In this talk, we will first review some recent results on the eigenvectors of random matrices under fixed-rank deformation, and then we will focus on the limit distribution of the leading eigenvectors of the Gaussian Unitary Ensemble (GUE) with fixed-rank (aka spiked) external source, in the critical regime of the Baik-Ben Arous-Peche (BBP) phase transition. The distribution is given in terms of a determinantal point process with extended Airy kernel. Our result can be regarded as an eigenvector counterpart of the BBP eigenvalue phase transition. The derivation of the distribution makes use of the recently rediscovered eigenvector-eigenvalue identity, together with the determinantal point process representation of the GUE minor process with external source. This is a joint work with Dong Wang (UCAS).

In mathematics, every mathematical object is generated along with a set of processes setting up boundaries and relationships as recently emphasized in Prof. June Huh's public lecture (July 13, 2022), commemorating his Fields Medal award. These days we live in the era of the 4th industrial revolution in which the advent of “the era of expanding technological super-gap on a global scale” is expected. More than ever including the era of Gauss (German: Gauß; 30 April 1777 – 23 February 1855) when he emphasized, "Mathematics is the queen of sciences, often condescending to render service to other sciences, but in all relations, she is entitled to the first rank," the role of mathematics is apparently getting much more important as time goes by in the era of the digital revolution. The importance of raising awareness of this cannot be overemphasized. In this talk according the above, three concrete examples are introduced to show how mathematics can practically contribute to the improvement of the human digital civilization in view of the processes setting up boundaries and relationships: 1) mathematics and "the smallest object" in physics, 2) first-principles(ab initio) in physics and mathematics, and 3) building up and utilizing our own first-principles allowing to flexibly cross boundaries between academic fields, which often makes it much easier for us to deal with various important problems. As for the practical examples, some of our recent works are briefly introduced as well, including mathematical conceptualizaiton of metaverse, construction of "physical system for linguistic data" with its ab initio-based utilization, etc; we might as well say that a sort of "Academic Continuation (analogous to analytic continuation)" is applied in each case. From this talk, we learn to boldly seek out useful mathematical connections crossing boundaries as above, more enriching the digital revolution; various academic/theoretical fields considered different from each other actually share an amount of common/similar mathematical structures.

Unlike Green's functions for elliptic equations in divergence form, Green's function for elliptic operators in nondivergence form do not possess nice pointwise bounds even in the case when the coefficients are uniformly continuous. In this talk, I will describe how to construct and get pointwise estimates for elliptic PDEs in non-divergence form with coefficients satisfying the so called Dini mean oscillation condition. I will also mention the parallel result for parabolic equations in non-divergence form.

We study the problem of maximizing a continuous DR-submodular function that is not necessarily smooth. We prove that the continuous greedy algorithm achieves a [(1-1/e)OPT-ε] guarantee when the function is monotone and Hölder-smooth, meaning that it admits a Hölder-continuous gradient. For functions that are non-differentiable or non-smooth, we propose a variant of the mirror-prox algorithm that attains a [(1/2)OPT-ε] guarantee. We apply our algorithmic frameworks to robust submodular maximization and distributionally robust submodular maximization under Wasserstein ambiguity. In particular, the mirror-prox method applies to robust submodular maximization to obtain a single feasible solution whose value is at least [(1/2)OPT-ε]. For distributionally robust maximization under Wasserstein ambiguity, we deduce and work over a submodular-convex maximin reformulation whose objective function is Hölder-smooth, for which we may apply both the continuous greedy method and the mirror-prox method. This is joint work with Duksang Lee, a fifth-year Ph.D. student at KAIST Math, and Nam Ho-Nguyen from the University of Sydney.

We’re all familiar with sleep, but how can we mathematically model it? And what determines how long and when we sleep? In this talk I’ll introduce the nonsmooth coupled oscillator systems that form the basis of current models of sleep-wake regulation and discuss their dynamical behaviour. I will describe how we are using models to unravel environmental, societal and physiological factors that determine sleep timing and outline how we are using models to inform the quantitative design of light interventions for mental health disorders and address contentious societal questions such as whether to move school start time for adolescents.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

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서울대학교 법과대학 학사 사법시험 32회 합격 판사(서울지방법원 등) 현) 김&장 법률사무소

TBA

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

Cellular, chemical, and population processes are all often represented via networks that describe the interactions between the different population types (typically called the “species”). If the counts of the species are low, then these systems are often modeled as continuous-time Markov chains on the d-dimensional integer lattice (with d being the number of species), with transition rates determined by stochastic mass-action kinetics. A natural (broad) mathematical question is: how do the qualitative properties of the dynamical system relate to the graph properties of the network? For example, it is of particular interest to know which graph properties imply that the stochastically modeled reaction network is positive recurrent, and therefore admits a stationary distribution. After a general introduction to the models of interest, I will discuss this problem, giving some of the known results. I will also discuss recent progress on the Chemical Recurrence Conjecture, which has been open for decades, which is the following: if each connected component of the network is strongly connected, then the associated stochastic model is positive recurrent.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

In computer science, random expressions are commonly used to analyze algorithms, either to study their average complexity, or to generate benchmarks to test them experimentally. In general, these approaches only consider the expressions as purely syntactic trees, and completely ignore their semantics — i.e. the mathematical object represented by the expression. However, two different expressions can be equivalent (for example “0*(x+y)” and “0” represent the same expression, the null expression). Can these redundancies question the relevance of the analyses and tests that do not take into account the semantics of the expressions? I will present how the uniform distribution over syntactic expression becomes completely degenerate when we start taking into account their semantics, in a very simple but common case where there is an absorbing element. If time permits it, I will briefly explain why the BST distribution offers more hope. This is a joint work with Cyril Nicaud and Pablo Rotondo.

Wigner's jellium is a model for a gas of electrons. The model consists of unit negatively charged particles lying in a sea of neutralizing homogeneous positive charges spread out according to Lebesgue measure. The key challenge in analyzing this system stems from the long-range Coulomb interactions. While the motivation for the jellium stems from physics, Coulomb systems appear in a variety of different research fields such as random matrix theory. In the first part of this talk, I will review key limit results for classical Coulomb systems in large domains. In the second part, I will present some recent advances for quantum Coulomb systems.

Order types are a combinatorial classification of finite point sets used in discrete and computational geometry. This talk will give an introduction to these objects and their analogue for the projective plane, with an emphasis on their symmetry groups. This is joint work with Emo Welzl: https://arxiv.org/abs/2003.08456

We determine the maximum number of copies of $K_{s,s}$ in a $C_{2s+2}$-free $n$-vertex graph for all integers $s \ge 2$ and sufficiently large $n$. Moreover, for $s\in\{2,3\}$ and any integer $n$ we obtain the maximum number of cycles of length $2s$ in an $n$-vertex $C_{2s+2}$-free bipartite graph. This is joint work with Ervin Győri (Renyi Institute), Zhen He (Tsinghua University), Zequn Lv (Tsinghua University), Casey Tompkins (Renyi Institute), Kitti Varga (Technical University of Budapest BME), and Xiutao Zhu (Nanjing University).

The driving passion of molecular cell biologists is to understand the molecular mechanisms that control important aspects of cell physiology, but this ambition is – paradoxically – limited by the very wealth of molecular details currently known about these mechanisms. Their complexity overwhelms our intuitive notions of how molecular regulatory networks might respond under normal and stressful conditions. To make progress we need a new paradigm for connecting molecular biology to cell physiology. I will outline an approach that uses precise mathematical methods to associate the qualitative features of dynamical systems, as conveyed by ‘bifurcation diagrams’, with ‘signal–response’ curves measured by cell biologists.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

Cell growth, DNA replication, mitosis and division are the fundamental processes by which life is passed on from one generation of eukaryotic cells to the next. The eukaryotic cell cycle is intrinsically a periodic process but not so much a ‘clock’ as a ‘copy machine’, making new daughter cells as warranted. Cells growing under ideal conditions divide with clock-like regularity; however, if they are challenged with DNA-damaging agents or mitotic spindle disruptors, they will not progress to the next stage of the cycle until the damage is repaired. These ‘decisions’ (to exit and re-enter the cell cycle) are essential to maintain the integrity of the genome from generation to generation. A crucial challenge for molecular cell biologists in the 1990s was to unravel the genetic and biochemical mechanisms of cell cycle control in eukaryotes. Central to this effort were biochemical studies of the clock-like regulation of ‘mitosis promoting factor’ during synchronous mitotic cycles of fertilized frog eggs and genetic studies of the switch-like regulation of ‘cyclin-dependent kinases’ in yeast cells. The complexity of these control systems demands a dynamical approach, as described in the first lecture. Using mathematical models of the control systems, I will uncover some of the secrets of cell cycle ‘clocks’ and ‘switches’.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

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This lecture explores a list of topics and areas that have led my research in computational mathematics and deep learning in recent years. Numerical approaches in computational science are crucial for understanding real-world phenomena, and deep neural networks have achieved state-of-the-art performance in a variety of fields. The exponential growth and the extreme success of deep learning and scientific computing have seen application across a multitude of disciplines. In this lecture, I will focus on recent advancements in scientific computing and deep learning such as adversarial examples, nanophotonics, and numerical PDEs.

This talk is about the complex dynamics, which cares the iteration of holomorphic map (usually a rational map on the Riemann sphere), and the shift locus is a nice set of polynomials that all critical points escape to infinity under iteration. Understanding the shape and topology of shift locus is a challenge for decades, and accumulated works are done by Blanchard, Branner, Hubbard, Keen, McMullen, and recently Calegari introduce a nice lamination model. In this talk I will explain the basic complex dynamics and introduce the topology of the shift locus of cubic polynomials done by Calegari's paper 'Sausages and Butcher paper' and if time allows, I will end this talk with the connection to the Big mapping class group, the MCG of Sphere - Cantor set.

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서울대 의학 학사/박사 한국학중앙연구원 철학 박사 현) 이화여대 의과대학교수

This series of talks is intended to be a gentle introduction to the random walk theory on infinite groups and hyperbolic spaces. We will touch upon keywords including hyperbolicity, stationary measure, boundaries and limit laws. Those who are interested in geometric group theory or random walks are welcomed to join.

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This is a casual seminar among TARGET students, but other graduate students are also welcomed.

For a graph $F$, the Turán number is the maximum number of edges in an $n$-vertex simple graph not containing $F$. The celebrated Erdős-Stone-Simonovits Theorem gives that \[ \text{ex}(n,F)=\bigg(1-\frac{1}{\chi(F)-1}+o(1)\bigg)\binom{n}{2},\] where $\chi(F)$ is the chromatic number of $H$. This theorem asymptotically solves the problem when $\chi(F)\geqslant 3$. In case of bipartite graphs $F$, not even the order of magnitude is known in general. In this talk, I will introduce some recent progress on Turán numbers of bipartite graphs and related generalizations and discuss several methods developed in recent years. Finally, I will introduce some interesting open problems on this topic.

This series of talks is intended to be a gentle introduction to the random walk theory on infinite groups and hyperbolic spaces. We will touch upon keywords including hyperbolicity, stationary measure, boundaries and limit laws. Those who are interested in geometric group theory or random walks are welcomed to join.

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This is a casual seminar among TARGET students, but other graduate students are also welcomed.

Compositional data analysis with a high proportion of zeros has gained increasing popularity, especially in chemometrics and human gut microbiomes research. Statistical analyses of this type of data are typically carried out via a log-ratio transformation after replacing zeros with small positive values. We should note, however, that this procedure is geometrically improper, as it causes anomalous distortions through the transformation. We propose a radial transformation that does not require zero substitutions and more importantly results in essential equivalence between domains before and after the transformation. We show that a rich class of kernels on hyperspheres can successfully define a kernel embedding for compositional data based on this equivalence. The applicability of the proposed approach is demonstrated with kernel principal component analysis.

We consider a deep generative model for nonparametric distribution estimation problems. The true data-generating distribution is assumed to possess a certain low-dimensional structure. Under this assumption, we study convergence rates of estimators obtained by likelihood approaches and generative adversarial networks (GAN). The convergence rate depends only on the noise level, intrinsic dimension and smoothness of the underlying structure. The true distribution may or may not possess the Lebesgue density, depending on the underlying structure. For the singular case (no Lebesgue density), the convergence rate of GAN is strictly better than that of the likelihood approaches. Our lower bound of the minimax optimal rates shows that the convergence rate of GAN is close to the optimal rate. If the true distribution allows a smooth Lebesgue density, an estimator obtained by a likelihood approach achieves the minimax optimal rate.

What allowed for many developments in algebraic geometry and commutative algebra was a discovery of the notion of a Frobenius splitting, which, briefly speaking, detects how pathological positive characteristic Fano and Calabi-Yau varieties can be. Recently, Yobuko introduced a more general concept, a quasi-F-splitting, which captures much more refined arithmetic invariants. In my talk, I will discuss on-going projects in which we develop the theory of quasi-F-splittings in the context of birational geometry and derive applications, for example, to liftability of singularities. This is joint work with Tatsuro Kawakami, Hiromu Tanaka, Teppei Takamatsu, Fuetaro Yobuko, and Shou Yoshikawa. * Zoom information will not be provided. Please send an email to Jinhyung Park if you are interested in.

In real world, people are interested in causality rather than association. For example, pharmaceutical companies want to know effectiveness of their new drugs against diseases. South Korea Government officials are concerned about the effects of recent regulation with respect to an electric car subsidy from United States. Due to this reason, causal inference has been received much attention in decades and it is now a big research field in statistics. In this seminar, I will talk about basic idea and theory in the causal inference. Real data examples will be discussed.

Van der Waerden's theorem states that any coloring of $\mathbb{N}$ with a finite number of colors will contain arbitrarily long monochromatic arithmetic progressions. This motivates the definition of the van der Waerden number $W(r,k)$ which is the smallest $n$ such that any $r$-coloring of $\{1,2,\cdots,n\}$ guarantees the presence of a monochromatic arithmetic progression of length $k$. It is natural to ask what other arithmetic structures exhibit van der Waerden-type results. One notion, introduced by Landman and Robertson, is that of a $D$-diffsequence, which is an increasing sequence $a_1 Host: Sang-il Oum     English     2022-09-02 18:06:28 Detailed