphs, exhibit mean field behavior; for instance, its spectral dimension is 4/3. In this talk, I will talk about this conjecture for a canonical dependent percolation model.

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