I will explain how to put certain natural geometric structures on Tate-Shafarevich groups and other related groups attached to abelian varieties over function fields. We can refine arithmetic duality theorems by taking these geometric structures into account. This has applications to Weil-etale cohomology, the Birch-Swinnerton-Dyer conjecture and Iwasawa theory. Partially based on joint work with Geisser and with Lai, Longhi, Tan and Trihan.

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Please contact Wansu Kim at for Zoom meeting info and any inquiry.

The extremal function $c(H)$ of a graph $H$ is the supremum of densities of graphs not containing $H$ as a minor, where the density of a graph is the ratio of the number of edges to the number of vertices. Myers and Thomason (2005), Norin, Reed, Thomason and Wood (2020), and Thomason and Wales (2019) determined the asymptotic behaviour of $c(H)$ for all polynomially dense graphs $H$, as well as almost all graphs of constant density. We explore the asymptotic behavior of the extremal function in the regime not covered by the above results, where in addition to having constant density the graph $H$ is in a graph class admitting strongly sublinear separators. We establish asymptotically tight bounds in many cases. For example, we prove that for every planar graph $H$, \[c(H) = (1+o(1))\max (v(H)/2, v(H)-\alpha(H)),\] extending recent results of Haslegrave, Kim and Liu (2020). Joint work with Sergey Norin and David R. Wood.

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https://youtube.com/c/ibsdimag

(KAIX Distinguished Lectures Series)

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1st talk 10:00-11:00 short break and Q&A 11:00-11:15 2nd talk 11:15-12:15 Q&A 12:15-12:30

Recently, deep learning approaches have become the main research frontier for image reconstruction and enhancement problems thanks to their high performance, along with their ultra-fast inference times. However, due to the difficulty of obtaining matched reference data for supervised learning, there has been increasing interest in unsupervised learning approaches that do not need paired reference data. In particular, self-supervised learning and generative models have been successfully used for various inverse problem applications. In this talk, we overview these approaches from a coherent perspective in the context of classical inverse problems and discuss their various applications. In particular, the cycleGAN approach and a recent Noise2Score approach for unsupervised learning will be explained in detail using optimal transport theory and Tweedie’s formula with score matching.

This is joint work with Kenjiro Ishizuka (Kyoto). We study global behavior of solutions to the nonlinear Klein-Gordon equation with a damping and a focusing nonlinearity on the Euclidean space. Recently, Cote, Martel and Yuan proved the soliton resolution conjecture completely in the one-dimensional case: every global solution in the energy space is asymptotic to a superposition of solitons getting away from each other as time tends to infinity. The next question is to see which initial data evolve into each of the asymptotic forms. The asymptotic decomposition is very sensitive to initial perturbation because all the solitons are unstable. We consider the simplest non-trivial setting in general space dimensions: the global behavior of solutions starting near a superposition of two ground states. Cote, Martel, Yuan and Zhao proved that the solutions asymptotic to 2-solitons form a codimension-2 manifold in the energy space. Our question is what happens for the other initial data in the neighborhood. As an answer, we give a complete classification of those solutions into 5 types of global behavior. Two of them are asymptotic to the positive ground state and the negative one respectively. They form two codimension-1 manifolds that are joined at their boundary by the Cote-Martel-Yuan-Zhao manifold of 2-solitons. The connected union of those three manifolds separates the remainder of the neighborhood into the open set of global decaying solutions and that of blow-up. The main difficulty to prove it is in controlling the direction of instability in two dimensions attached to the two soliton components, because the soliton interactions are not integrable in time, breaking the simple superposition of the linearized approximation around each soliton. It is resolved by showing that the non-integrable interactions do not essentially affect the direction of instability, using the reflection symmetry of the equation and the 2-solitons. I will also explain the difficulty for the 3-solitons due to a more dramatic phenomenon, which may be called soliton merger.

I am going to present an algorithm for computing a feedback vertex set of a unit disk graph of size k, if it exists, which runs in time $2^{O(\sqrt{k})}(n + m)$, where $n$ and $m$ denote the numbers of vertices and edges, respectively. This improves the $2^{O(\sqrt{k}\log k)}(n + m)$-time algorithm for this problem on unit disk graphs by Fomin et al. [ICALP 2017].

The cohomology of Shimura varieties have rich structures and have been studied for many years. Some new vanishing theorems were proved in the last few years and especially the one by Caraiani-Scholze is crucial in arithmetic applications. I will survey these results, and discuss further development.

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(Please contact Wansu Kim at for Zoom meeting info and any inquiry.)

In his famous 1900 presentation, Hilbert proposed so-called the Hilbert’s 6thproblem, namely “Mathematical Treatment of the Axioms of Physics”. He mentioned that “Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua.” In this lecture, we present some recent development of the Hilbert’s 6th problem in the Boltzmann theory when the various fluid models have natural “singularities” such as unbounded vorticity and formation of boundary layers.

Majority dynamics on a graph $G$ is a deterministic process such that every vertex updates its $\pm 1$-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O'Donnell, Tamuz and Tan conjectured that, in the Erd\H{o}s--R\'enyi random graph $G(n,p)$, the random initial $\pm 1$-assignment converges to a $99\%$-agreement with high probability whenever $p=\omega(1/n)$. This conjecture was first confirmed for $p\geq\lambda n^{-1/2}$ for a large constant $\lambda$ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for $p< \lambda n^{-1/2}$. We break this $\Omega(n^{-1/2})$-barrier by proving the conjecture for sparser random graphs $G(n,p)$, where $\lambda' n^{-3/5}\log n \leq p \leq \lambda n^{-1/2}$ with a large constant $\lambda'>0$.

The temporal credit assignment, the problem of determining which actions in the past are responsible for the current outcome (long-term cause and effect), is difficult to solve because one needs to backpropagate the error signal through space and time. Despite its computational challenges, humans are very good at solving this problem. Our lab uses reinforcement learning theory and algorithms to explore the nature of computations underlying the brain’s ability to solve the temporal credit assignment. I will outline two-fold approaches to this issue: (1) training a computational model from human behavioral data without underfitting and overfitting (Brain → AI) and (2) using the trained model to manipulate the way the human brain solves the temporal credit assignment problem (AI → brain).

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Education/employments PhD, KAIST (2009)Postdoc, MIT (2010-2011), Caltech (2011-2015)Faculty, KAIST (2015-now) Honors/awards IBM Academic Research Award (2021)Google Faculty Research Award (2017)Della-Martin Fellowship (2014) KAIST Breakthroughs (2020)KAIST Songam Distinguished Research Award (2019)KAIST Top 10 Technologies (2019)KAIST Institute Faculty Award (2019) KIIS Young Investigator Award (2016)ICROS Young Investigator Award (2016)

Anyons are quasiparticles in two dimensions. They do not belong to the two classes of elementary particles, bosons and fermions. Instead, they obey Abelian or non-Abelian fractional statistics. Their quantum mechanical states are determined by fusion or braiding, to which braid groups and conformal field theories are naturally applied. Some of non-Abelian anyons are central in realization of topological qubits and topological quantum computing. I will introduce the basic properties of anyons and their recent experimental signatures observed in systems of topological order such as fractional quantum Hall systems and topological superconductors.

The purpose of this talk is to mathematically investigate the formation of a plasma sheath, and to analyze the Bohm criterions which are required for the formation. Bohm derived originally the (hydrodynamic) Bohm criterion from the Euler–Poisson system. Boyd and Thompson proposed the (kinetic) Bohm criterion from kinetic point of view, and then Riemann derived it from the Vlasov–Poisson system. We study the solvability of boundary value problems of the Vlasov–Poisson system. On the process, we see that the kinetic Bohm criterion is a necessary condition for the solvability. The argument gives a simpler derivation of the criterion. Furthermore, the hydrodynamic criterion can be derived from the kinetic criterion. It is of great interest to find the relation between the solutions of the Vlasov–Poisson and Euler–Poisson systems. To clarify the relation, we also investigate the hydrodynamic limit of solutions of the Vlasov–Poisson system.

In hyperbolic 3 manifolds, by Marden, Thurston and Bonahon, every immersed surface of which the fundamental group is invectively embedded in the 3-manifold group is quasi-fuchsian or doubly degenerated. Surface subgroups of 3-manifold groups play an important rule in 3-manifold theory. For instance, some collection of immersed surfaces give a rise to a CAT(0) cube complex. Especially, in the usual construction of the CAT(0) cube complex, each immersed surface composing the collection is quasi-fuchsian. In this talk, I introduce the work by Cooper, Long and Reid. In hyperbolic mapping tori, the work gives a criterion to determine whether the given immersed surface is quasi-fuchsian or not. The criterion is given in terms of laminations induced in immersed surfaces.

Many of real-world data, e.g., the VGGFace2 dataset, which is a collection of multiple portraits of individuals, come with nested structures due to grouped observation. The Ornstein auto-encoder (OAE) is an emerging framework for representation learning from nested data, based on an optimal transport distance between random processes. An attractive feature of OAE is its ability to generate new variations nested within an observational unit, whether or not the unit is known to the model. A previously proposed algorithm for OAE, termed the random-intercept OAE (RIOAE), showed an impressive performance in learning nested representations, yet lacks theoretical justification. In this work, we show that RIOAE minimizes a loose upper bound of the employed optimal transport distance. After identifying several issues with RIOAE, we present the product-space OAE (PSOAE) that minimizes a tighter upper bound of the distance and achieves orthogonality in the representation space. PSOAE alleviates the instability of RIOAE and provides more flexible representation of nested data. We demonstrate the high performance of PSOAE in the three key tasks of generative models: exemplar generation, style transfer, and new concept generation. This is a joint work with Dr. Youngwon Choi (UCLA) and Sungdong Lee (SNU).

Questions about the mechanistic operation of biological systems are naturally formulated as stochastic processes, but confronting such models with data can be challenging. In this talk, I describe the essence of the difficulty, highlighting both the technical issues and the importance of the “plug-and-play property”. I then illustrate some effective approaches to efficient inference based on such models. I conclude by sketching promising new developments and describing some open problems.

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Zoom link: 709 120 4849 (pw: 1234)

The objective of the study is to evaluate neural circuitry supporting a cognitive control task, and associated practice-related changes via acquisition of blood oxygenation level dependent (BOLD) signal collected using functional magnetic resonance imaging (fMRI). FMR images are acquired from participants engaged in antisaccade (generating a glance away from a cue) performance at two scanning sessions: 1) pre-practice before any exposure to the task, and 2) post-practice, after one week of daily practice on antisaccades, prosaccades (glancing towards a target) or fixation (maintaining gaze on a target). The three practice groups are compared across the two sessions, and analyses are conducted via the application of a model-free clustering technique based on wavelet analysis. This series of procedures is developed to address analysis problems inherent in fMRI data and is composed of several steps: data aggregation, no trend test, decorrelation, principal component analysis and K-means clustering. Also, we develop a semiparametric approach under shape invariance to quantify and test the differences in sessions and groups using the property that brain signals from a task-related experiment may exhibit a similar pattern in regions of interest across participants. We estimate the common function with local polynomial regression and estimate the shape invariance model parameters using evolutionary optimization methods. Using the proposed approach, we compare BOLD signals in multiple regions of interest for the three practice groups at the two sessions and quantify the effects of task practice in these groups.

I will talk about data science and Big Data, and how I view statistics in the data science and Big Data era. Next, I will briefly introduce my research areas in statistics. Finally, I will present some of my interdisciplinary research on functional magnetic resonance imaging data analysis.

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Direct ZOOM link

While the presence of immune cells within solid tumours was initially viewed positively, as the host fighting to rid itself of a foreign body, we now know that the tumour can manipulate immune cells so that they promote, rather than inhibit, tumour growth. Immunotherapy aims to correct for this by boosting and/or restoring the normal function of the immune system. Immunotherapy has delivered some extremely promising results. However, the complexity of the tumour-immune interactions means that it can be difficult to understand why one patient responds well to immunotherapy while another does not. In this talk, we will show how mathematical, statistical and topological methods can contribute to resolving this issue and present recent results which illustrate the complementary insight that different approaches can deliver.

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Zoom link: 709 120 4849 (pw: 1234)

A particularly important substructure in modeling joint linear chance-constrained programs with random right-hand sides and finite sample space is the intersection of mixing sets with common binary variables (and possibly a knapsack constraint). In this talk, we first explain basic mixing sets by establishing a strong and previously unrecognized connection to submodularity. In particular, we show that mixing inequalities with binary variables are nothing but the polymatroid inequalities associated with a specific submodular function. This submodularity viewpoint enables us to unify and extend existing results on valid inequalities and convex hulls of the intersection of multiple mixing sets with common binary variables. Then, we study such intersections under an additional linking constraint lower bounding a linear function of the continuous variables. This is motivated from the desire to exploit the information encoded in the knapsack constraint arising in joint linear CCPs via the quantile cuts. We propose a new class of valid inequalities and characterize when this new class along with the mixing inequalities are sufficient to describe the convex hull. This is based on joint work with Fatma Fatma Kılınç-Karzan and Simge Küçükyavuz.

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https://youtube.com/c/ibsdimag

Our current approach to cancer treatment has been largely driven by finding molecular targets, those patients fortunate enough to have a targetable mutation will receive a fixed treatment schedule designed to deliver the maximum tolerated dose (MTD). These therapies generally achieve impressive short-term responses, that unfortunately give way to treatment resistance and tumor relapse. The importance of evolution during both tumor progression, metastasis and treatment response is becoming more widely accepted. However, MTD treatment strategies continue to dominate the precision oncology landscape and ignore the fact that treatments drive the evolution of resistance. Here we present an integrated theoretical/experimental/clinical approach to develop treatment strategies that specifically embrace cancer evolution. We will consider the importance of using treatment response as a critical driver of subsequent treatment decisions, rather than fixed strategies that ignore it. We will also consider using mathematical models to drive treatment decisions based on limited clinical data. Through the integrated application of mathematical and experimental models as well as clinical data we will illustrate that, evolutionary therapy can drive either tumor control or extinction using a combination of drug treatments and drug holidays. Our results strongly indicate that the future of precision medicine shouldn’t be in the development of new drugs but rather in the smarter evolutionary, and model informed, application of preexisting ones.