---

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

---

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

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.

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.)

---

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

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.

---

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

---

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

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.)

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.

---

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

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.

---

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

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

---

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.

---

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.

---

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