In this talk, we propose the Landau-Lifshitz type system augmented with Chern-Simons gauge terms, which can be considered as the geometric analog of so-called the Chern-Simons-Schrodinger equations. We first derive its self-dual equations through the energy minimization so that we can provide $N$-equivariant solitons. We next deliver basic ideas of constructing $N$-equivariant solitary waves for non-self-dual cases and investigating their qualitative properties.

It has been told that deep learning is a black box. The universal approximation theorem was the key theorem which makes the stories going on. On the other hand, in the perspective of the function classes generated by deep neural network, it can be analyzed by in terms of the choice of the various activation functions. The piecewise linear functions, fourier series, wavelets and many other classes would be considered for the purpose of the tasks such as classification, prediction and generation models which heavily depend on the data sets. It might be a challenging problem for mathematicians to develop a new optimization theory depending on the various function classes.

Property testers are probabilistic algorithms aiming to solve a decision problem efficiently in the context of big-data. A property tester for a property P has to decide (with high probability correctly) whether a given input graph has property P or is far from having property P while having local access to the graph. We study property testing of properties that are definable in first-order logic (FO) in the bounded-degree model. We show that any FO property that is defined by a formula with quantifier prefix ∃*∀* is testable, while there exists an FO property that is expressible by a formula with quantifier prefix ∀*∃* that is not testable. In the dense graph model, a similar picture is long known (Alon, Fischer, Krivelevich, Szegedy, Combinatorica 2000), despite the very different nature of the two models. In particular, we obtain our lower bound by a first-order formula that defines a class of bounded-degree expanders, based on zig-zag products of graphs. This is joint work with Isolde Adler and Pan Peng.

Global wellposedness and asymptotic stability of the Boltzmann equation with specular reflection boundary condition in 3D non-convex domain is an outstanding open problem in kinetic theory. Motivated by Guo’s L^2-L^\infty theory, the problem was completely solved for general C^3 domain, but it is still widely open for general non-convex domains. The problem was solved in cylindrical domain with analytic non-convex cross section. Generalizing previous work, we study the problem in general solid torus, a solid torus with general analytic convex cross-section. This is the first results for the domain which contains essentially 3D non-convex structure. This is a joint work with Chanwoo Kim and Gyeonghun Ko.

The past decade has witnessed a great advancement on the Tate conjecture for varieties with Hodge number $h^{2,0} = 1$. Charles, Madapusi-Pera and Maulik completely settled the conjecture for K3 surfaces over finite fields, and Moonen proved the Mumford-Tate (and hence also Tate) conjecture for more or less arbitrary $h^{2,0} = 1$ varieties in characteristic $0$. In this talk, I will explain that the Tate conjecture is true for mod $p$ reductions of complex projective $h^{2,0} = 1$ varieties when $p \gg 0$, under a mild assumption on moduli. By refining this general result, we prove that in characteristic $p \geq 5$ the BSD conjecture holds for a height $1$ elliptic curve $E$ over a function field of genus 1, as long as $E$ is subject to the generic condition that all singular fibers in its minimal compactification are irreducible. We also prove the Tate conjecture over finite fields for a class of surfaces of general type and a class of Fano varieties. The overall philosophy is that the connection between the Tate conjecture over finite fields and the Lefschetz $(1,1)$-theorem over $C$ is very robust for $h^{2,0} = 1$ varieties, and works well beyond the hyperk\”{a}hler world. This is based on joint work with Paul Hamacher and Xiaolei Zhao.

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

The past decade has witnessed a great advancement on the Tate conjecture for varieties with Hodge number $h^{2,0} = 1$. Charles, Madapusi-Pera and Maulik completely settled the conjecture for K3 surfaces over finite fields, and Moonen proved the Mumford-Tate (and hence also Tate) conjecture for more or less arbitrary $h^{2,0} = 1$ varieties in characteristic $0$. In this talk, I will explain that the Tate conjecture is true for mod $p$ reductions of complex projective $h^{2,0} = 1$ varieties when $p \gg 0$, under a mild assumption on moduli. By refining this general result, we prove that in characteristic $p \geq 5$ the BSD conjecture holds for a height $1$ elliptic curve $E$ over a function field of genus 1, as long as $E$ is subject to the generic condition that all singular fibers in its minimal compactification are irreducible. We also prove the Tate conjecture over finite fields for a class of surfaces of general type and a class of Fano varieties. The overall philosophy is that the connection between the Tate conjecture over finite fields and the Lefschetz $(1,1)$-theorem over $C$ is very robust for $h^{2,0} = 1$ varieties, and works well beyond the hyperk\”{a}hler world. This is based on joint work with Paul Hamacher and Xiaolei Zhao.

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

(학사과정 학생 개별연구 결과 발표 세미나) Čech cohomology is the direct limit of cohomology taken from the cochain complex obtained by an open cover and a sheaf. In this talk we will derive some important results about Riemann surfaces such as Riemann-Roch theorem and Serre Duality, regarding low level Čech cohomologies. We will also discuss some basic structure and properties of Riemann surfaces using these results, focusing on genus and the embeddings.

We show a flow-augmentation algorithm in directed graphs: There exists a polynomial-time algorithm that, given a directed graph G, two integers $s,t\in V(G)$, and an integer $k$, adds (randomly) to $G$ a number of arcs such that for every minimal st-cut $Z$ in $G$ of size at most $k$, with probability $2^{−\operatorname{poly}(k)}$ the set $Z$ becomes a minimum $st$-cut in the resulting graph. The directed flow-augmentation tool allows us to prove fixed-parameter tractability of a number of problems parameterized by the cardinality of the deletion set, whose parameterized complexity status was repeatedly posed as open problems: (1) Chain SAT, defined by Chitnis, Egri, and Marx [ESA'13, Algorithmica'17], (2) a number of weighted variants of classic directed cut problems, such as Weighted st-Cut, Weighted Directed Feedback Vertex Set, or Weighted Almost 2-SAT. By proving that Chain SAT is FPT, we confirm a conjecture of Chitnis, Egri, and Marx that, for any graph H, if the List H-Coloring problem is polynomial-time solvable, then the corresponding vertex-deletion problem is fixed-parameter tractable. Joint work with Stefan Kratsch, Marcin Pilipczuk, Magnus Wahlström.

For a given stable subalgebra of the formal power series ring, its Laurent extension, or others, we define an operator algebra over the subalgebra. One of the important operator algebras is the Weyl algebra or its generalization. We define generalized radical Weyl algebras (GRWA) and define the generalized radical Weyl algebra modules, we prove that the algebras and modules are simple. An automorphism of the GRWA define a twisted simple module as well. Since GRWA is an associative algebra, it has an ${\Bbb F}$-subalgebra which is a Lie algebra with respect to the commutator and we show that the Lie algebra is simple. We consider some other generalized Weyl algebra and its descended consequences as well.

We call an order type inscribable if it is realized by a point configuration where all extreme points are all on a circle. In this talk, we investigate inscribability of order types. We first show that every simple order type with at most 2 interior points is inscribable, and that the number of such order types is $\Theta(\frac{4^n}{n^{3/2}})$. We further construct an infinite family of minimally uninscribable order types. The proof of uninscribability mainly uses Möbius transformations. We also suggest open problems around inscribability. This is a joint work with Michael Gene Dobbins.

Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its “expectation-threshold,” which is a natural (and often easy to calculate) lower bound on the threshold. In the first talk on Monday, I will introduce the Kahn-Kalai Conjecture with some motivating examples and then briefly talk about the recent resolution of the Kahn-Kalai Conjecture due to Huy Pham and myself. In the second talk on Tuesday, I will discuss our proof of the conjecture in detail.

Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its “expectation-threshold,” which is a natural (and often easy to calculate) lower bound on the threshold. In the first talk on Monday, I will introduce the Kahn-Kalai Conjecture with some motivating examples and then briefly talk about the recent resolution of the Kahn-Kalai Conjecture due to Huy Pham and myself. In the second talk on Tuesday, I will discuss our proof of the conjecture in detail.

In this talk, I will introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimeswith odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-timetailson stationary spacetimes. Moreover, the method also applies to dynamical spacetimes. In this case, I will explain how the late-timetailsare in general different(!) from the stationary case in the presence of dynamical and/or nonlinear perturbations of the problem. This is joint work with Jonathan Luk(Stanford).

In this talk, I will discuss some recent developments on the long-term dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariantsymmetry. CSS is a gauge-covariant 2D cubic nonlinear Schrödingerequation, which admits the L2-scaling/pseudoconformalinvariance and soliton solutions.I will first discuss soliton resolution for this model, which is a remarkable consequence of the self-duality and non-local nonlinearity that are distinguished features of CSS. Next, I will discuss the blow-up dynamics (singularity formation) for CSS and introduce an interesting instability mechanism (rotational instability) of finite-time blow-up solutions. This talk is based on joint works with SoonsikKwon and Sung-JinOh.

Recently, mapping a signal/image into a low rank Hankel/Toeplitz matrix has become an emerging alternative to the traditional sparse regularization, due to its ability to alleviate the basis mismatch between the true support in the continuous domain and the discrete grid. In this talk, we introduce a novel structured low rank matrix framework to restore piecewise smooth functions. Inspired by the total generalized variation to use sparse higher order derivatives, we derive that the Fourier samples of higher order derivatives satisfy an annihilation relation, resulting in a low rank multi-fold Hankel matrix. We further observe that the SVD of a low rank Hankel matrix corresponds to a tight wavelet frame system which can represent the image with sparse coefficients. Based on this observation, we also propose a wavelet frame analysis approach based continuous domain regularization model for the piecewise smooth image restoration.

The degree-shifting action on the cohomology of locally symmetric spaces, which has its origins in the representation theory of real reductive groups, enjoys a surprising connection with arithmetic, as expected by the so-called motivic action conjectures of A. Venkatesh. Although these conjectures are expected to hold in great generality, there is a disparity between the algebraic and non-algebraic locally symmetric spaces. We will discuss the nature of the degree-shifting action in both cases (For those who cannot attend the in-person seminar, we will also stream the seminar talk via Zoom. Please contact Wansu Kim for the Zoom connection details.)

Reconstruction of gene regulatory networks (GRNs) is a powerful approach to capture a prioritized gene set controlling cellular processes. In our previous study, we developed TENET a GRN reconstructor from single cell RNA sequencing (scRNAseq). TENET has a superior capability to identify key regulators compared with other algorithms. However, accurate inference of gene regulation is still challenging. Here, we suggest an integrative strategy called TENET+ by combining single cell transcriptome and chromatin accessibility data. By applying TENET+ to a paired scRNAseq and scATACseq dataset of human peripheral blood mononuclear cells, we found critical regulators and their epigenetic regulations for the differentiations of CD4 T cells, CD8 T cells, B cells and monocytes. Interestingly, TENET+ predicted LRRFIP1 and ZBTB16 as top regulators of CD4 and CD8 T cells which were not predicted in a motif-based tool SCENIC. In sum, TENET+ is a tool predicting epigenetic gene regulatory programs in unbiased way, suggesting that novel epigenetic regulations can be identified by TENET+.

Reconstruction of gene regulatory networks (GRNs) is a powerful approach to capture a prioritized gene set controlling cellular processes. In our previous study, we developed TENET a GRN reconstructor from single cell RNA sequencing (scRNAseq). TENET has a superior capability to identify key regulators compared with other algorithms. However, accurate inference of gene regulation is still challenging. Here, we suggest an integrative strategy called TENET+ by combining single cell transcriptome and chromatin accessibility data. By applying TENET+ to a paired scRNAseq and scATACseq dataset of human peripheral blood mononuclear cells, we found critical regulators and their epigenetic regulations for the differentiations of CD4 T cells, CD8 T cells, B cells and monocytes. Interestingly, TENET+ predicted LRRFIP1 and ZBTB16 as top regulators of CD4 and CD8 T cells which were not predicted in a motif-based tool SCENIC. In sum, TENET+ is a tool predicting epigenetic gene regulatory programs in unbiased way, suggesting that novel epigenetic regulations can be identified by TENET+.

Reconstruction of gene regulatory networks (GRNs) is a powerful approach to capture a prioritized gene set controlling cellular processes. In our previous study, we developed TENET a GRN reconstructor from single cell RNA sequencing (scRNAseq). TENET has a superior capability to identify key regulators compared with other algorithms. However, accurate inference of gene regulation is still challenging. Here, we suggest an integrative strategy called TENET+ by combining single cell transcriptome and chromatin accessibility data. By applying TENET+ to a paired scRNAseq and scATACseq dataset of human peripheral blood mononuclear cells, we found critical regulators and their epigenetic regulations for the differentiations of CD4 T cells, CD8 T cells, B cells and monocytes. Interestingly, TENET+ predicted LRRFIP1 and ZBTB16 as top regulators of CD4 and CD8 T cells which were not predicted in a motif-based tool SCENIC. In sum, TENET+ is a tool predicting epigenetic gene regulatory programs in unbiased way, suggesting that novel epigenetic regulations can be identified by TENET+.

This talk will highlight recent results establishing a beautiful computational phase transition for approximate counting/sampling in (binary) undirected graphical models (such as the Ising model or on weighted independent sets). The computational problem is to sample from the equilibrium distribution of the model or equivalently approximate the corresponding normalizing factor known as the partition function. We show that when correlations die off on the infinite D-regular tree then the Gibbs sampler has optimal mixing time of O(n log n) on any graph of maximum degree D, whereas when the correlations persist (in the limit) then the sampling/counting problem are NP-hard to approximate. The Gibbs sampler is a simple Markov Chain Monte Carlo (MCMC) algorithm. Key to these mixing results are a new technique known as spectral independence which considers the pairwise influence of vertices. We show that spectral independence implies an optimal convergence rate for a variety of MCMC algorithms.