# 세미나 및 콜로퀴엄

구글 Calendar나 iPhone 등에서 구독하면 세미나 시작 전에 알림을 받을 수 있습니다.

For more than one hundred years, the Poincare conjecture was a driving force for topologists and its study led to many progresses on topology. It was finally solved by Perelman using differential geometric methods. In this lecture, I will tell what is the Poincare conjecture and a brief history of pursuing it. I will explain geometric ideas involved in solving the conjecture, particularly, geometrization of 3-spaces. I will end up with some speculations on future developments in geometry. This lecture is aimed at general audience.

In these two lectures, I will first introduce the Ricci flow. I will discuss how it can be applied to geometrization of 3-manifolds. Next, I will give a tour of the Analytic Minimal Model Program whose goal is to classify Kaehler manifolds birationally through geometric methods. I will discuss some results and open problems. Finally, I will discuss other and newer curvature flows and their applications.

In these two lectures, I will first introduce the Ricci flow. I will discuss how it can be applied to geometrization of 3-manifolds. Next, I will give a tour of the Analytic Minimal Model Program whose goal is to classify Kaehler manifolds birationally through geometric methods. I will discuss some results and open problems. Finally, I will discuss other and newer curvature flows and their applications.

We prove that for each compact connected one-manifold M and for each real number a >=1, there exists a finitely generated group G inside Diff^a(M) such that G admits no injective homomorphisms into the group \cup_{b>a} Diff^b(M). This is a joint work with Thomas Koberda.

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.

Let A be an Abelian variety over a field K. The group A(K) of K-rational points on A, known as the Mordell-Weil group of A, is known to be finitely generated if K is an algebraic number field of finite degree. It is known to be of infinite rank if K is a certain type of algebraic number field of infinite degree. If K is "too large", then A(K) contains a non-trivial divisible subgroup. I will discuss some reasonable conditions on K which allow A(K) to contain no non-trivial divisible subgroups, and give some examples of such K.

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.

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.

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.

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.

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.

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. $k$.

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.

In this talk, we introduce some recent results on a 1D stochastic particle model, the totally asymmetric simple exclusion process (TASEP). Contrary to the usual TASEP, in the TASEP with second class particles, first class particles have priority over second class particles when they move. In this talk, we introduce some techniques to find the exact formulas of the transition probabilities and the block probabilities.

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E6-1, ROOM 3433
Discrete Math
Otfried Cheong (School of Computing, KAIST)
The reverse Kakeya problem

We prove a generalization of Pal’s 1921 conjecture that if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360 degrees inside Q. We also prove a lower bound of Ω(m n^{2}) on the number of combinatorially distinct maximal placements of a convex m-gon P in a convex n-gon Q. This matches the upper bound proven by Agarwal et al.

Michael Artin and Barry Mazur's classical comparison theorem tells us that for a pointed connected finite type $C$-scheme $X$, there is a map from the singular complex associated to the underlying topological spaces of the analytification of $X$ to the 'etale homotopy type of $X$, and it induces an isomorphism on profinite completions. I'll begin with a brief review on Artin-Mazur's 'etale homotopy theory of schemes, and explain how I extended it to algebraic stacks under model category theory. Finally, I'll provide a formal proof of the comparison theorem for algebraic stacks using a new characterization of profinite completions.

The present talk introduces a localized version of la méthode des multiplicateurs (known as method of Lagrange multipliers) and its recent applications in computational engineering. We will, first, offer a brief review of a variational formulation for the partitioned equations of motion for multi-physics and/or multi-domains utilizing the method of localized Lagrange multipliers, with some of its earlier applications: pore fluid-soil, structure-control, acoustic-structure, structural-thermal and structure-electromagnetic problems. We then focus on recent advances: regularization for stiff coupled systems, reduced-order modeling, nonmatching interfaces, a direct generation of inverse mass matrices for explicit transient analysis, and uncertainty quantification analysis. The presentation concludes with potential areas of further developments in partitioned analysis employing the method of localized Lagrange multipliers.

We start by introducing general determinantal point processes in one dimension and their relation to random matrices following Borodin. Several examples with increasing level of complexity will be discussed as the classical Gaussian Unitary Ensemble, products of several independent and uncorrelated Gaussian random matrices and the effect of introducing correlations. We will then display the corresponding double contour integral representation of the respective kernels and discuss the issue of universality in the limit of large matrix size. This is based on several joint works with Eugene Strahov as well as a work including also Tomasz Checinski and Dang-Zheng Liu.

For a Hecke character of a totally real field, we consider its twist by a line of characters of p-power order. Following the method of Rohrlich approximate functional equation for family of the twisted L-values. We develop a method to count units whose residue have bounded norms in the p-adic expansion w.r.t. a nonsingular cone, namely (C,p)-adic expansion, we obtain the nonvanishing of the L-values when the conductor goes to the infinity. Finally, we discuss how to apply the result to generation of the coefficient field. This is a joint work with Jungyun Lee and Hae-Sang Sun.

Let $F(t,X)$ be an irreducible polynomial in two variables over a number field $k$. Famously, Hilbert's irreducibility theorem asserts that there exist infinitely many $t_0\in k$ such that $f(t_0,X)$ remains irreducible.

In fact, stronger versions of the theorem assert that the ``exceptional" Hilbert set $\mathcal{R}_f:=

\{t_0\in k\mid f(t_0,X) \text{ is reducible}\}$ is small in several well-defined ways.

We will focus on polynomials of the form $F(t,X)=F_1(X)-tF_2(X)$, i.e. $t=f(x):=F_1(x)/F_2(x)$ for a root $x$ of $F$. Using the classification of monodromy groups, we show the following:\\

If $f=f_1\circ ... \circ f_r$ is a decomposition of $f$ into indecomposable rational functions, and all $f_i$ are ``sufficiently generic" and of sufficiently large degree, then up to finitely many values, the set $\mathcal{R}_f$ consists only of the $k$-rational values of $f_1$.\\

This generalizes in several ways previous finiteness results, such as M\"uller's results on reducible {\it integral} specializations.

This talk is based on joint work in progress with Danny Neftin.

Reorganization of neuronal circuits through experience-dependent modification of synaptic connections has been thought to be one of the basic mechanisms for learning and memory. This idea is supported by in-vitro experimental works that show long-term changes of synaptic strengths in different slice preparations. However, a single neuron receives inputs from many neurons in cortical circuits, and it is difficult to identify the rule governing synaptic plasticity of an individual synapse from in vivo studies.

In this talk, I would discuss a novel method to infer synaptic plasticity rules and principles of neural dynamics from neural activities obtained in vivo. The method was applied to the data obtained in monkeys performing visual learning tasks. This study can connect several experimental works of learning and long-term memory at cellular and system level, and could be applicable to other cortical circuits to further our understanding the interactions between circuit dynamics and synaptic plasticity rules.

We consider the 3D axisymmetric Euler equations on exterior domains $ { (x,y,z) : (1 + epsilon |z|)^2 le x^2 + y^2 } $ for any $epsilon > 0$ so that we can get arbitrarily close to the exterior of a cylinder. We construct a strong local well-posedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local well-posedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. This is joint work with Tarek Elgindi.

Memory refers to the ability to hold information in time long after the stimulus is off, and is essential for a variety of adaptive behaviors including integration, learning and generalization. Persistent changes in the activity or connectivity of the systems that lasts longer than the triggering events have been suggested as a substrate for memory. In this talk, I would discuss requirements of the memory system and theoretical principles that can allow the brain to construct persistent states for memory. I would review dominant theories based on attractor dynamics suggested for various types of memory as well as reviewing alternative theories. Also, I would discuss open problems and experimental evidence or tests that can distinguish different mechanisms.

Networked systems, including social, biological, and computer networks, are subjects of study in many disciplines. One of the key properties of the network is the community structure, which refers to the occurrence of natural division of a network into groups of nodes that are more densely connected internally. In this talk, I will briefly introduce methods for finding communities with emphasis on the spectral method using adjacency matrices. In addition, I will also describe how the sparsity of the network affect the community detection problem.

Coordinated and/or cooperative control of multiple unmanned vehicles has been spotlighted as a means to accomplish complex mission objectives in a cost- and resource-effective way, and a rich set of theories and algorithms have been proposed. This talk briefly introduces recent advances in the coordinated decision making for networked autonomous vehicles, in the context of mission & task allocation and informative planning of mobile sensor networks. The particular emphasis is on how advanced mathematical frameworks such as game theory have been adopted in analyzing/synthesizing such coordinated systems. In addition, a potential link of network analysis methodologies with secure and resilient coordination of networked vehicles will be discussed.

We study the asymptotic translation length on curve complexes of the pseudo-Anosov surface homeomorphisms. We first show that the minimal asymptotic translation length of Torelli groups and pure braid groups are asymptotically 1/chi(S) where chi(S) is the Euler characteristic of the surface. If the time permits, we also discuss the asymptotic translation length of pseudo-Anosov monodromies of primitive elements in Thurston’s fibered cone. This talk represents joint work with Hyunshik Shin and Chenxi Wu.

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E6-1, ROOM 3433
Discrete Math
김린기 (KAIST)
Characterization of forbidden subgraphs for bounded star-chromatic number

The chromatic number of a graph is the minimum k such that the graph has a proper k-coloring. It is known that if T is a tree, then every graph with large chromatic number contains T as a subgraph. In this talk, we discuss this phenomena for star-coloring (a proper coloring forbidding a bicolored path on four vertices) and acyclic-coloring (a proper coloring forbidding bicolored cycles). Specifically, we will characterize all graphs T such that every graph with sufficiently large star-chromatic number (acyclic-chromatic number) contains T as a subgraph.