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.

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.

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.

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.)