학과 세미나 및 콜로퀴엄
Deep learning has emerged as a dominant approach in machine learning and has achieved remarkable success in various domains such as computer vision and natural language processing. Its influence has progressively extended to numerous research areas within the fields of science and engineering. In this presentation, I will outline our work on the design and training of a foundation model, named PDEformer, which aims to serve as a flexible and efficient solver across a spectrum of parametric PDEs. PDEformer is specifically engineered to facilitate a range of downstream tasks, including but not limited to parameter estimation and system identification. Its design is tailored to accommodate applications necessitating repetitive solving of PDEs, where a balance between efficiency and accuracy is sought.
This is a joint workshop with the Serabol program.
We discuss how optimal transport, which is a theory for matching different distributions in a cost effective way, is applied to the supercooled Stefan problem, a free boundary problem that describes the interface dynamics of supercooled water freezing into ice. This problem exhibits a highly unstable behaviour and its mathematical study has been limited mostly to one space dimension, and widely open for multi-dimensional cases. We consider a version of optimal transport problem that considers stopping of the Brownian motion, whose solution is then translated into a solution to the supercooled Stefan problem in general dimensions.
In this talk, we consider the Boltzmann equation in general 3D toroidal domains with a specular reflection boundary condition. So far, it is a well-known open problem to obtain the low-regularity solution for the Boltzmann equation in general non-convex domains because there are grazing cases, such as inflection grazing. Thus, it is important to analyze trajectories which cause grazing. We will provide new analysis to handle these trajectories in general 3D toroidal domains.
Post-critically finite (PCF) rational maps are a fascinating class of dynamical systems with rich mathematical structures. In this minicourse, we explore the interplay between topology, geometry, and dynamics in the study of PCF rational maps.
[Lecture 3: Geometry of PCF rational maps]
Geometry of PCF rational maps The topological models for PCF rational maps we discuss define canonical quasi-symmetric classes of metrics on their Julia sets. We investigate the conformal dimensions of Julia sets, which measure their geometric complexity and provide insights into the underlying dynamics. Through this exploration, we uncover the intricate relationship between the topology, geometry, and dynamics of PCF rational maps.
The finite quotient groups of étale fundamental groups of algebraic curves in positive characteristic are precisely determined, but without explicit construction of quotient maps, by well-known results of Raynaud, Harbater and Pop, previously known as Abhyankar's conjecture. Katz, Rojas León and Tiep have been studying the constructive side of this problem using certain "easy to remember" local systems. In this talk, I will discuss the main results and methods of this project in the case of a specific type of local systems called hypergeometric sheaves.
Post-critically finite (PCF) rational maps are a fascinating class of dynamical systems with rich mathematical structures. In this minicourse, we explore the interplay between topology, geometry, and dynamics in the study of PCF rational maps.
[Lecture 2: Topology of PCF rational maps]
W.Thurston's and D.Thurston's characterizations provide powerful frameworks for understanding the topological dynamics of rational maps. We delve into these characterizations, exploring their implications for the dynamics of PCF rational maps. Additionally, we discuss finite subdivision rules and topological surgeries, such as matings, tunings, and decompositions, as tools for constructing and analyzing PCF rational maps in topological ways.
Quantum embedding is a fundamental prerequisite for applying quantum machine learning techniques to classical data, and has substantial impacts on performance outcomes. In this study, we present Neural Quantum Embedding (NQE), a method that efficiently optimizes quantum embedding beyond the limitations of positive and trace-preserving maps by leveraging classical deep learning techniques. NQE enhances the lower bound of the empirical risk, leading to substantial improvements in classification performance. Moreover, NQE improves robustness against noise. To validate the effectiveness of NQE, we conduct experiments on IBM quantum devices for image data classification, resulting in a remarkable accuracy enhancement. In addition, numerical analyses highlight that NQE simultaneously improves the trainability and generalization performance of quantum neural networks, as well as of the quantum kernel method.
We provide general upper and lower bounds for the Gromov–Hausdorff distance d_GH(S^m, S^n) between spheres S^m and S^n (endowed with the round metric) for 0 <= m < n <= 1. Some of these lower bounds are based on certain topological ideas related to the Borsuk–Ulam theorem. Via explicit constructions of (optimal) correspondences, we prove that our lower bounds are tight in the cases of d_GH(S^0, S^n), d_GH(S^m, S^\infty), d_GH(S^1, S^2), d_GH(S^1, S^3), and d_GH(S^2, S^3). We also formulate a number of open questions.
This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.
Reinforcement learning (RL) has become one of the most central problems in machine learning, showcasing remarkable success in recommendation systems, robotics and super-human level game plays. Yet, existing literature predominantly focuses on (almost) fully observable environments, overlooking the complexities of real-world scenarios where crucial information remains hidden.
In this talk, we consider reinforcement learning in partially observable systems through the proposed framework of the Latent Markov Decision Process (LMDP). In LMDPs, an MDP is randomly drawn from a set of possible MDPs at the beginning of the interaction, but the context -- the latent factors identifying the chosen MDP -- is not revealed to the agent. This opacity poses new challenges for decision-making, particularly in scenarios like recommendation systems without sensitive user data, or medical treatments for undiagnosed illnesses. Despite the significant relevance of LMDPs to real-world problems, existing theories rely on restrictive separation assumptions -- an unrealistic constraint in practical applications. We present a series of new results addressing this gap: from leveraging higher-order information to develop sample-efficient RL algorithms, to establishing lower bounds and improved results under more realistic assumptions within Latent MDPs.
Post-critically finite (PCF) rational maps are a fascinating class of dynamical systems with rich mathematical structures. In this minicourse, we explore the interplay between topology, geometry, and dynamics in the study of PCF rational maps.
[Lecture 1: What are PCF rational maps?]
We begin by introducing PCF rational maps, highlighting their significance in complex dynamics.
1. 데이터 분석 업무의 이해(김준범)- 데이터 분석가의 역할 소개
2. 초거대 언어 모델 동향(김정섭)-GPT-3 부터 Llama-3까지 이미 우리 삶 속에 깊숙이 자리잡은 초거대 언어 모델의 동향
3. 데이터 분석가에서 공직으로 오게된 과정과 앞으로의 계획(심규석)-
삼성화재에서의 데이터 분석 및 AI 모델링 업무, 행정안전부에서의 데이터 분석과제 기획·관리 및 공무원의 데이터 분석 역량지원 업무 전반에 관한 설명과 함께 각 기관을 지원하게 된 동기, 지원방법, 준비사항 등
The qualitative theory of dynamical systems mainly provides a mathematical framework for analyzing the long-time behavior of systems without necessarily finding solutions for the given ODEs. The theory of dynamical systems could be related to deep learning problems from various perspectives such as approximation, optimization, generalization, and explainability.
In this talk, we first introduce the qualitative theory of dynamical systems. Then, we present numerical results as the application of the qualitative theory of dynamical systems to deep learning problems.
This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.
EO strata are subvarieties in the moduli space of g-dimensional abelian varieties in characterstic p which classify points with given isomorphism type of p-torson subgroups.
We are interested in how automorphism groups of points vary in supersingular EO strata. We show that when g is even and p>3, there is an open dense of the maximal supersingular EO stratum in which every point has automorphism group {\pm 1}, and prove Oor's conjecture in this case.
This is joint work in progress with Valentijn Karemaker.
This lecture explores the topics and areas that have guided my research in computational mathematics and machine learning in recent years. Numerical methods in computational science are essential for comprehending real-world phenomena, and deep neural networks have achieved state-of-the-art results in a range of fields. The rapid expansion and outstanding success of deep learning and scientific computing have led to their applications across multiple disciplines, ranging from fluid dynamics to material sciences. In this lecture, I will focus on bridging machine learning with applied mathematics, specifically discussing topics such as scientific machine learning, numerical PDEs, and mathematical approaches of machine learning, including generative models and adversarial examples.
The Tomas-Stein inequality is a fundamental inequality in Fourier Analysis. It measures the L^4 norm of the Fourier transform of the sphere surface measure in terms of the L^2 norm. It is possible because the sphere has a positive Gaussian curvature. In this talk we will present what is an extremizer problem to this inequality and what is the progress of this problem.
We introduce a general equivalence problems for geometric structures arising from minimal rational curves on uniruled complex projective manifolds. To study these problems, we need approaches fusing differential geometry and algebraic geometry. Among such geometric structures, those associated to homogeneous manifolds are particularly accessible to differential-geometric methods of Cartan geometry. But even in these cases, only a few cases have been worked out so far. We review some recent developments.
Deep learning techniques are increasingly applied to scientific problems, where the precision of networks is crucial. Despite being deemed as universal function approximators, neural networks, in practice, struggle to reduce the prediction errors below O(10−5) even with large network size and extended training iterations. To address this issue, we developed the multi-stage neural networks that divides the training process into different stages, with each stage using a new network that is optimized to fit the residue from the previous stage. Across successive stages, the residue magnitudes decreases substantially and follows an inverse power-law relationship with the residue frequencies. The multi-stage neural networks effectively mitigate the spectral biases associated with regular neural networks, enabling them to capture the high frequency feature of target functions. We demonstrate that the prediction error from the multi-stage training for both regression problems and physics-informed neural networks can nearly reach the machine-precision O(10−16) of double-floating point within a finite number of iterations. Such levels of accuracy are rarely attainable using single neural networks alone.
Link prediction (LP), inferring the connectivity between nodes, is a significant research area in graph data, where a link represents essential information on relationships between nodes. Although graph neural network (GNN)-based models have achieved high performance in LP, understanding why they perform well is challenging because most comprise complex neural networks. We employ persistent homology (PH), a topological data analysis method that helps analyze the topological information of graphs, to explain the reasons for the high performance. We propose a novel method that employs PH for LP (PHLP) focusing on how the presence or absence of target links influences the overall topology. The PHLP utilizes the angle hop subgraph and new node labeling called degree double radius node labeling (Degree DRNL), distinguishing the information of graphs better than DRNL. Using only a classifier, PHLP performs similarly to state-of-the-art (SOTA) models on most benchmark datasets. Incorporating the outputs calculated using PHLP into the existing GNN-based SOTA models improves performance across all benchmark datasets. To the best of our knowledge, PHLP is the first method of applying PH to LP without GNNs. The proposed approach, employing PH while not relying on neural networks, enables the identification of crucial factors for improving performance.
https://arxiv.org/abs/2404.15225
I tell a personal story of how a mathematician working in complex algebraic geometry had come to discover the relevance of Cartan geometry, a subject in differential geometry, in an old problem in algebraic geometry, the problem of deformations of Grassmannians as projective manifolds, which originated from the work of Kodaira and Spencer. In my joint work with Ngaiming Mok, we used the theory of minimal rational curves to study such deformations and it reduced the question to a problem in Cartan geometry.
In nonstationary bandit learning problems, the decision-maker must continually gather information and adapt their action selection as the latent state of the environment evolves. In each time period, some latent optimal action maximizes expected reward under the environment state. We view the optimal action sequence as a stochastic process, and take an information-theoretic approach to analyze attainable performance. We bound per-period regret in terms of the entropy rate of the optimal action process. The bound applies to a wide array of problems studied in the literature and reflects the problem’s information structure through its information-ratio.
This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.
Hamiltonian time-series data are observations derived from a Hamiltonian dynamical system. Our goal is to analyze the time-series data using the topological information of Hamiltonian dynamical systems. Exact Multi-parameter Persistent Homology is one aspect of this analysis, in this case, the Hamiltonian system is composed of uncoupled one-dimensional harmonic oscillators. This is a very simple model. However, we can induce the exact persistence barcode formula from it. From this formula, we can obtain a calculable and interpretable analysis. Filtration is necessary to extract the topological information of data and to define persistent homology. However, in many cases, we use static filtrations, such as the Vietoris-Rips filtration. My ongoing research is on topological optimization, which involves finding a filtration in Exact Multi-parameter Persistent Homology that minimizes the cross-entropy loss function for the classification of time-series data.
In recent years, ``stealthy'' particle systems have gained considerable attention in condensed matter physics. These are particle systems for which the diffraction spectrum or structure function (i.e. the Fourier transform of the truncated pair correlation function) vanishes in a neighbourhood of the origin in the wave space. These systems are believed to exhibit the phenomenon of ``cloaking'', i.e. being invisible to probes of certain frequencies. They also exhibit the phenomenon of hyperuniformity, namely suppressed fluctuations of particle counts, a property that has been shown to arise in a wide array of settings in chemistry, physics and biology. We will demonstrate that stealthy particle systems (and their natural extensions to stealthy stochastic processes) exhibit a highly rigid structure; in particular, their entropy per unit volume is degenerate, and any spatial void in such a system cannot exceed a certain size. Time permitting, we will also discuss the intriguing correlation geometry of such systems and its interplay with the analytical
properties of their diffraction spectrum. Based on joint works with Joel Lebowitz and Kartick Adhikari.
In this presentation, we discuss comprehensive frequency domain methods for estimating and inferring the second-order structure of spatial point processes. The main element here is on utilizing the discrete Fourier transform (DFT) of the point pattern and its tapered counterpart. Under second-order stationarity, we show that both the DFTs and the tapered DFTs are asymptotically jointly independent Gaussian even when the DFTs share the same limiting frequencies. Based on these results, we establish an α-mixing central limit theorem for a statistic formulated as a quadratic form of the tapered DFT. As applications, we derive the asymptotic distribution of the kernel spectral density estimator and establish a frequency domain inferential method for parametric stationary point processes. For the latter, the resulting model parameter estimator is computationally tractable and yields meaningful interpretations even in the case of model misspecification. We investigate the finite sample performance of our estimator through simulations, considering scenarios of both correctly specified and misspecified models. Joint work with Yongtao Guan @CUHK-Shenzhen.
The r-th cactus variety of a subvariety X in a projective space generalises secant variety of X and it is defined using linear spans of finite schemes of degree r. It's original purpose was to study the vanishing sets of catalecticant minors. We propose adding a scheme structure to the cactus variety and we define it via relative linear spans of families of finite schemes over a potentially non-reduced base. In this way we are able to study the vanishing scheme of the catalecticant minors. For X which is a sufficiently large Veronese reembedding of projective variety, we show that r-th cactus scheme and the zero scheme of appropriate catalecticant minors agree on an open and dense subset which is the complement of the (r-1)-st cactus variety/scheme. As an application, we can describe the singular locus of (in particular) secant varieties to high degree Veronese varieties.
Based on a joint work with Hanieh Keneshlou.
In the past decade, machine learning methods (MLMs) for solving partial differential equations (PDEs) have gained significant attention as a novel numerical approach. Indeed, a tremendous number of research projects have surged that apply MLMs to various applications, ranging from geophysics to biophysics. This surge in interest stems from the ability of MLMs to rapidly predict solutions for complex physical systems, even those involving multi-physics phenomena, uncertainty, and real-world data assimilation. This trend has led many to hopeful thinking MLMs as a potential game-changer in PDE solving. However, despite the hopeful thinking on MLMs, there are still significant challenges to overcome. These include limits compared to conventional numerical approaches, a lack of thorough analytical understanding of its accuracy, and the potentially long training times involved. In this talk, I will first assess the current state of MLMs for solving PDEs. Following this, we will explore what roles MLMs should play to become a conventional numerical scheme.
I will discuss some recent progress on the freeness problem for groups of 2x2 rational matrices generated by two parabolic matrices. In particular, I will discuss recent progress on determining the structural properties of such groups (beyond freeness) and when they have finite index in the finitely presented group SL(2,Z[1/m]), for appropriately chosen m.
In this talk, we focus on the global existence of volume-preserving mean curvature flows. In the isotropic case, leveraging the gradient flow framework, we demonstrate the convergence of solutions to a ball for star-shaped initial data. On the other hand, for anisotropic and crystalline flows, we establish the global-in-time existence for a class of initial data with the reflection property, utilizing explicit discrete-in-time approximation methods.
The size and complexity of recent deep learning models continue to increase exponentially, causing a serious amount of hardware overheads for training those models. Contrary to inference-only hardware, neural network training is very sensitive to computation errors; hence, training processors must support high-precision computation to avoid a large performance drop, severely limiting their processing efficiency. This talk will introduce a comprehensive design approach to arrive at an optimal training processor design. More specifically, the talk will discuss how we should make important design decisions for training processors in more depth, including i) hardware-friendly training algorithms, ii) optimal data formats, and iii) processor architecture for high precision and utilization.
This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.
We begin the first talk by introducing the concept of an h-principle that is mostly accessible through the two important methods. One of the methods is the convex integration that was successfully used by Mueller and Sverak and has been applied to many important PDEs. The other is the so-called Baire category method that was mainly studied by Dacorogna and Marcellini. We compare these methods in applying to a toy example.
In the second talk of the series, we exhibit several examples of application of convex integration to important PDE problems. In particular, we shall sketch some ideas of proof such as in the p-Laplace equation and its parabolic analogue, Euler-Lagrange equation of a polyconvex energy, gradient flow of a polyconvex energy and polyconvex elastodynamics.
One of the classical and most fascinating problems at the intersection between combinatorics and number theory is the study of the parity of the partition function. Even though p(n) in widely believed to be equidistributed modulo 2, progress in the area has proven exceptionally hard. The best results available today, obtained incrementally over several decades by Serre, Soundarajan, Ono and many otehrs, do not even guarantee that, asymptotically, p(n) is odd for /sqrt{x} values of n/neq x,
In this talk, we present a new, general conjectural framework that naturally places the parity of p(n) into the much broader, number-theoretic context of eta-eqotients. We discuss the history of this problem as well as recent progress on our "master conjecture," which includes novel results on multi-and regular partitions. We then show how seemingly unrelated classes of eta-equotients carry surprising (and surprisingly deep) connections modulo 2 to the partition function. One instance is the following striking result: If any t-multiparition function, with t/neq 0(mod 3), is odd with positive density, then so is p(n). (Note that proving either fact unconditionally seems entirely out of reach with current methods.) Throughout this talk, we will give a sense of the many interesting mathematical techniques that come into play in this area. They will include a variety of algebraic and combinatorial ideas, as well as tools from modular forms and number theory.
In this talk, we consider some polynomials which define Gaussian Graphical models in algebraic statistics. First, we briefly introduce background materials and some preliminary on this topic. Next, we regard a conjecture due to Sturmfels and Uhler concerning generation of the prime ideal of the variety associated to the Gaussian graphical model of any cycle graph and explain how to prove it. We also report a result on linear syzygies of any model coming from block graphs. The former work was done jointly with A. Conner and M. Michalek and the latter with J. Choe.
We introduce bordered Floer theory and its involutive version, as well as their applications to knot complements. We will sketch the proof that invariant splittings of CFK and those of CFD correspond to each other under the Lipshitz-Ozsvath-Thurston correspondence, via invariant splitting principle, which is an ongoing work with Gary Guth.