Deep generative models have received much attention recently since they can generate very realistic synthetic images. There are two regimes for the estimation of deep generaitve models. One is generative adversarial network (GAN) and the other is variational auto encoder (VAE), Even though GAN is known to generate more clean synthetic images, it suffers from numerical instability and mode collapsing problems. VAE is an useful alternative to GAN and an important advantage of VAE is that the representational learning (i.e. learning latent variables) is possible. In this talk, I explain my recent studies about VAE. The first topic is computation. Typically, the estimation of VAE is done by maximizing the ELBO – an upper bound of the marginal likelihood. However, it is known that ELBO is inferior to the maximum likelihood estimator. I propose an efficient EM algorithm for VAE which directly finds the maximizer of the likelihood (the maximum likelihood estimator, MLE). The second topic is theory. I explain how the MLE of VAE behaves asymptotically. I derive a convergence rate which depends on the noise level as well as the complexity of a deep architecture. A surprising observation is that the convergence rate of the MLE becomes slower when the noise level is too low. A new technique to modify the MLE when the noise level is small is proposed and is shown to outperform the original MLE by analyzing real data.

Oscillatory signals are ubiquitously observed in many different intracellular systems such as immune systems and DNA repair processes. While we know how oscillatory signals are created, we do not fully understand what a critical role they play to regulate signal pathway systems in cells. Recently by using a stochastic nucleosome system, we found that a special signal (NFkB signal) in an immune cell can enhance the variability of the immune response to inflammatory stimulations when the signal is oscillatory. Hence in this talk, we discuss the roles of oscillatory and non-oscillatory NFkB signals in an inflammatory system of immune cells as the main example for revealing the role of oscillatory signals. And then we will talk about how this finding can be generalized for other intra- or extra-cellular systems to study why cells use oscillations.

We introduce a new subclass of chordal graphs that generalizes split graphs, which we call well-partitioned chordal graphs. Split graphs are graphs that admit a partition of the vertex set into cliques that can be arranged in a star structure, the leaves of which are of size one. Well-partitioned chordal graphs are a generalization of this concept in the following two ways. First, the cliques in the partition can be arranged in a tree structure, and second, each clique is of arbitrary size. We mainly provide a characterization of well-partitioned chordal graphs by forbidden induced subgraphs and give a polynomial-time algorithm that given any graph, either finds an obstruction or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal. We demonstrate the algorithmic use of this graph class by showing that two variants of the problem of finding pairwise disjoint paths between k given pairs of vertices are in FPT, parameterized by k, on well-partitioned chordal graphs, while on chordal graphs, these problems are only known to be in XP. From the other end, we introduce some problems that are polynomial-time solvable on split graphs but become NP-complete on well-partitioned chordal graphs. This is joint work with Lars Jaffke, O-joung Kwon, and Paloma T. Lima.

In this talk, we present an algebraic and graph theoretic (data-based) image inpainting algorithm. The algorithm is designed to reconstruct area or volume data from one and two dimensional slice data. More precisely, given one or two dimensional slice data, our algorithm begins with a simple algebraic pre-smoothing of the data, constructs low dimensional representation of pre-smoothed data via Dynamic Mode Decomposition, performs initial area or volume reconstruction via interpolation, and finishes with smoothing the outcome using a constraint bilateral smoothing. Numerical experiments including MRI of a three year old and a CT scan of a Covid-19 patient, are presented to demonstrate the superiority of the proposed techniques in comparisons with other commercial and published methods. Some further applications we are currently doing will also be presented. This work is jointly done with Gwanghyun Jo and Ivan Ojeda-Ruiz.

Sheaf cohomology and direct images are fundamental objects in algebraic geometry. However, they are defined in an abstract way (as right derived functors), and thus they are often hard to compute in explicit examples. In this talk, we briefly review Bernstein-Gel'fand-Gel'fand (BGG) correspondence and resolutions over an exterior algebra. Then, we review Tate resolutions and how it can be used to understand a given coherent sheaf and its cohomology groups in terms of Beilinson monad. Finally, we discuss an algorithm to compute direct images using Eisenbud-Erman-Schreyer's generalization on products of projective spaces. A part of the talk is a joint work in progress with J. Barrott and F.-O. Schreyer.

This is part V of the lectures on the foundations of infinity-categories. After continued discussion about the role of model categories in the theory of infinity-categories, the simplicial and differential graded nerve constructions will be presented to provide a plethora of examples of infinity-categories. Finally, I'll talk about an analogy between the theories of ordinary categories and infinity-categories, which wraps up this series of talks.

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Zoom ID: 352 730 6970, Password: 9999.

Traditional clustering identifies groups of objects that share certain qualities. Tangles do the converse: they identify groups of qualities that typically occur together. They can thereby discover, relate, and structure types: of behaviour, political views, texts, or proteins. Tangles offer a new, quantitative, paradigm for grouping phenomena rather than things. They can identify key phenomena that allow predictions of others. Tangles also offer a new paradigm for clustering in large data sets. The mathematical theory of tangles has its origins in the theory of graph minors developed by Robertson and Seymour. It has recently been axiomatized in a way that makes it applicable to a wide range of contexts outside mathematics: from clustering in data science to predicting customer behaviour in economics, from DNA sequencing and drug development to text analysis and machine learning. This very informal talk will not show you the latest intricacies of abstract tangle theory (for which you can find links on the tangle pages of my website), but to win you over to join our drive to develop real tangle applications in areas as indicated above. We have some software to share, but are looking for people to try it out with us on real-world examples! Here are some introductory pages from a book I am writing on this, which may serve as an extended abstract: https://arxiv.org/abs/2006.01830

This talk will be presented online. Zoom link: 709 120 4849 (pw: 1234) Age brings the benefit of experience and looking back at my job as a professor, there are a couple of things that fall into the category “I wish someone had told me that earlier”. In this seminar, I would like to share some of the things I learned and which, I hope, will be useful for younger scientists. The questions I will touch upon include What is productivity, for a scientist? What are qualities of successful people? How can one create motivation and success? How to organize myself? (project management; getting things done) How to communicate effectively? Seeking fulfillment The seminar is targeted at PhD students, postdocs, and junior group leaders.

Bouchet introduced isotropic systems in 1983 unifying some combinatorial features of binary matroids and 4-regular graphs. The concept of isotropic system is a useful tool to study vertex-minors of graphs and yet it is not well known. I will give an introduction to isotropic systems.

This is part IV of the series of lectures on the foundations of infinity-categories. n the first half, we'll cover the precise definition of infinity-categories based on quasi-categories. Some relationship between infinity-categories and model categories will be presented to help better understand the theory of infinity-categories in the remaining half.

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Zoom ID: 352 730 6970, Password: 9999

In this talk, I will present a result on the existence of 2-dimensional subsonic steady compressible flows around a finite thin profile with a vortex line at the trailing edge, which is a special case in the celebrated lifting line theory by Prandtl. Such a flow pattern is governed the two-dimensional steady compressible Euler equations. The vortex line attached to the trailing edge is a free interface corresponding to a contact discontinuity. Such a flow pattern is obtained as a consequence of structural stability of a uniform contact discontinuity. The problem is formulated and solved by an application of the implicit function theorem in a suitable weighted space. The main difficulties are the possible singularities at the fitting of the profile and the vortex line and the subtle instability of the vortex line. Some ideas of the analysis will be presented. This talk is based on joint works with Jun Chen and Aibin Zang at Yichun University. The research is supported in part by Hong Kong Earmarked Research Grants CUHK 14305315, CUHK 14302819, CUHK 14300917, and CUHK 14302917.

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https://kaist.zoom.us/j/3098650340

This talk will be presented online. Zoom link: 709 120 4849 (pw: 1234) Abstract: Life science has been a prosperous subject for a long time, and is still developing with high speed now. One of its major aims is to study the mechanisms of various biological processes on the basis of biological big-data. Many statistics-based methods have been proposed to catch the essence by mining those data, including the popular category classification, variables regression, group clustering, statistical comparison, dimensionality reduction, and component analysis, which, however, mainly elucidate static features or steady behavior of living organisms due to lack of temporal data. But, a biological system is inherently dynamic, and with increasingly accumulated time-series data, dynamics-based approaches based on physical and biological laws are demanded to reveal dynamic features or complex behavior of biological systems. In this talk, I will present a new concept “dynamics-based data science” and the approaches for studying dynamical bio-processes, including dynamical network biomarkers (DNB), autoreservoir neural networks (ARNN) and partical cross-mapping. These methods are all data-driven or model-free approaches but based on the theoretical frameworks of nonlinear dynamics. We show the principles and advantages of dynamics-based data-driven approaches as explicable, quantifiable, and generalizable. In particular, dynamics-based data science approaches exploit the essential features of dynamical systems in terms of data, e.g. strong fluctuations near a bifurcation point, low-dimensionality of a center manifold or an attractor, and phase-space reconstruction from a single variable by delay embedding theorem, and thus are able to provide different or additional information to the traditional approaches, i.e. statistics-based data science approaches. The dynamical-based data science approaches will further play an important role in the systematical research of biology and medicine in future.

Elie Cartan's celebrated paper ˝Pfaffian systems in 5 variables˝ in 1910 studied the equivalence problem for general Pfaffian systems of rank 3 in 5 variables as the curved version of the Pfaffian system with G_2 symmetry. The G_2 case admits a beautiful geometric correspondence with Engel's PDE system. We give a historical overview of Cartan's paper and discuss recent works extending the correspondence to curved cases, which is based on ideas from geometric control theory and algebraic geometry.

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pw: 2021math

TBA

Let $X$ be an abelian variety of dimension $g$ over a field $k$. In general, the group $\textrm{Aut}_k(X)$ of automorphisms of $X$ over $k$ is not finite. But if we fix a polarization $\mathcal{L}$ on $X$, then the group $\textrm{Aut}_k(X,\mathcal{L})$ of automorphisms of the polarized abelian variety $(X,\mathcal{L})$ over $k$ is known to be finite. Then it is natural to ask which finite groups can be realized as the full automorphism group of a polarized abelian variety over some field $k.$ In one of earlier works, a complete classification of such finite groups was given for the case when $k$ is a finite field, $g$ is an odd prime, and $X$ is simple. One interesting thing is that the maximal such a finite group was a cyclic group of order $4g+2$ only when $g$ is a Sophie Germain prime. Another notable thing is the fact that the abelian variety that was constructed to achieve the maximal cyclic group splits over $\overline{k}$, an algebraic closure of $k.$ \\ In this talk, we provide a construction of an absolutely simple abelian variety of dimension $g$ ($g$ being a Sophie Germain prime) over a finite field $k$, which attains the maximal automorphism group. This can be regarded as the counterpart for the previous construction. Also, we briefly describe the asymptotic behavior of the characteristic of the base field $k$ for which we can give such a construction. Finally, if time permits, we take a closer look at the case of $g=5$ by introducing the Newton polygon of an abelian variety of dimension $5.$ (If you want to join the seminar, please contact Bo-Hae Im to get the zoom link.)

Ordered Ramsey numbers were introduced in 2014 by Conlon, Fox, Lee, and Sudakov. Their results included upper bounds for general graphs and lower bounds showing separation from classical Ramsey numbers. We show the first nontrivial results for ordered Ramsey numbers of specific small graphs. In particular we prove upper bounds for orderings of graphs on four vertices, and determine some numbers exactly using SAT solvers for lower bounds. These results are in the spirit of exact calculations for classical Ramsey numbers and use only elementary combinatorial arguments. This is joint work with Jeremy Alm, Kayla Coffey, and Carolyn Langhoff.

At Data Science Group, we try to offer computational models for challenging real-world problems. This talk will introduce two such problems that can benefit from collaboration with mathematicians and theorists. One is customs fraud detection, where the goal is to determine a small set of fraudulent transactions that will maximize the tax revenue when caught. We had previously shown a state-of-the-art deep learning model in collaboration with the World Customs Organization [KDD2020]. The next challenge is to consider semi-supervised (i.e., using very few labels) and unsupervised (i.e., no label information) settings that better suit developing countries' conditions. Another research problem is poverty mapping, where the goal is to infer economic index from high-dimensional visual features learned from satellite images. Several innovative algorithms have been proposed for this task [Science2016, AAAI2020, KDD2020]. I will introduce how we approach this problem under extreme conditions with little validation data, as in North Korea.

We will discuss about “Highly accurate fluorogenic DNA sequencing with information theory–based error correction”, Chen et al., Nature Biotechnology (2017) Eliminating errors in next-generation DNA sequencing has proved challenging. Here we present error-correction code (ECC) sequencing, a method to greatly improve sequencing accuracy by combining fluorogenic sequencing-by-synthesis (SBS) with an information theory–based error-correction algorithm. ECC embeds redundancy in sequencing reads by creating three orthogonal degenerate sequences, generated by alternate dual-base reactions. This is similar to encoding and decoding strategies that have proved effective in detecting and correcting errors in information communication and storage. We show that, when combined with a fluorogenic SBS chemistry with raw accuracy of 98.1%, ECC sequencing provides single-end, error-free sequences up to 200 bp. ECC approaches should enable accurate identification of extremely rare genomic variations in various applications in biology and medicine.

This is the part 3 of the lectures on infinity-categories: This talk will be focused on introducing quasi-categories as our model for infinity-categories. After reviewing some background material needed to define quasi-categories, we'll see how the definition works.

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Zoom ID: 352 730 6970, Password: 9999

In this talk, we consider stochastic systems with several stable sets. Typical examples are low-temperature physical systems and stochastic optimization algorithms. The macroscopic description of such systems is usually carried out via a so-called model reduction. We explain a necessary and sufficient condition for model reduction in terms of solutions of certain form of partial differential equations.

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pw: 2021math

Topological defect structure is one of the most interesting topics in natural sciences. Especially, the topological defect transition was highlighted by Nobel prize in 2016. But this topic is hard to understand and realize in the practical condition because the size and time-scale are huge in cosmos or so tiny in skyrmion system. So, we proposed to use liquid crystal (LC) materials to directly show this interesting topic, phase transition of topological defect.

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화학과&수리과학과 공동 주관 세미나

Thurston classified mappings from a given surface to itself. By iterating the surface mappings, one can view this as a dynamical system. Most of those surface mappings are so-called pseudo-Anosov. We briefly explain how we should understand these pseudo-Anosov maps and their physical meaning.

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화학과&수리과학과 공동 주관 세미나

Overparametrized neural networks have infinitely many solutions that achieve zero training loss, but gradient-based optimization methods succeed in finding solutions that generalize well. It is conjectured that the optimization algorithm and the network architecture induce an implicit bias towards favorable solutions, and understanding such a bias has become a popular topic. We study the implicit bias of gradient flow (i.e., gradient descent with infinitesimal step size) applied on linear neural network training. We consider separable classification and underdetermined linear regression problems where there exist many solutions that achieve zero training error, and characterize how the network architecture and initialization affects the final solution found by gradient flow. Our results apply to a general tensor formulation of neural networks that includes linear fully-connected networks and linear convolutional networks as special cases, while removing convergence assumptions required by prior research. We also provide experiments that corroborate our theoretical analysis.

This is part II of the lecture series in infinity-categories. I'll continue to talk about higher categories and some difficulty in defining them. In the end, a few models for infinity-categories will be introduced very roughly.

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Zoom ID: 352 730 6970, password: 9999

Bose-Einstein condensation (BEC) is one of the most famous phenomena, which cannot be explained by classical mechanics. Here, we discuss the time evolution of BEC in the mean-field limit. First, we review quantum mechanics briefly, and we understand the problem in a mathematically rigorous way. Then, we taste the idea of proof by using coherent state and the Fock space. Finally, some recent developments will be provided.

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.

Generative adversarial networks (GAN) are a widely used class of deep generative models, but their minimax training dynamics are not understood very well. In this work, we show that GANs with a 2-layer infinite-width generator and a 2-layer finite-width discriminator trained with stochastic gradient ascent-descent have no spurious stationary points. We then show that when the width of the generator is finite but wide, there are no spurious stationary points within a ball whose radius becomes arbitrarily large (to cover the entire parameter space) as the width goes to infinity.

Many types of diffusion equations have been used to describe diverse natural phenomena. The classical heat equation describes the heat propagation in homogeneous media, and the heat equation with fractional time derivative describes anomalous diffusion, especially sub-diffusion, caused by particle sticking and trapping effects. On the other hand, space-fractional diffusion equations are related to diffusion of particles with long range jumps. In this talk, I will introduce the following: 1. Elementary notion of stochastic parabolic equations 2. Stochastic processes with jumps and their related PDEs and Stochastic PDEs 3. Some regularity results of PDEs and Stochastic PDEs with non-local operators

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.

Abstract: Millions of individuals track their steps, heart rate, and other physiological signals through wearables. This data scale is unprecedented; I will describe several of our apps and ongoing studies, each of which collects wearable and mobile data from thousands of users, even in > 100 countries. This data is so noisy that it often seems unusable and in desperate need of new mathematical techniques to extract key signals used in the (ode) mathematical modeling typically done in mathematical biology. I will describe several techniques we have developed to analyze this data and simulate models, including gap orthogonalized least squares, a new ansatz for coupled oscillators, which is similar to the popular ansatz by Ott and Antonsen, but which gives better fits to biological data and a new level-set Kalman Filter that can be used to simulate population densities. My focus applications will be determining the phase of circadian rhythms, the scoring of sleep and the detection of COVID with wearables.

Deep neural networks have shown amazing success in various domains of artificial intelligence (e.g. vision, speech, language, medicine and game playing). However, classical tools for analyzing these models and their learning algorithms are not sufficient to provide explanations for such success. Recently, the infinite-width limit of neural networks has become one of key breakthroughs in our understanding of deep learning. This limit is unique in giving an exact theoretical description of large scale neural networks. Because of this, we believe it will continue to play a transformative role in deep learning theory. In this talk, we will first review some of the interesting theoretical questions in the deep learning community. Then we will review recent progress in the study of the infinite-width limit of neural networks focused around Neural Network Gaussian Process (NNGP) and Neural Tangent Kernel (NTK). This correspondence allows us to understand wide neural networks as different kernel based machine learning models and provides 1) exact Bayesian inference without ever initializing or training a network and 2) closed form solution of network function under gradient descent training. We will discuss recent advances, applications and remaining challenges of the infinite-width limit of neural networks.

Infinity-category theory is a generalization of the ordinary category theory, where we extend the categorical perspective into the homotopical one. Putting differently, we study objects of interest and "mapping spaces" between them. This theory goes back to Boardman and Vogt, and more recently, Joyal, Lurie, and many others laid its foundation. Despite its relatively short history, it has found applications in many fields of mathematics. For example, number theory, mathematical physics, algebraic K-theory, and derived/spectral algebraic geometry: more concretely, p-adic Hodge theory, Geometric Langlands, the cobordism hypothesis, topological modular forms, deformation quantization, and topological quantum field theory, just to name a few. The purpose of this series of talks on infinity-categories is to make it accessible to those researchers who are interested in the topic. We’ll start from scratch and try to avoid (sometimes inevitable) technical details in developing the theory. That said, a bit of familiarity to the ordinary category theory is more or less necessary. Overall, this series has an eye toward derived/spectral algebraic geometry, but few experience in algebraic geometry would hardly matter. Therefore, everyone is welcome to join us. This is the first in the series. We’ll catch a glimpse of infinity-category theory through some motivational examples.

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Zoom ID: 352 730 6970, password: 9999

It is a gentle introduction to the mean curvature flow and its application to knot theory for undergraduate students. J.W.Alexander discovered a knotted sphere embedded in 3-dimensional Euclidean space in 1924. This example has provoked curiosity to find simple conditions under which embedded spheres are unknotted. In this talk we will sketch theorems and conjectures in the mean curvature flow for the knot theory, in analogy to the Ricci flow for the smooth 4-dimensional Poincare conjecture.

We will discuss about “Synthetic multistability in mammalian cells”, Zhu et al., bioRxiv (2021) In multicellular organisms, gene regulatory circuits generate thousands of molecularly distinct, mitotically heritable states, through the property of multistability. Designing synthetic multistable circuits would provide insight into natural cell fate control circuit architectures and allow engineering of multicellular programs that require interactions among cells in distinct states. Here we introduce MultiFate, a naturally-inspired, synthetic circuit that supports long-term, controllable, and expandable multistability in mammalian cells. MultiFate uses engineered zinc finger transcription factors that transcriptionally self-activate as homodimers and mutually inhibit one another through heterodimerization. Using model-based design, we engineered MultiFate circuits that generate up to seven states, each stable for at least 18 days. MultiFate permits controlled state-switching and modulation of state stability through external inputs, and can be easily expanded with additional transcription factors. Together, these results provide a foundation for engineering multicellular behaviors in mammalian cells.

In this talk, I start by giving a brief overview of the practice in deep learning with focus on learning (optimization) and model selection (hyperparameter optimization). In particular, I will describe and discuss black-box optimization approaches to model selection, followed by discussion on how these two stages in deep learning can be collapsed into a single optimization problem, often referred to as bilevel optimization. This allows us to extend the applicability of gradient-based optimization into model selection, although existing gradient-based model selection, or hyperparameter optimization, approaches have been limited because they require an extensive numbers of so-called roll-out. I will then explain how we can view gradient-based optimization as a recurrent network and how this enables us to view hyperparameter optimization as training a recurrent network. This is an insight that leads to a novel paradigm of online hyperparameter optimization which does not require any simulated roll-out.

Abstract: The large deviation problem for the spectrum of random matrices has attracted immense interest. It was first studied for GUE and GOE, which are exactly solvable, and subsequently studied for Wigner matrices with general distributions. Once the sparsity is induced (i.e. each entry is multiplied by the independent Bernoulli distribution, Ber(p)), eigenvalues can exhibit a drastically different behavior. For a large class of Wigner matrices, including Gaussian ensembles and the adjacency matrix of Erdos-Renyi graphs, dense behavior ceases to hold near the constant average degree of sparsity, p~1/n (up to a poly-logarithmic factor). In this talk, I will talk about the spectral large deviation for Gaussian ensembles with a sparsity p=1/n. Joint work with Shirshendu Ganguly.

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ZOOM회의정보 link:https://zoom.us/j/94727585394?pwd=QlBSRUNTQi9UWXNLSTlPOTgrRnhhUT09 회의 ID: 947 2758 5394 암호: saarc

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.

The compressible Euler system was first formulated by Euler in 1752, and was complemented by Laplace and Clausius in the 19th century, by introducing the energy conservation law and the concept of entropy based on thermodynamics. The most important feature of the Euler system is the finite-time breakdown of smooth solutions, especially, appearance of a shock wave as severe singularity to irreversibility(-in time) and discontinuity(-in space). Therefore, a fundamental question (since Riemann 1858) is on what happens after a shock occurs. This is the problem on well-posedness (that is, existence, uniqueness, stability) of weak solutions satisfying the 2nd law of thermodynamics, which is called entropy solution. This issue has been conjectured as follows: Well-posedness of entropy solutions for CE can be obtained in a class of vanishing viscosity limits of solutions to the Navier-Stokes system. This conjecture for the fundamental issue remains wide open even for the one-dimensional CE. This talk will give an overview of the conjecture, and recent progress on it.

The Hybridizable discontinuous Galerkin (HDG) methods retain the advantage of the discontinuous Galerkin (DG) methods such as flexibility in meshing and preserving local conservation of physical quantities and overcome to shortcomings of the DG by reducing the globally coupled degree of freedom. I will design a multiscale method within the HDG framework. The main concept of the multiscale HDG method is to deriving upscale structure of the method and to generate multiscale spaces defined on the coarse edges that provide a reduced dimensional approximation for numerical traces. Eigenvalue problems plays a significant role in generating a multiscale space. Also, error analysis and a representative number of numerical examples will be given.

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.