Abstract: The biological master clock, the suprachiasmatic nucleus (SCN) of a mouse, contains many (~20,000) clock cells heterogeneous, particularly with respect to their circadian period. Despite the inhomogeneity, within an intact SCN, they maintain a very high degree of circadian phase coherence, which is generally rendered visible as system-wide propagating phase waves. The phase coherence is vital for mammals sustaining various circadian rhythmic activities. It is supposedly achieved not by one but a few different cell-to-cell coupling mechanisms: Among others, action potential (AP)-mediated connectivity is known to be essential. However, due to technical difficulties and limitations in experiments, so far, very little information is available about the (connectome) morphology of the AP-mediated SCN neural connectivity. With that limited amount of information, here we exhaustively and systematically explore a large (~25,000) pool of various model network morphologies to come up with the most realistic case for the SCN. All model networks within this pool reflect an actual indegree distribution as well as a physical range distribution of afferent clock cells, which were acquired in earlier optogenetic connectome experiments. Subsequently, our network selection scheme is based on a collection of multitude criteria, testing the properties of SCN circadian phase waves in perturbed (or driven) as well as in their natural states: Key properties include, 1) degree of phase synchrony (or dispersal) and direction of wave propagation, 2) entrainability of the model oscillator networks to an external circadian forcing (mimicking the light modulation subject to the geophysical circadian rhythm), and 3) emergence of “phase-singularities” following a global perturbation and their decay. The selected network morphologies require several common features that 1) the indegree – outdegree relation must have a positive correlation; 2) the cells in the SCN core region have a larger total (in+out) degree than that of the shell region; 3) core to shell (or shell to core) connections should be much less than core to core (and shell to shell) connections. Taken all together, our comprehensive test results strongly suggest that degree distribution over the whole SCN is not uniform but position-dependent and raise a question of whether this inhomogeneous degree distribution is related to the distribution of known subpopulations of SCN cells.

The linear bandit problem has received a lot of attention in the past decade due to its applications in new recommendation systems and online ad placements where the feedback is binary such as thumbs up/down or click/no click. Linear bandits, however, assume the standard linear regression model and thus are not well-suited for binary feedback. While logistic linear bandits, the logistic regression counterpart of linear bandits, are more attractive for these applications, developments have been slow and practitioners often end up using linear bandits for binary feedback -- this corresponds to using linear regression for classification tasks.In this talk, I will present recent breakthroughs in logistic linear bandits leading to tight performance guarantees and lower bounds. These developments are based on self-concordant analysis, improved fixed design concentration inequalities, and novel methods for the design of experiments. I will also discuss open problems and conjectures on concentration inequalities. This talk will be based on our recent paper accepted to ICML'21 (https://arxiv.org/abs/2011.11222).

TBA

피부암은 미국에서 가장 흔한 암 중의 하나이며 국내에서도 고령화, 오존층 변화와 자외선 노출 빈도의 증가에 따른 발병 율이 증가 하고 있다. 피부암 중의 약 2.1%인 전이 율이 높아 치료하기가 쉽지 않은 악성 흑색종(Malignant melanoma)이며 이와 달리 양성 흑색종 (NMSK: Nonmelanoma Skin Cancer)은 암으로 분류되긴 하지만, 피부 내에서만 자라고, 전이 율이 적어 초기에 발견하여 암 부위만 제거하면 10년 이상 생존율이 89%이상인 양성 암이다. 따라서, 초기에 아주 작은 수지만 위험한 악성 흑색종과 많은 수의 크게 위험하지 않은 양성 흑색종을 구분하여 적절히 치료하는 것은 피부암 치료에서 아주 중요한 부분이라고 할 수 있다. 흑색종 진단에 많이 사용되어져 왔던 전통적인 특징점(Feature)들로는 ABCD criteria(그림 1)과 같이 의심영역의 비대칭성(A: Asymmetry), 경계의 불규칙성(B:Border irregularity), 색조의 다양성(C: Color variegation) 등이 있다. 피부암 진단에 대한 연구는 많은 진전이 있었고, 2018년의 International Skin Imaging Collaboration (ISIC) 는 피부암 판별 경진대회를 열어서 사실상 이 분야의 벤치마크가 되었다. 다양한 영상 분류 알고리즘과 기법이 시도되었고 특히 CNN을 이용한 진단 결과들은 피부과 전문의 진단결과와 비슷하거나 더 좋은 결과를 내기도 했다. CNN의 이런 좋은 성과에도 불구하고 의학영역은 환자의 목숨을 다루기 때문에 높은 정확도와 함께 고려해야 할 것은 설명 가능한 심층학습의 필요성이다. 예를 들어, 만약 의학적 사고가 발생하였을 때 의료 행위에 대한 과실 책임에 대한 문제가 대두 되고 있다. 즉 의료사고에 대한 책임을 누가 질 것인가?, 의료 사고의 발생 원인이 무엇이며 어떻게 설명할 수 있는가?, 만약 결과가 좋다면 어떻게 이 방법을 발전시킬 수 있을 것인가? 의 문제들이다. 이를 위해 Grad-CAM 기법, Image Occulsion 기법, LIME (Local interpretable model-agnostic explanations)기법 등이 사용되고 있다. 이런 기법들을 이용하여 잘 알려진 진단 결과가 우수한 심층학습을 대상으로 어떤 특징 점들이 진단 결과의 우수성에 영향을 미치는 지 조사하려고 한다. 이 기법을 통해 전통적으로 알려진 특징 점인 A(비대칭성), B(경계 불규칙성), C(색조 다양성), D(크기) 와 설명 가능한 심층학습 기법으로 새로운 특징 점을 발견할 수 있을지 알아보고자 한다. Meeting ID: 857 8963 8008 Passcode: 932161

In this lecture, I will present recent works on the asymptotical stability of wave patterns for viscous conservation laws under periodic perturbations.

We define a localised Euler class for isotropic sections, and isotropic cones, in SO(N) bundles. We use this to give an algebraic definition of Borisov-Joyce sheaf counting invariants on Calabi-Yau 4-folds. When a torus acts, we prove a localisation result. This talk is based on the joint work with Richard P. Thomas.

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Zoom ID: 352-730-6970, PW:9999 The above times are in Korean Standard Time. This is the same time as 9AM of June 18, 2021 (Friday) in the UK (GMT+1)

Bayesian statistical approaches have been developed for various applications due to their flexibility. I will cover different application areas of Bayesian methods including applications to analyses of complex count data, decision making problems, and analyses of survival times data. For part I, I will discuss two recently completed projects and comment on some on-going and future projects. Part II will cover a gentle introduction to survival analyses focusing on Bayesian approaches and discuss its extensions for joint analysis with recurrent events data or longitudinal data. For part III, I will cover a general Bayesian decision making framework and their applications to clinical trial design and data analysis.

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Zoom ID : 901 228 2472 (Password : 123456)

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties. We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese, and consider various techniques to provide equations on the secants.In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

It is known that the stochastic Potts model exhibits exponentially slow mixing at the low temperature regime. In this talk, we explain precise quantitative results regarding this slow mixing behavior of the Potts model at large lattices. This talk is based on the joint work with Seonwoo Kim.

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties. We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese, and consider various techniques to provide equations on the secants.In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

Bayesian statistical approaches have been developed for various applications due to their flexibility. I will cover different application areas of Bayesian methods including applications to analyses of complex count data, decision making problems, and analyses of survival times data. For part I, I will discuss two recently completed projects and comment on some on-going and future projects. Part II will cover a gentle introduction to survival analyses focusing on Bayesian approaches and discuss its extensions for joint analysis with recurrent events data or longitudinal data. For part III, I will cover a general Bayesian decision making framework and their applications to clinical trial design and data analysis.

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Zoom ID : 901 228 2472 (Password : 123456)

심사위원장 정연승, 심사위원 김성호, 황강욱, 전현호, 최태련(고려대 통계학과)

TBA

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties. We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese, and consider various techniques to provide equations on the secants.In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

Bayesian statistical approaches have been developed for various applications due to their flexibility. I will cover different application areas of Bayesian methods including applications to analyses of complex count data, decision making problems, and analyses of survival times data. For part I, I will discuss two recently completed projects and comment on some on-going and future projects. Part II will cover a gentle introduction to survival analyses focusing on Bayesian approaches and discuss its extensions for joint analysis with recurrent events data or longitudinal data. For part III, I will cover a general Bayesian decision making framework and their applications to clinical trial design and data analysis.

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Zoom ID : 901 228 2472 (Password : 123456)

심사위원장 백형렬, 심사위원 박정환, 최서영, 임선희(서울대학 수리과학부), 김상현(고등과학원 수학부)

An intersection digraph is a digraph where every vertex $v$ is represented by an ordered pair $(S_v, T_v)$ of sets such that there is an edge from $v$ to $w$ if and only if $S_v$ and $T_w$ intersect. An intersection digraph is reflexive if $S_v\cap T_v\neq \emptyset$ for every vertex $v$. Compared to well-known undirected intersection graphs like interval graphs and permutation graphs, not many algorithmic applications on intersection digraphs have been developed. Motivated by the successful story on algorithmic applications of intersection graphs using a graph width parameter called mim-width, we introduce its directed analogue called `bi-mim-width’ and prove that various classes of reflexive intersection digraphs have bounded bi-mim-width. In particular, we show that as a natural extension of $H$-graphs, reflexive $H$-digraphs have linear bi-mim-width at most $12|E(H)|$, which extends a bound on the linear mim-width of $H$-graphs [On the Tractability of Optimization Proble

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Zoom ID: 934 3222 0374 (ibsdimag)

심사위원장 김재경, 심사위원 김용정, 황강욱, 정연승, 이승규(고려대 세종캠퍼스 응용수리과학부)

MUltiple SIgnal Classification (MUSIC) is a well-known, non-iterative imaging technique in inverse scattering problem. Throughout various researches, it has been confirmed that MUSIC is very fast, effective, and stable. Due to this reason MUSIC has been applied to various inverse scattering problems however, it has not yet been designed and used to identify unknown anomalies from measured scattering parameters (S-parameters) in microwave imaging. In this presentation, we apply MUSIC in microwave imaging for a fast identification of arbitrary shaped anomalies from real-data and establish a mathematical theory for illustrating the feasibilities and limitations of MUSIC. Simulations results with real-data are shown for supporting established theoretical results.

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갑작스러운 연사 사정으로 세미나 일정이 취소되었습니다. 양해 부탁드립니다.

We discuss sharp regularity results for a larger class of of elliptic and parabolic equations in divergence form and present new and interesting features from the view point of regularity theory

I will discuss various near-rationality concepts for smooth projective varieties. I will introduce the standard norm variety associated with a symbol in mod-l Milnor K-theory. The standard norm varieties played an important role in Vovedsky's proof of the Bloch-Kato conjecture. I will then describe known near-rationality results for standard norm varieties and outline an argument showing that a standard norm variety is universally R-trivial over an algebraically closed field of characteristic 0. The talk is based on joint work with Chetan Balwe and Amit Hogadi.

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Zoom ID: 352 730 6970, Password : 9999. All times are local in KST.

심사위원장 김경국, 심사위원 황강욱, 전현호, 정연승, 송은혜(Department of Industrial & Manufacturing Engineering, Pennsylvania State University)

Diophantine approximation is a branch of number theory where one studies approximation of irrational numbers by rationals and quality of such approximations. In this talk, we will consider intrinsic Diophantine approximation, which deals with approximating irrational points in a closed subset $X$ in $\mathbb{R}^n$ via rational points lying in $X$. First, we consider $X = S^1$, the unit circle in $\mathbb{R}^2$ centered at the origin. We give a complete description of an initial discrete part of the Lagrange spectrum of $S^1$ in the sense of intrinsic Diophantine approximation. This is an analogue of the classical result of Markoff in 1879, where he characterized the most badly approximable real numbers via the periods of their continued fraction expansions. Additionally, we present similar results for approximations of $S^1$ by a few different sets of rational points. This is joint work with Dong Han Kim (Dongguk University, Seoul). (Contact Bo-Hae Im if you plan to join the seminar.)

A knot is a smooth embedding of an oriented circle into the three-sphere, and two knots are concordant if they cobound a smoothly embedded annulus in the three-sphere times the interval. Concordance gives an equivalence relation, and the set of equivalence classes forms a group called the concordance group. This group was introduced by Fox and Milnor in the 60's and has played an important role in the development of low-dimensional topology. In this talk, I will present some known results on the structure of the group. Also, I will talk about a knot that has infinite order in the concordance group, though it bounds a smoothly embedded disk in a rational homology ball. This is joint work with Jennifer Hom, Sungkyung Kang, and Matthew Stoffregen.

Let $\mathbb{F}_q$ be a finite field of order $q$ which is a prime power. In the finite field setting, we say that a function $\phi\colon \mathbb{F}_q^d\times \mathbb{F}_q^d\to \mathbb{F}_q$ is a Mattila-Sjölin type function in $\mathbb{F}_q^d$ if for any $E\subset \mathbb{F}_q^d$ with $|E|\gg q^{\frac{d}{2}}$, we have $\phi(E, E)=\mathbb{F}_q$. The main purpose of this talk is to present the existence of such a function. More precisely, we will construct a concrete function $\phi: \mathbb{F}_q^4\times \mathbb{F}_q^4\to \mathbb{F}_q$ with $q\equiv 3 \mod{4}$ such that if $E\subset \mathbb F_q^4$ with $|E|>q^2,$ then $\phi(E,E)=\mathbb F_q$. This is a joint work with Daewoong Cheong, Thang Pham, and Chun-Yen Shen.

One famous conjecture in quantum chaos and random matrix theory is the so-called phase transition conjecture of random band matrices. It predicts that the eigenvectors' localization-delocalization transition occurs at some critical bandwidth $W_c(d)$, which depends on the dimension $d$. The well-known Anderson model and Anderson conjecture have a similar phenomenon. It is widely believed that $W_c(d)$ matches $1/\lambda_c(d)$ in the Anderson conjecture, where $\lambda_c(d)$ is the critical coupling constant. Furthermore, this random matrix eigenvector phase transition coincides with the local eigenvalue statistics phase transition, which matches the Bohigas-Giannoni-Schmit conjecture in quantum chaos theory. We proved the eigenvector's delocalization property for most of the general $d>=8$ random band matrix as long as the size of this random matrix does not grow faster than its bandwidth polynomially. In other words, as long as bandwidth $W$ is larger than $L^\epislon$ for some $\epislon>0$, and matrix size $L$. It is joint work with H.T. Yau (Harvard) and F. Yang (Upenn).

심사위원장 이지운, 심사위원 강완모, 황강욱, 윤세영(AI 대학원), 강남규(고등과학원)

심사위원장 이지운, 심사위원 강완모, 황강욱, 윤세영(AI 대학원), 강남규(고등과학원)

The rapid development of high-throughput sequencing technology in recent years is providing unprecedented opportunities to profile microbial communities from a variety of environments, but analysis of such multivariate taxon count data remains challenging. I present two flexible Bayesian methods to analyze complex count data with application to microbiome study. The first project is to develop a Bayesian sparse multivariate regression method that model the relationship between microbe abundance and environmental factors. We extend conventional nonlocal priors, and construct asymmetric non-local priors for regression coefficients to efficiently identify relevant covariates and their effect directions. The developed Bayesian sparse regression model is applied to analyze an ocean microbiome dataset collected over time to study the association of harmful algal bloom conditions with microbial communities. For the second project, we develop a Bayesian nonparametric regression model for count data with excess zeros. The approach provides straightforward community-level insights into how characteristics of microbial communities such as taxa richness and diversity are related to covariates. The baseline counts of taxa in samples are carefully constructed to obtain improved estimates of differential abundance. We apply the model to a chronic wound microbiome dataset, comparing the microbial communities present in chronic wounds versus in healthy skin.

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Co-authors; Kurtis Shuler (Sandia National Lab), Marilou Sison-Mangus (Ocean Sciences, UCSC), Irene A. Chen (Chemistry and Biochemistry, UCLA), Samuel Verbanic (Chemistry and Biochemistry, UCLA)

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties. We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese, and consider various techniques to provide equations on the secants.In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

Non-Markov models of stochastic biochemical kinetics often incorporate explicit time delays to effectively model large numbers of intermediate biochemical processes. Analysis and simulation of these models, as well as the inference of their parameters from data, are fraught with difficulties because the dynamics depends on the system’s history. Here we use an artificial neural network to approximate the time-dependent distributions of non-Markov models by the solutions of much simpler time-inhomogeneous Markov models; the approximation does not increase the dimensionality of the model and simultaneously leads to inference of the kinetic parameters. The training of the neural network uses a relatively small set of noisy measurements generated by experimental data or stochastic simulations of the non-Markov model. We show using a variety of models, where the delays stem from transcriptional processes and feedback control, that the Markov models learnt by the neural network accurately reflect the stochastic dynamics across parameter space.

심사위원장 권순식, 심사위원 김용정, 이지운, 허형진(중앙대 수학과), 오성진(University of California at Berkeley, 수학과)

The logarithmic analog of SH(k) is logSH(k). In logSH(k), topological cyclic homology is representable. Furthermore, the cyclotomic trace map from K to TC is representable too when k is a perfect field with resolution of singularities. In the second talk, I will explain the construction of logSH(k) and how we can achieve these results. This work is joint with Federico Binda and Paul Arne Østvær.

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Zoom ID: 352 730 6970, Password: 9999.

We show that log-concavity is the weakest power concavity preserved by the Dirichlet heat flow in convex domains in ${\bf R}^N$, where $N\ge 2$. Jointly with what we already know, i.e. that log-concavity is the strongest power concavity preserved by the heat flow, we see that log-concavity is the only power concavity preserved by the Dirichlet heat flow. This is a joint work with Paolo Salani (Univ. of Florence) and Asuka Takatsu (Tokyo Metropolitan Univ.)

We will discuss about “Independent Markov Decomposition: Towards modeling kinetics of biomolecular complexes”, Hempel et. al., bioRxiv, 2021 In order to advance the mission of in silico cell biology, modeling the interactions of large and complex biological systems becomes increasingly relevant. The combination of molecular dynamics (MD) and Markov state models (MSMs) have enabled the construction of simplified models of molecular kinetics on long timescales. Despite its success, this approach is inherently limited by the size of the molecular system. With increasing size of macromolecular complexes, the number of independent or weakly coupled subsystems increases, and the number of global system states increase exponentially, making the sampling of all distinct global states unfeasible. In this work, we present a technique called Independent Markov Decomposition (IMD) that leverages weak coupling between subsystems in order to compute a global kinetic model without requiring to sample all combinatorial states of subsystems. We give a theoretical basis for IMD and propose an approach for finding and validating such a decomposition. Using empirical few-state MSMs of ion channel models that are well established in electrophysiology, we demonstrate that IMD can reproduce experimental conductance measurements with a major reduction in sampling compared with a standard MSM approach. We further show how to find the optimal partition of all-atom protein simulations into weakly coupled subunits.

A rooted digraph is a vertex-flame if for every vertex v there is a set of internally disjoint directed paths from the root to v whose set of terminal edges covers all ingoing edges of v. It was shown by Lovász that every finite rooted digraph admits a spanning subdigraph which is a vertex-flame and large, where the latter means that it preserves the local connectivity to each vertex from the root. A structural generalisation of vertex-flames and largeness to infinite digraphs was given by Joó and the analogue of Lovász’ result for countable digraphs was shown. In this talk, I present a strengthening of this result stating that in every countable rooted digraph each vertex-flame can be extended to a large vertex-flame. Joint work with Joshua Erde and Attila Joó.

다중관점기하학(multiple view geometry)에서는 카메라 관점의 변화에 따라 피사체의 영상이 어떻게 변화하는 지를 분석하고, 여러 관점에서 얻은 2차원 영상으로부터 3차원 모델을 재구성하는 기법을 연구한다. 본 강연에서는 다중관점기하학의 기본적인 개념과 테크닉을 설명하고, 이를 의료영상, 자율주행, 스마트헬스케어 등 다양한 산업분야에 적용하는 것에 대하여 소개한다.

For the mass-critical/supercritical pseudo-relativistic nonlinear Schrödinger equation, Bellazzini, Georgiev and Visciglia constructed a class of stationary solutions as local energy minimizers under an additional kinetic energy constraint, and they showed the orbital stability of the energy minimizer manifold. In this talk, by proving its local uniqueness, we show the orbital stability of the solitary wave, not that of the energy minimizer set. The key new aspect is reformulation of the variational problem in the non-relativistic regime, which is, we think, more natural because the proof heavily relies on the sub-critical nature of the limiting model. By this approach, the meaning of the additional constraint is clarified, a more suitable Gagliardo-Nirenberg inequality is introduced, and the non-relativistic limit is proved. Finally, using the non-relativistic limit, we obtain the local uniqueness and the non-degeneracy of the minimizer. This talk is based on joint work with Sangdon Jin.

We will describe a construction of infinitesimal invariants of thickened one dimensional cycles in three dimensional space, which are the simplest cycles that are not in the Milnor range. This is an analog of a construction of J. Park in the context of additive Chow groups. The construction allows us to prove the infinitesimal version of the strong reciprocity conjecture for thickenings of all orders. Classical analogs of our invariants are based on the dilogarithm function and our invariant could be seen as their infinitesimal version. Despite this analogy, the infinitesimal version cannot be obtained from their classical counterparts through a limiting process.

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Zoom ID: 352 730 6970. Password : 9999. All times are in KST.

심사위원장: 곽시종, 심사위원: 이용남, 백상훈, 박의성(고려대 수학과), 박진형(서강대 수학과)

Let G be a finite group. The minimal ramification problem famously asks about the minimal number µ(G) of ramified primes in any Galois extension of Q with group G. A conjecture due to Boston and Markin predicts the value of µ(G). I will report on recent progress on this problem, as well as several other problems which may be described as minimal ramification problems in a wider sense, notably: what is the smallest number k = k(G) such that there exists a G-extension of Q with discriminant not divisible by any (k + 1)-th power?, and: what is the smallest number m = m(G) such that there exists a G-extension of Q with all ramification indices dividing m? Apart from partial results over Q, I will present function field analogs. (If you would like to join the seminar, contact Bo-Hae Im to get the zoom link.)

A signed graph is a graph in which each edge has a positive or negative sign. Calling two graphs switch equivalent if one can get from one to the other by the iteration of the local action of switching all signs on edges incident to a given vertex, we say that there is a switch homomorphism from a signed graph $G$ to a signed graph $H$ if there is a sign preserving homomorphism from $G’$ to $H$ for some graph $G’$ that is switch equivalent to $G$. By reductions to CSP this problem, and its list version, are known to be either polynomial time solvable or NP-complete, depending on $H$. Recently those signed graphs $H$ for which the switch homomorphism problem is in $P$ were characterised. Such a characterisation is yet unknown for the list version of the problem. We talk about recent work towards such a characterisation and about how these problems fit in with bigger questions that still remain around the recent CSP dichotomy theorem.

The standard machine learning paradigm optimizing average-case performance performs poorly under distributional shift. For example, image classifiers have low accuracy on underrepresented demographic groups, and their performance degrades significantly on domains that are different from what the model was trained on. We develop and analyze a distributionally robust stochastic optimization (DRO) framework over shifts in the data-generating distribution. Our procedure efficiently optimizes the worst-case performance, and guarantees a uniform level of performance over subpopulations. We characterize the trade-off between distributional robustness and sample complexity, and prove that our procedure achieves this optimal trade-off. Empirically, our procedure improves tail performance, and maintains good performance on subpopulations even over time.

This is part VI of the lectures on infinity-categories. I'll keep talking about the simplicial nerve construction in contrast to the ordinary nerve functor. To finish off this whole series, some overview of the theory of infinity-categories will be given, including how similar and different it is to the ordinary category theory.

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Zoom ID: 352 730 6970, Password: 9999

Morel and Voevodsky constructed the A^1-motivic homotopy category SH(k). One purpose of motivic homotopy theory is to incorporate various cohomology theories into a single framework so that one can discuss relations between cohomology theories more efficiently. However, there are still lots of non A^1-invariant cohomology theories, e.g. Hodge cohomology and topological cyclic homology. There is no way to represent these in SH(k). In the first talk, I will explain the construction of logDM^{eff}(k) for a perfect field k, which is strictly larger than Voevodsky's triangulated categories of motives DM^{eff}(k) because Hodge cohomology is representable in logDM^{eff}(k). This work is joint with Federico Binda and Paul Arne Østvær.

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Zoom ID: 352 730 6970, Password: 9999

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties. We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese, and consider various techniques to provide equations on the secants. In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

A notion of sublinear expander has played a central role in the resolutions of a couple of long-standing conjectures in embedding problems in graph theory, including e.g. the odd cycle problem of Erdos and Hajnal that the harmonic sum of odd cycle length in a graph diverges with its chromatic number. I will survey some of these developments.