TBA

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

Fano varieties are algebraic varieties with positive curvature; they are basic building blocks of algebraic varieties. Great progress has been recently made by Xu et al. to construct moduli spaces of Fano varieties by using K-stability (which is related to the existence of Kähler-Einstein metrics). These moduli spaces are called K-moduli. In this talk I will explain how to easily deduce some geometric properties of K-moduli by using toric geometry and deformation theory. In particular, I will show how to construct a 1-dimensional component of K-moduli which parametrises certain K-polystable del Pezzo surfaces. * ZOOM information will not be provided. Please send an email to Jinhyung Park if you are interested in.

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심사위원장: 백형렬, 심사위원:최서영, 응우옌 응옥 쿠옹, 김상현(KIAS), 이계선(서울대학교)

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심사위원장: 백형렬, 심사위원:최서영, 응우옌 응옥 쿠옹, 김상현(KIAS), 이계선(서울대학교)

Deep generative models (DGM) have been an intersection between the probabilistic modeling and the machine learning communities. Particularly, DGM has impacted the field by introducing VAE, GAN, Flow, and recently Diffusion models with its capability to learn the data density of datasets. While there are many model variations in DGM, there are also common fundamental theories, assumptions and limitations to study from the theoretic perspectives. This seminar provides such general and fundamental challenges in DGMs, and later we particularly focus on the key developments in diffusion models and their mathematical properties in detail.

Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It now has been applied to solve scientific problems, which has become an emerging field, Scientific Machine Learning (SciML). Many ML techniques, however, are very complex and sophisticated, commonly requiring many trial-and-error and tricks. These result in a lack of robustness and interpretability, which are critical factors for scientific applications. This talk centers around mathematical approaches for SciML, promoting trustworthiness. The first part is about how to embed physics into neural networks (NNs). I will present a general framework for designing NNs that obey the first and second laws of thermodynamics. The framework not only provides flexible ways of leveraging available physics information but also results in expressive NN architectures. The second part is about the training of NNs, one of the biggest challenges in ML. I will present an efficient training method for NNs - Active Neuron Least Squares (ANLS). ANLS is developed from the insight gained from the analysis of gradient descent training.

TBD

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

Let $E$ be a number field and $X$ a smooth geometrically connected variety defined over a characteristic $p$ finite field. Given an $n$-dimensional pure $E$-compatible system of semisimple $\lambda$-adic representations of the \'etale fundamental group of $X$ with connected algebraic monodromy groups $\bG_\lambda$, we construct a common $E$-form $\bG$ of all the groups $\bG_\lambda$ and in the absolutely irreducible case, a common $E$-form $\bG\hookrightarrow\GL_{n,E}$ of all the tautological representations $\bG_\lambda\hookrightarrow\GL_{n,E_\lambda}$. Analogous rationality results in characteristic $p$ assuming the existence of crystalline companions in $\mathrm{\textbf{F-Isoc}}^{\dagger}(X)\otimes E_{v}$ for all $v|p$ and in characteristic zero assuming ordinariness are also obtained. Applications include a construction of $\bG$-compatible system from some $\GL_n$-compatible system and some results predicted by the Mumford-Tate conjecture. (If you would like to join this seminar please contact Bo-Hae Im to get the zoom link.)

A linear $3$-graph is called a (3-)hypertree if there exists exactly one path between each pair of two distinct vertices.  A linear $3$-graph is called a Steiner triple system if each pair of two distinct vertices belong to a unique edge. A simple greedy algorithm shows that every $n$-vertex Steiner triple system $G$ contains all hypertrees $T$ of order at most $\frac{n+3}{2}$. On the other hand, it is not immediately clear whether one can always find larger hypertrees in $G$. In 2011, Goodall and de Mier proved that a Steiner triple system $G$ contains at least one spanning tree. However, one cannot expect the Steiner triple system to contain all possible spanning trees, as there are many Steiner triple systems that avoid numerous spanning trees as subgraphs. Hence it is natural to wonder how much one can improve the bound from the greedy algorithm. Indeed, Elliott and Rödl conjectured that an $n$-vertex Steiner triple system $G$ contains all hypertrees of order at most $(1-o(1))n$. We prove the conjecture by Elliott and Rödl. This is joint work with Jaehoon Kim, Joonkyung Lee, and Abhishek Methuku.

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서울대학교 산업공학과 학사 행정고시 37회 재경직 합격 KDI 국제정책대학원 석사 서울지방국세청 조사과장 국세청 기획재정담당관

TBA

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

The development and analysis of efficient methods and techniques for solving an inverse scattering problem are of great interest due to their potential in various applications, such as nondestructive testing, biomedical imaging, radar imaging, and structural imaging, among others. Sampling-type imaging methods allow us to non-iteratively retrieve the support of (possibly multiconnected) targets with low computational cost, assuming no a priori information about the targets. A sampling method tests a region of interest with its associated indicator function; the indicator function blows up if a test location is in support of inhomogeneities. Even though the sampling methods show promising results in ideal (multistatic, full-aperture, sufficiently many receivers) measurement configuration, one can frequently encounter limited measurement cases in practical applications. This presentation introduces the sampling-type imaging methods in two-dimensional limited-aperture, monostatic, and bistatic measurement cases. We identify the asymptotic structure of imaging methods to explore the applicability and intrinsic properties.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87958862292

Geometric group theory concerns about how to see geometric properties in finitely generated groups. Defining Cayley graph of a finitely generated group with respect to finite generating set gives a perspective to describe geometric properties of finitely generated groups. Once we get a geometric perspective, we can classify finitely generated groups via quasi-isometry, since two Cayley graphs are quasi-isometric. In this talk, we will explain some basic notions appeared in geometric group theory (for example, quasi-isometry, hyperbolic groups, Švarc–Milnor lemma) and some theorems related to (relative) hyperbolicity of groups.

In this talk, we will introduce the absolute coregularity of Fano varieties. The coregularity measures the singularities of the anti-pluricanonical sections. Philosophically, most Fano varieties have coregularity 0. In the talk, we will explain some theorems that support this philosophy. We will show that a Fano variety of coregularity 0 admits a non-trivial section in |-2K_X|, independently of the dimension of X. This is joint work with Fernando Figueroa, Stefano Filipazzo, and Junyao Peng. * ZOOM information will not be provided. Please send an email to Jinhyung Park if you are interested in.

By a seminal result of Valiant, computing the permanent of (0, 1)-matrices is, in general, #P-hard. In 1913 Pólya asked for which (0, 1)-matrices A it is possible to change some signs such that the permanent of A equals the determinant of the resulting matrix. In 1975, Little showed these matrices to be exactly the biadjacency matrices of bipartite graphs excluding $K_{3,3}$ as a matching minor. This was turned into a polynomial time algorithm by McCuaig, Robertson, Seymour, and Thomas in 1999. However, the relation between the exclusion of some matching minor in a bipartite graph and the tractability of the permanent extends beyond K3,3. Recently it was shown that the exclusion of any planar bipartite graph as a matching minor yields a class of bipartite graphs on which the permanent of the corresponding (0, 1)-matrices can be computed efficiently. In this paper we unify the two results above into a single, more general result in the style of the celebrated structure theorem for single-crossing minor-free graphs. We identify a class of bipartite graphs strictly generalising planar bipartite graphs and $K_{3,3}$ which includes infinitely many non-Pfaffian graphs. The exclusion of any member of this class as a matching minor yields a structure that allows for the efficient evaluation of the permanent. Moreover, we show that the evaluation of the permanent remains #P-hard on bipartite graphs which exclude $K_{5,5}$ as a matching minor. This establishes a first computational lower bound for the problem of counting perfect matchings on matching minor closed classes. As another application of our structure theorem, we obtain a strict generalisation of the algorithm for the k-vertex disjoint directed paths problem on digraphs of bounded directed treewidth. This is joint work with Archontia Giannopoulou and Dimitrios Thilikos.

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서울대학교 산업공학과 학/석사 삼성전자 연구원 조지아텍 박사 삼성전자 종합기술원 전문연구원 현) 삼성전자 종합기술원 머신러닝 랩장

Stochastic finite-sum optimization problems are ubiquitous in many areas such as machine learning, and stochastic optimization algorithms to solve these finite-sum problems are actively studied in the literature. However, there is a major gap between practice and theory: practical algorithms shuffle and iterate through component indices, while most theoretical analyses of these algorithms assume uniformly sampling the indices. In this talk, we talk about recent research efforts to close this theory-practice gap. We will discuss recent developments in the theoretical convergence analysis of shuffling-based optimization methods. We will first consider minimization algorithms, mainly focusing on stochastic gradient descent (SGD) with shuffling; we will then briefly talk about some new progress on minimax optimization methods.

Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. Although there are some recent works on uncertainty quantification (UQ) in NNs, there is no systematic investigation of suitable methods towards quantifying the total uncertainty effectively and efficiently even for function approximation, and there is even less work on solving partial differential equations and learning operator mappings between infinite-dimensional function spaces using NNs. In this talk, we will present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as evaluation metrics and post-hoc improvement approaches. To demonstrate the applicability and reliability of our framework, we will also present an extensive comparative study in which various methods are tested on prototype problems, including problems with mixed input-output data, and stochastic problems in high dimensions.

In this talk, we discuss the Cauchy problem for the Vlasov-Riesz system, which is a Vlasov equation featuring interaction potentials generalizing various previously studied cases, including the Coulomb and Manev potentials. For the first time, we extend the local theory of classical solutions to interaction potentials which are more singular than that for the Manev. Then, we obtain finite-time singularity formation for solutions with various attractive interaction potentials, extending the well-known singularity formation result for attractive Vlasov-Poisson. Our local well-posedness and singularity formation results extend to cases with linear diffusion and damping in velocity.

In recent years, community detection has been an active research area in various fields including machine learning and statistics. While a plethora of works has been published over the past few years, most of the existing methods depend on a predetermined number of communities. Given the situation, determining the proper number of communities is directly related to the performance of these methods. Currently, there does not exist a golden rule for choosing the ideal number, and people usually rely on their background knowledge of the domain to make their choices. To address this issue, we propose a community detection method that is equipped with data-adaptive methods of finding the number of the underlying communities. Central to our method is fused l-1 penalty applied on an induced graph from the given data. The proposed method shows promising results.

A theorem of Khare-Wintenberger and Kisin (once Serre’s modularity conjecture) says that every two-dimensional odd absolutely irreducible representation \bar\rho of the Galois group of the rationals over a finite field comes from a modular form f, that is, \bar\rho ~ \bar\rho_f. The conjecture even provides a recipe for the weight, level and character of f, but does not give any information about the slope of f. In this talk, based on joint work with Kumar, we provide conditions on f - the main one being that the weight of f is close to 0 - which guarantee that the slope of a modular form g giving rise to the twist of \bar\rho_f by the cyclotomic character has slope one more than the slope of f. This provides a global explanation of some local patterns mentioned in our first talk. The proof uses the theta operator and Coleman-Hida families of overconvergent forms. (This is the second of the two KAIX Invited Lectures.)

Wearable analytics hold far more potential than sleep tracking or step counting. In recent years, a number of applications have emerged which leverage the massive quantities of data being amassed by wearables around the world, such as real-time mood detection, advanced COVID screening, and heart rate variability analysis. Yet packaging insights from research for success in the consumer market means prioritizing design and understandability, while also seamlessly managing the sometimes-unreliable stream of data from the device. In this presentation, I will discuss my own experiences building apps which interface with wearable data and process the data using mathematical modeling, as well as recent work extending to other wearable streams and environmental controls.

Cells make individual fate decisions through linear and nonlinear regulation of gene network, generating diverse dynamics from a single reaction pathway. In this colloquium, I will present two topics of our recent work on signaling dynamics at cellular and patient levels. The first example is about the initial value of the model, as a mechanism to generate different dynamics from a single pathway in cancer and the use of the dynamics for stratification of the patients [1-3]. Models of ErbB receptor signaling have been widely used in prediction of drug sensitivity for many types of cancers. We trained the ErbB model with the data obtained from cancer cell lines and predicted the common parameters of the model. By simulation of the ErbB model with those parameters and individual patient transcriptome data as initial values, we were able to classify the prognosis of breast cancer patients and drug sensitivity based on their in silico signaling dynamics. This result raises the question whether gene expression levels, rather than genetic mutations, might be better suited to classify the disease. Another example is about the regulation of transcription factors, the recipients of signal dynamics, for target gene expression [4-6]. By focusing on the NFkB transcription factor, we found that the opening and closing of chromatin at the DNA regions of the putative transcription factor binding sites and the cooperativity in their interaction significantly influenced the cell-to cell heterogeneity in gene expression levels. This study indicates that the noise in gene expression is rather strongly regulated by the DNA side, even though the signals are similarly regulated in a cell population. Overall these mechanisms are important in our understanding the cell as a system for encoding and decoding signals for fate decisions and its application to human diseases.

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

The zig-zag conjecture predicts that the reductions of two-dimensional irreducible p-adic crystalline representations of half-integral slope and exceptional weights - weights which are two more than twice the slope modulo (p-1) - have reductions which are given by an alternating sequence of irreducible and reducible representations. Some partial progress was made towards this conjecture over the years by Buzzard-Gee (slope 1/2), Bhattacharya-G-Rozensztajn (slope 1) and G-Rai (slope 3/2). In this talk I shall use work of Breuil-Mézard and Guerberoff-Park in the semi-stable case and a limiting argument connecting crystalline and semi-stable representations due to Chitrao-G-Yasuda to show that zig-zag holds for half-integal slopes bounded by (p-1)/2, at least on the inertia subgroup, if the weight is sufficiently close to a weight bounded by p+1. (This is the first of the two KAIX Invited Lectures.)

Shift workers experience profound circadian disruption due to the nature of their work, which often has them working at times when their internal clock is sending a strong signal for sleep. Mathematical models can be used to generate recommendations for shift workers that shift their body’s clock to better align with their work schedules, to help them sleep, feel, and perform better. In this talk, I will discuss our recent mobile app, Shift, which pulls wearable data from user’s devices and generates personalized recommendations to help them manage shift work schedules. I will also discuss how this product was designed, how it can interface with Internet of Things devices, and how its insights can be useful for other groups beyond shift workers.

Shift workers experience profound circadian disruption due to the nature of their work, which often has them working at times when their internal clock is sending a strong signal for sleep. Mathematical models can be used to generate recommendations for shift workers that shift their body’s clock to better align with their work schedules, to help them sleep, feel, and perform better. In this talk, I will discuss our recent mobile app, Shift, which pulls wearable data from user’s devices and generates personalized recommendations to help them manage shift work schedules. I will also discuss how this product was designed, how it can interface with Internet of Things devices, and how its insights can be useful for other groups beyond shift workers.

Let $\mathcal{F}$ be a family of graphs, and let $p$ and $r$ be nonnegative integers. The $(p,r,\mathcal{F})$-Covering problem asks whether for a graph $G$ and an integer $k$, there exists a set $D$ of at most $k$ vertices in $G$ such that $G^p\setminus N_G^r[D]$ has no induced subgraph isomorphic to a graph in $\mathcal{F}$, where $G^p$ is the $p$-th power of $G$ and $N^r_G[D]$ is the set of all vertices in $G$ at distance at most $r$ from $D$ in $G$. The $(p,r,\mathcal{F})$-Packing problem asks whether for a graph $G$ and an integer $k$, $G^p$ has $k$ induced subgraphs $H_1,\ldots,H_k$ such that each $H_i$ is isomorphic to a graph in $\mathcal{F}$, and for distinct $i,j\in \{1, \ldots, k\}$, the distance between $V(H_i)$ and $V(H_j)$ in $G$ is larger than $r$. The $(p,r,\mathcal{F})$-Covering problem generalizes Distance-$r$ Dominating Set and Distance-$r$ Vertex Cover, and the $(p,r,\mathcal{F})$-Packing problem generalizes Distance-$r$ Independent Set and Distance-$r$ Matching. By taking $(p',r',\mathcal{F}')=(pt, rt, \mathcal{F})$, we may formulate the $(p,r,\mathcal{F})$-Covering and $(p, r, \mathcal{F})$-Packing problems on the $t$-th power of a graph. Moreover, $(1,0,\mathcal{F})$-Covering is the $\mathcal{F}$-Free Vertex Deletion problem, and $(1,0,\mathcal{F})$-Packing is the Induced-$\mathcal{F}$-Packing problem. We show that for every fixed nonnegative integers $p,r$ and every fixed nonempty finite family $\mathcal{F}$ of connected graphs, the $(p,r,\mathcal{F})$-Covering problem with $p\leq2r+1$ and the $(p,r,\mathcal{F})$-Packing problem with $p\leq2\lfloor r/2\rfloor+1$ admit almost linear kernels on every nowhere dense class of graphs, and admit linear kernels on every class of graphs with bounded expansion, parameterized by the solution size $k$. We obtain the same kernels for their annotated variants. As corollaries, we prove that Distance-$r$ Vertex Cover, Distance-$r$ Matching, $\mathcal{F}$-Free Vertex Deletion, and Induced-$\mathcal{F}$-Packing for any fixed finite family $\mathcal{F}$ of connected graphs admit almost linear kernels on every nowhere dense class of graphs and linear kernels on every class of graphs with bounded expansion. Our results extend the results for Distance-$r$ Dominating Set by Drange et al. (STACS 2016) and Eickmeyer et al. (ICALP 2017), and the result for Distance-$r$ Independent Set by Pilipczuk and Siebertz (EJC 2021). This is joint work with Jinha Kim and O-joung Kwon.

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온라인(ZOOM) https://kaist.zoom.us/j/7011567816

In extremal graph theory, one big question is finding a condition of the number of edges that guarantees the existence of a particular substructure in a graph. In the first half of this talk, I'll talk about the history of such problems, especially focusing on clique subdivisions. In the last half of the talk, I'll introduce my recent result with Jaehoon Kim, Younjin Kim, and Hong Liu, which states that if a graph G has no dense small subgraph, then G has a clique subdivision of size almost linear in its average degree and discuss some applications and further open questions.

The classification of terminal Fano 3-folds has been tackled from different directions: for instance, using the Minimal Model Program, via explicit Birational Geometry, and via Graded Rings methods. In this talk I would like to introduce the Graded Ring Database - an upper bound to the numerics of Fano 3-folds - and discuss the role it plays in the classification and construction of codimension 4 Fano 3-folds having Fano index 2.

Castelnuovo-Mumford regularity, simply regularity, is one of the most interesting invariants in projective algebraic geometry, and the regularity conjecture due to Eisenbud and Goto says that the regularity can be controlled by the degree for any projective variety. But counterexamples to the conjecture have been constructed by some methods. In this talk we review the counterexample constructions including the Rees-like algebra method by McCullough and Peeva and the unprojection method.

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.