# 세미나 및 콜로퀴엄

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Principal component analysis (PCA) is a well-known tool in multivariate statistics. One significant challenge in using PCA is the choice of the number of principal components. In order to address this challenge, we propose an exact distribution-based method for hypothesis testing and construction of confidence intervals for signals in a noisy matrix with infinite samples. Assuming Gaussian noise, we derive exact results based on the conditional distribution of the singular values of a Gaussian matrix by utilizing a post-selection inference framework. In simulation studies we find that our proposed methods compare well to existing approaches.

Artificial neural network theory has been developing rapidly in recent 10 years. This talk introduces the basic concepts of deep learning such as back propagation, gradient descent, batch normalization, and introduces the general concepts and latest trends of deep neural network theory such as Autoencoder, Restricted Boltzmann machine, CNN, RNN and reinforcement learning.

The renormalized volume is an invariant of a conformally compact Einstein manifold, which has been studied extensively in several research areas: conformal geometry, global analysis, and mathematical physics. In this coloquium talk, I will explain the basic notion and properties of the renormalized volume of 3-dimensional hyperbolic manifolds of infinite volume, and its relation with the Liouville theory for conformal boundary Riemann surface.

Given plural datasets, Canonical Correlation Analysis (CCA) investigates the linear transformation of the variables which reduces the correlation structure to the simplest possible form, and addresses the relationships between the variables among the datasets. We propose a novel method for testing the statistical significance of canonical correlation coefficients between two datasets. Utilizing post-selection inference framework, our proposed method provides exact type I error as well as steady detection power with Gaussian assumption. Simulation results compare well with existing approaches.

Let $C$ be a conjugacy class of $S_n$ and $K$ an $S_n$-field. Let $n_{K,C}$ be the smallest prime which is ramified or whose Frobenius automorphism Frob$_p$ does not belong to $C$. Under some technical conjectures, we show that the average of $n_{K,C}$ is a constant. We explicitly compute the constant.

For $S_3$ and $S_4$-fields, our result is unconditional. Let $N_{K,C}$ be the smallest prime for which Frob$_p$ belongs to $C$.

We obtain the average of $N_{K,C}$ under some technical conjectures. When $C$ is the union of all the conjugacy classes not contained in $A_n$ and $n=3,4$, our result is unconditional. This is a joint work with Henry Kim.

I shall discuss the role of geometry in creating the space time that is fundamental to the physics of general relativity. I shall also discuss fundamental concepts such as mass, linear momentum and angular momentum in general relativity. The lack of continuous symmetries in general spacetime makes it difficult to define such quantities and I shall explain how the difficulty can be overcome by the works of Brown-York, Liu-Yau and Wang -Yau.

Many modern applications such as machine learning, inverse problems, and control require solving large-dimensional optimization problems. First-order methods such as a gradient method are widely used to solve such large-scale problems, since their computational cost per iteration mildly depends on the problem dimension. However, they suffer from slow convergence rates, compared to second-order methods such as Newton's method. Therefore, accelerating a gradient method has received a great interest in the optimization community, and this led to the development and extension of a conjugate gradient method, a heavy-ball method, and Nesterov's fast gradient method, which we review in this talk. This talk will then present new proposed accelerated gradient methods, named optimized gradient method (OGM) and OGM-G, that have the best known worst-case convergence rates for smooth convex optimization among any accelerated gradient methods.

This talk is about improving the efficiency of the optimization methods that is mainly characterized by their worst-case convergence rates. In particular, this talk will present how the proposed accelerated gradient methods, named optimized gradient method (OGM) and OGM-G, with the best-known worst-case convergence rates for smooth convex optimization are developed by optimizing the efficiency of the first-order methods in terms of the cost function and the norm of the gradient respectively. This is based on the performance estimation problem approach that casts the worst-case convergence rate analysis into a finite-dimensional semidefinite optimization problem. This talk will further discuss about extending the approach to more general classes of problems such as nonsmooth composite convex optimization problems and monotone inclusion problems.

Mazur and Rubin introduced the so-called Selmer-companion curves in 2015. Let $E$ be an elliptic curve over a number field $K$. Suppose there is a function that sends a quadratic character of $K$ to the $p$-Selmer rank of $E$ twisted by that character. How much information of $E$ can be read off from the function? In this talk, we give a sketch of proof of a conjecture on $p$-Selmer near-companion posed by Mazur and Rubin when $p=2$. If time allows, we will discuss an application of the technique in the proof to the Shafarevich-Tate groups.

This is a gentle introduction to the Langlands program based on selected historical developments. In particular attention will be drawn to the birth of the Langlands program in a letter of Langlands to Weil in 1967. Time permitting a snapshot of some current developments will be given.

The Dirichlet prime number theorem states that if two integers a and b are relatively prime, then there are infinitely many primes of the form an+b. It is a natural question after the Dirichlet theorem to find an upper bound for the smallest prime having the form an+b. Linnik answered the question, which is called 'Linnik problem' now. This problem can be stated in terms of a certain character, and so the Linnik problem is extended to a question on automorphic representations. In this talk, I will talk about application of Langalnds program to the Linnik problem for automorphic representations.

The theory of error-correcting codes, also known as Coding Theory, was invented by R. Hamming and C. Shannon around 1948. Since then, we can communicate information or data reliably. It has been applied to satellite communication, mobile phone, compact disc, high definition TV, and Artificial Intelligence.

In this talk, we will give two other applications of error-correcting codes. One is a game based on the binary Hamming [7,4,3] code. The other is code-based cryptography based on product codes. We only assume graduate level Algebra.

For del Pezzo surfaces, it is known that Q-Gorenstein degenerations and three block collections are controlled by the same Markov type equations. Besides the projective planes, there has been no explicit theory describing such an intimacy. I will introduce our ongoing attempts to relate three block collections and Q-Gorenstein degenerations of del Pezzo surfaces.

Let X be a smooth projective variety in the N-dimensional projective space, embedded by a very ample line bundle L. An ACM bundle E on X is a locally free sheaf which does not have intermediate cohomology with respect to L. If furthermore E is linear, i.e., the minimal free resolution of E on mathbb{P}^N is completely linear, then it is called Ulrich. Not only they provide constructive examples of `good' vector bundles on X, but also they encode a lot of algebro-geometric information on X via their several connections to different topics. Unfortunately, in spite of their importance, almost nothing is known except for a very few examples. In this talk, we first review basic properties and classical results, mostly on hypersurfaces. Then we discuss several open problems, mostly motivated from cubic hypersurfaces.

A sub-Riemannian space is a manifold with a selected distribution (of "allowed movement directions" represented by the spanning vector fields) of the tangent bundle, which spans by nested commutators, up to some finite order, the whole tangent bundle. Such geometries naturaly arise in nonolonomic mechanics, robotics, thermodynamics, quantum mechanics, neurobiology etc. and are closely related to optimal control problems on the corresponding configuration space. As is well known, there exists an intrinsic Carnot-Caratheodory metric generated by the «allowed» vector fiels. Studying the Gromov's tangent cone to the corresponding metric space is widely used to construct efficient motion planning algorithms for related optimal control systems. We generalize this construction to weighted vector fields, which provides applications to optimal control theory of systems nonlinear on control parameters. Such construction requires, in particular, an extension of Gromov's theory to quasimetric spaces, since the intrinsic C-C metric doesn't exist in this case.

A sub-Riemannian space is a manifold with a selected distribution (of "allowed movement directions" represented by the spanning vector fields) of the tangent bundle, which spans by nested commutators, up to some finite order, the whole tangent bundle. Such geometries naturaly arise in nonolonomic mechanics, robotics, thermodynamics, quantum mechanics, neurobiology etc. and are closely related to optimal control problems on the corresponding configuration space. As is well known, there exists an intrinsic Carnot-Caratheodory metric generated by the «allowed» vector fiels. Studying the Gromov's tangent cone to the corresponding metric space is widely used to construct efficient motion planning algorithms for related optimal control systems. We generalize this construction to weighted vector fields, which provides applications to optimal control theory of systems nonlinear on control parameters. Such construction requires, in particular, an extension of Gromov's theory to quasimetric spaces, since the intrinsic C-C metric doesn't exist in this case.

언어/Language: 영어/English

Chemical processes occurring in living cells are often complex stochastic process composed of numerous enzymatic reactions whose rates are coupled to cell state variables, the majority of which are hidden and uncontrollable. Despite advances in single-cell technologies, the lack of an accurate kinetic theory describing intracellular reactions has restricted a robust, quantitative understanding of biological phenomena. In this talk, after a brief review of the assumptions underlying the conventional chemical kinetics and the Pauli’s master equation, I will discuss a new chemical kinetic theory for intracellular enzyme reactions composed of arbitrary stochastic elementary processes and its application to modern single enzyme experiments. Next, I will present a derivation of the Chemical Fluctuation Theorem (CFT), which provides an accurate relationship between the environment-coupled chemical dynamics of enzyme reactions comprising gene expression and gene expression variability among cells with the same gene. Time permitted, I will also present the application of the CFT to a unified, quantitative explanation ofmRNA noise for various gene expression systems and predictions for the dependence of the mRNA noise on the mRNA lifetime distribution, whose correctness is confirmed against stochastic simulation. This work suggests promising, new directions for quantitative investigation into cellular control over biological functions by making the complex dynamics of intracellular reactions accessible to rigorous mathematical deductions [1].

Ref [1]: Park et al., Nature Communications 9, 297 (2018).

Recall that Bloch's higher cycle group $Z^*(X;r)$ of an algebraic scheme $X$ is a free abelian group (graded by codimension) generated by integral closed subschemes of $X\times\Delta^r$ meeting all the faces properly. Given a closed subset $D$ of $X$, we consider the subgroup $Z^*(X,D;r)$ of $Z^*(X;r)$ generated by those cycles which do not meet $D\times\Delta^r$. Then it is assembled into a simplicial abelian group $Z^*(X,D;-)$ and we denote its $n$-th homotopy group by $CH^*(X,D;n)$. In this talk, I explain that $CH^*(X,D;n)$ is related to the relative homotopy K-theory $KH(X,D)$ as Bloch's higher Chow group $CH^*(X;n)$ is related to the K-theory $K(X)$. More precisely, under some general hypotheses, we establish an Atiyah-Hirzebruch type spectral sequence relating them.

Hyperrings generalize commutative rings in such a way that addition is ``multi-valued''. In this talk, we illustrate how the notion of algebraic geometry over hyperrings provides a natural framework to show that certain topological spaces (underlying topological spaces of (1) schemes, (2) Berkovich analytification of schemes, and (3) real schemes) are homeomorphic to sets of rational points of (Grothendieck) schemes over hyperfields.

From any monoid scheme $X$ (also known as an $mathbb{F}_1$-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) $X_S$ by scalar extension to an idempotent semifield $S$. We prove that for a given irreducible monoid scheme $X$ (with some mild conditions) and an idempotent semifield $S$, the Picard group $Pic(X)$ of $X$ is stable under scalar extension to $S$. In other words, we show that the two groups $Pic(X)$ and $Pic(X_S)$ are isomorphic. A similar argument to our proof can be applied to provide a new proof of the theorem by J.~Flores and C.~Weibel stating that the Picard group of a toric monoid scheme associated to a fan is stable under the scalar extension to a field. $k$.

A phase-field method is a useful mathematical tools for solving an interfacial dynamics problems such as solidification, multiphase fluid flows, image inpainting, volume reconstruction, and etc. The method replaces a boundary condition at the interface with a partial differential equation. One of the most commonly used equation is the Cahn-Hilliard equation, which is the 4th order nonlinear partial differential equation. Eyre proposed the convex splitting scheme to overcome its severe time-step restriction and it has been widely used in last two decays; however, it was reported that there is a time-step rescaling problem. In this talk, we analyze the effective time-step size of a nonlinear convex splitting scheme for the Cahn-Hilliard equation to choose proper time-step size for own purpose.

In this talk, we consider the following Cauchy problem for 1 < 2

i@tu + ()

2 u = (j (1) j juj2)u; u(0; ) = ':

We prove that there exists a globally well-posed solution to (1) and the solution

scatters to free waves asymptotically as t ! 1 whenever the initial data is radial

and suciently small in L2(R3). This result is shown to be optimal by proving the

discontinuity of the

ow map in the super-critical range. We employ the standard

contraction argument in a function space constructed based on the space of bounded

quadratic variation V 2. The main ingredients for the proof are L2(R1+3) bilinear

estimates for free solutions and its transference to adapted V 2 spaces. This is joint

work with Sebastian Herr.

I would like to explicitly bound the lengths of each singularities of class T on nonrational normal projective surfaces W with many singularities of class T and K_W ample. For the case when W has only one singularity, I will briefly introduce [Rana-Urzúa 2017] for algebraic-geometric approach and [Evans-Smith 2017] for symplectic-topological approach. The potential proof would combine symplectic techniques with the algebraic ones in [RU 17]. This may answer effectiveness of bounds (see [Alexeev 1994], [Alexeev-Mori 2004], [Y. Lee 1999]) for those surfaces. This is a joint work in progress with Heesang Park and Giancarlo Urzúa.

We discuss a priori estimates and the existence of solutions to the modied Benjamin-Ono

equation (mBO)

@tu + H@xxu = @x(u3=3); (x; t) 2 K R;

u(0; x) = u0(x) 2 HsR

(K);

(1)

with K 2 fR;Tg. Localization of time to small frequency-dependent time intervals recovers

control of solutions at low regularities and yields a priori estimates and existence of solutions for 1=4 < s < 1=2. Previously, this was carried out on the real line in [1]. We prove the same results for periodic solutions after observing that the localization to short time intervals recovers dispersive properties from Euclidean space. The strategy can also be adjusted to deal with periodic solutions to the modied Korteweg-de Vries equation (cf. [2]).

This talk concerns the Boussinesq abcd system originally derived by Bona, Chen and Saut [J. Nonlinear.

Sci. (2002)] as rst order 2-wave approximations of the incompressible and irrotational, two dimensional

water wave equations in the shallow water wave regime. Among many particular regimes, the Hamiltonian

generic regime is characterized by the setting b = d > 0 and a; c < 0. It is known that the system in

this regime is globally well-posed for small data in the energy space H1 H1 by Bona, Chen and Saut

[Nonlinearity (2004)]. In this talk, we are going to discuss about the decay and the scattering problem

in this regime, which is characterized as having (quadratic) long-range nonlinearities, very weak linear

decay O(t1=3) and existence of non scattering solutions (solitary waves). More precisely, we will see

that for a suciently dispersive abcd systems (characterized only in terms of parameters a; b and c), all

small solutions must decay to zero, locally strongly in the energy space, in proper subset of the light cone

jxj jtj. No parity nor extra decay assumptions are needed to prove decay, only small solutions in the

energy space.

This is joint work with Claudio Mu~noz, Felipe Poblete and Juan C. Pozo.

1

Since Belavin, Polyakov, and Zamolodchikov introduced conformal field theory as an operator algebra formalism which relates some conformally invariant critical clusters in two-dimensional lattice models to the representation theory of Virasoro algebra, it has been applied in string theory and condensed matter physics. In mathematics, it inspired development of algebraic theories such as Virasoro representation theory and the theory of vertex algebras. After reviewing its development and presenting its rigorous model in the context of probability theory and complex analysis, I discuss its application to the theory of Schramm-Loewner evolution.

Let $X$ be a projective normal $\mathbb{Q}$-factorial variety of Picard number~$1$ and $S$ be a prime divisor on $X$. The affine variety $X\setminus S$ is called an affine Fano variety if the pair $(X, S)$ has purely log terminal singularities and $-(K_X+S)$ is ample. Furthermore, the affine Fano variety~$X\setminus S$ is said to be super-rigid if the following two conditions hold.

_{1},…, H

_{k}, a graph G is (H

_{1},…, H

_{k})-free if there is a k-edge-colouring of G with no H

_{i}in colour-i for all i in {1,2,…,k}. Fix a function f(n), the Ramsey-Turán function rt(n,H

_{1},…,H

_{k},f(n)) is the maximum size of an n-vertex (H

_{1},…, H

_{k})-free graph with independence number at most f(n). We determine rt(n,K

_{3},K

_{s},δn) for s in {3,4,5} and sufficiently small δ, confirming a conjecture of Erdős and Sós from 1979. It is known that rt(n,K

_{8},f(n)) has a phase transition at f(n)=Θ(√(n\log n)). We prove that rt(n,K

_{8},o(√(n\log n)))=n

^{2}/4+o(n

^{2}), answering a question of Balogh, Hu and Simonovits. The proofs utilise, among others, dependent random choice and results from graph packings. Joint work with Jaehoon Kim and Younjin Kim.

It is well known that the Cone Theorem in birational geometry played a crucial role in the development of the minimal model program. In this talk, we discuss a generalization of the Cone Theorem to subcones of the Mori cone. This is an on-going joint work with Yoshinori Gongyo.

We discuss the refinements of the Birch and Swinnerton-Dyer conjecture \`{a} la Mazur-Tate and Kurihara, which concerns the behavior of Fitting ideals of Selmer groups of elliptic curves over finite subextensions in the cyclotomic \mathbb{Z}_p-extension of \mathbb{Q}. This is joint work with Masato Kurihara.

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E6-1, ROOM 3433
Discrete Math
Mark Siggers (Kyungpook National University)
The reconfiguration problem for graph homomorpisms

For problems with a discrete set of solutions, a reconfiguration problem defines solutions to be adjacent if they meet some condition of closeness, and then asks for two given solutions it there is a path between them in the set of all solutions.

There is a rich theory of harmonic mappings between Riemannian manifolds, going back to the celebrated Eels-Sampson theorem

which guarantees the existence of harmonic maps between negatively curved manifolds. Recently, the study of twisted harmonic maps has generated much interest in higher Teichmüller theory, as it is the key to the nonabelian Hodge correspondence between the character variety of a surface group and the moduli space of Higgs bundles. In this talk, I will present a computer software that I have developed with J. Gaster, whose purpose is to compute and investigate equivariant harmonic maps between hyperbolic surfaces. I will also discuss the theoretical aspects of this project. Basic information and screenshots of this software can be found here: http://math.newark.rutgers.edu/~bl498/software.html#hitchin

A Bi-Lagrangian structure in a smooth manifold consists of a symplectic form and a pair of transverse Lagrangian foliatio ns.

Equivalently, it can be defined as a para-Kähler structure, that is the para-complex equivalent of a Kähler structure. Bi-Lagrangian manifolds have interesting features that I will discuss in both the real and complex settings. I will proceed to show that the complexification of a real-analytic Kähler manifold has a natural complex bi-L agrangian structure, and showcase its properties. I will then specialize this discussion to moduli spaces of geometric structures on surfaces, which have a rich symplectic geometry. I will show that some of the recognized geometric features of these moduli spaces are formal consequences of the general theory, while revealing other new geometric features; as well as deriving several well-known results of Teichmüller theory by pure differential geometric machinery. Time permits, I will also mention the construction of an almost hyper-Hermitian structure in the complexification of any real-analytic Kähler manifold, and compare it to the Feix-Kaledin hyper-Kähler structure. This is joint work with Andy Sanders.