# 세미나 및 콜로퀴엄

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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.

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.

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.

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.

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.

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.

Given a planar subdivision with n vertices, each face having a cone of possible directions of travel, our goal is to decide which vertices of the subdivision can be reached from a given starting point s. We give an O(n log n)-time algorithm for this problem, as well as an Ω(n log n) lower bound in the algebraic computation tree model. We prove that the generalization where two cones of directions per face are allowed is NP-hard.

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.

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 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.