(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

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

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.

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.

A signed graph is a pair $(G,\Sigma)$ where $G$ is a graph and $\Sigma$ is a subset of edges of $G$. A cycle $C$ of $G$ is a subset of edges of $G$ such that every vertex of the subgraph of $G$ induced by $C$ has an even degree. We say that $C$ is even in $(G,\Sigma)$ if $|C \cap \Sigma|$ is even; otherwise, $C$ is odd. A matroid $M$ is an even-cycle matroid if there exists a signed graph $(G,\Sigma)$ such that circuits of $M$ precisely corresponds to inclusion-wise minimal non-empty even cycles of $(G,\Sigma)$. For even-cycle matroids, two fundamental questions arise: (1) what is the relationship between two signed graphs representing the same even-cycle matroids? (2) how many signed graphs can an even-cycle matroid have? For (a), we characterize two signed graphs $(G_1,\Sigma_1)$ and $(G_2,\Sigma_2)$ where $G_1$ and $G_2$ are $4$-connected that represent the same even-cycle matroids. For (b), we introduce pinch-graphic matroids, which can generate exponentially many representations even when the matroid is $3$-connected. An even-cycle matroid is a pinch-graphic matroid if there exists a signed graph with a pair of vertices such that every odd cycle intersects with at least one of them. We prove that there exists a constant $c$ such that if a matroid is even-cycle matroid that is not pinch-graphic, then the number of representations is bounded by $c$. This is joint work with Bertrand Guenin and Irene Pivotto.

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줌정보 https://zoom.us/j/8456734198?pwd=d094SExIRW5HeElWSEVnampjdWZyZz09 회의 ID: 845 673 4198 암호: math

The Weighted $\mathcal F$-Vertex Deletion for a class $\mathcal F$ of graphs asks, given a weighted graph $G$, for a minimum weight vertex set $S$ such that $G-S\in\mathcal F$. The case when $\mathcal F$ is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted $\mathcal F$-Vertex Deletion. Only three cases of minor-closed $\mathcal F$ are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. We study the problem for the class $\mathcal F$ of $\theta_c$-minor-free graphs, under the equivalent setting of the Weighted c-Bond Cover, and present a constant-factor approximation algorithm using the primal-dual method. For this, we leverage a structure theorem implicit in [Joret et al., SIDMA’14] which states the following: any graph $G$ containing a $\theta_c$-minor-model either contains a large two-terminal protrusion, or contains a constant-size $\theta_c$-minor-model, or a collection of pairwise disjoint constant-sized connected sets that can be contracted simultaneously to yield a dense graph. In the first case, we tame the graph by replacing the protrusion with a special-purpose weighted gadget. For the second and third case, we provide a weighting scheme which guarantees a local approximation ratio. Besides making an important step in the quest of (dis)proving a constant-factor approximation for Weighted $\mathcal F$-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families. This is joint work with Euiwoong Lee and Dimitrios M. Thilikos.

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YouTube Live at https://youtube.com/ibsdimag

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줌정보 https://zoom.us/j/8456734198?pwd=d094SExIRW5HeElWSEVnampjdWZyZz09 회의 ID: 845 673 4198 암호: math

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줌정보 https://zoom.us/j/8456734198?pwd=d094SExIRW5HeElWSEVnampjdWZyZz09 회의 ID: 845 673 4198 암호: math

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자세한 내용은 아래 링크를 참고해주세요. https://saarc.kaist.ac.kr/boards/view/seminars/49

We call an induced cycle of length at least four a hole. The parity of a hole is the parity of its length. Forbidding holes of certain types in a graph has deep structural implications. In 2006, Chudnovksy, Seymour, Robertson, and Thomas famously proved that a graph is perfect if and only if it does not contain an odd hole or a complement of an odd hole. In 2002, Conforti, Cornuéjols, Kapoor, and Vuškovíc provided a structural description of the class of even-hole-free graphs. I will describe the structure of all graphs that contain only holes of length $\ell$ for every $\ell \geq 7$ (joint work with Jake Horsfield, Myriam Preissmann, Paul Seymour, Ni Luh Dewi Sintiari, Cléophée Robin, Nicolas Trotignon, and Kristina Vuškovíc. Analysis of how holes interact with graph structure has yielded detection algorithms for holes of various lengths and parities. In 1991, Bienstock showed it is NP-Hard to test whether a graph G has an even (or odd) hole containing a specified vertex $v \in V(G)$. In 2002, Conforti, Cornuéjols, Kapoor, and Vuškovíc gave a polynomial-time algorithm to recognize even-hole-free graphs using their structure theorem. In 2003, Chudnovsky, Kawarabayashi, and Seymour provided a simpler and slightly faster algorithm to test whether a graph contains an even hole. In 2019, Chudnovsky, Scott, Seymour, and Spirkl provided a polynomial-time algorithm to test whether a graph contains an odd hole. Later that year, Chudnovsky, Scott, and Seymour strengthened this result by providing a polynomial-time algorithm to test whether a graph contains an odd hole of length at least $\ell$ for any fixed integer $\ell \geq 5$. I will present a polynomial-time algorithm (joint work with Paul Seymour) to test whether a graph contains an even hole of length at least $\ell$ for any fixed integer $\ell \geq 4$.

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YouTube Live at https://youtube.com/ibsdimag

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자세한 내용은 아래 링크를 참고해주세요. https://saarc.kaist.ac.kr/boards/view/seminars/49

We show that for pairs (Q,R) and (S,T) of disjoint subsets of vertices of a graph G, if G is sufficiently large, then there exists a vertex v in V(G)−(Q∪R∪S∪T) such that there are two ways to reduce G by a vertex-minor operation while preserving the connectivity between Q and R and the connectivity between S and T. Our theorem implies an analogous theorem of Chen and Whittle (2014) for matroids restricted to binary matroids. Joint work with Sang-il Oum.

(전체일정: 7/28, 7/29, 8/3, 8/5) In 2d first-passage percolation, we let (t_e) be a family of i.i.d. weights associated to the edges of the square lattice, and let T = T(x,y) be the induced weighted graph metric on Z^2. If we let p be the probability that a weight takes the value 0, then there is a transition in the large-scale behavior of T depending on the value of p. Specifically, when p < 1/2, T grows linearly, but when p > 1/2, T is stochastically bounded. In these lectures, I will describe some of the standard questions of FPP in the case p < 1/2, and then focus on the "critical" case, where p = 1/2. Regarding this case, I will show some of my work over the last few years including exact asymptotics for T, universality results, and recent work on a dynamical version of the model. The work I will present was done in collaboration with J. Hanson, D. Harper, W.-K. Lam, P. Tang, and X. Wang. Lec 4: Critical FPP: the general case

(전체일정: 7/26, 7/27, 8/2, 8/4) Lec 4: Information Percolation In this final lecture, we discuss the celebrated technique known as the information percolation introduced by Lubetzky and Sly. Then, we explain the application of this technique to Glauber dynamics of the Ising model on lattice, and of the Random cluster model. The last result is a joint work with Shirshendu Ganguly.

(전체일정: 7/28, 7/29, 8/3, 8/5) In 2d first-passage percolation, we let (t_e) be a family of i.i.d. weights associated to the edges of the square lattice, and let T = T(x,y) be the induced weighted graph metric on Z^2. If we let p be the probability that a weight takes the value 0, then there is a transition in the large-scale behavior of T depending on the value of p. Specifically, when p < 1/2, T grows linearly, but when p > 1/2, T is stochastically bounded. In these lectures, I will describe some of the standard questions of FPP in the case p < 1/2, and then focus on the "critical" case, where p = 1/2. Regarding this case, I will show some of my work over the last few years including exact asymptotics for T, universality results, and recent work on a dynamical version of the model. The work I will present was done in collaboration with J. Hanson, D. Harper, W.-K. Lam, P. Tang, and X. Wang. Lec 3: Critical FPP: the Bernoulli case

(전체일정: 7/26, 7/27, 8/2, 8/4) Lec 3: Cut-off phenomenon for mean-field spin systems In this third lecture, we explain the cut-off phenomenon for the Glauber dynamics of the mean-field Ising or Potts model and general strategy to prove it.