# 세미나 및 콜로퀴엄

2019-04
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2019-05
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구글 Calendar나 iPhone 등에서 구독하면 세미나 시작 전에 알림을 받을 수 있습니다.

We discuss several problems regarding non-compact complete singularities of geometric PDEs including the mean curvature flow. Depending on the equations, the linearized operator would be a self-adjoint operator. Then, the eigen-decomposition allows us to reduce the equation into a system of ODEs. By solving the ODEs under prescribed asymptotic behavior, we improve their asymptotic behavior so that we classify the singularities.

Host: 이용남     미정     2019-05-27 09:15:50

We will discuss how to use PDEs in Geometry, especially for students working in Analysis. We will begin by considering how the geometric PDEs differ from the standard PDEs. Then, we will study some particular examples where the maximum principle, the spectral analysis, and the contraction mapping have been applied in geometric analysis. Also, we would talk about the recent trend in the collaborations between Geometry and Analysis.

Host: 이용남     미정     2019-05-27 09:18:42

(This is a reading seminar for graduate students.) In the setting of Quillen's $K$-theory, the $K$-groups are defined for the category of (algebraic) vector bundles. However, even in SGA 6, the need of replacing vector bundles by complexes quasi-isomorphic to bounded complexes of vector bundles were noticed. Such complexes are called perfect complexes and their K-theory provide better local-to-global properties, for example, Nisnevich descent or the localization sequence. In this talk, the basis notions of perfect complexes and pseudo-coherent complexes will be investigated and characterized categorically. We embrace as many technical details as possible.

Host: 박진현     Contact: 박진현 (2734)     한국어     2019-05-14 16:42:13

Many structures in mathematics have both covariant and contravariant operations, which can often be described using categories of spans or correspondences. In some cases there are two compatible classes of covariant operations, such as the additive and multiplicative transfers (or norms) that occur in representation theory and motivic homotopy theory. I will explain how these can be encoded using higher categories of "bispans", and discuss how this can be used to describe operations on motivic spectra. This is joint work with Elden Elmanto.

Host: 박진현     Contact: 박진현 (2734)     영어     2019-03-12 13:24:30

In numerical computations for wave propagation problems in unbounded domains, it is essential to use an accurate domain truncation technique, which allows wave fields to propagate through artificial boundaries resulting from domain truncation with reflections as small as we wish. In this talk, we will discuss two high-order  absorbing boundary conditions as domain truncation techniques, so-called, perfectly matched layers and complete radiation boundary conditions, for time-harmonic problems in waveguide geometry. Unlike exterior problems, waveguides involve three different types of modes including propagating modes, evanescent modes and cutoff modes. We will present a general idea on parameter selection of the absorbing boundary conditions to minimize reflection errors of three different types of modes.

Host: 이창옥     미정     2019-05-21 13:16:43

b-coloring of a graph G is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The b-Coloring problem asks whether a graph G has a b-coloring with k colors.
The b-chromatic number of a graph G, denoted by χb(G), is the maximum number k such that G admits a b-coloring with k colors. We consider the complexity of the b-Coloring problem, whenever the value of k is close to one of two upper bounds on χb(G): The maximum degree Δ(G) plus one, and the m-degree, denoted by m(G), which is defined as the maximum number i such that G has ivertices of degree at least i1. We obtain a dichotomy result stating that for fixed k{Δ(G)+1p,m(G)p}, the problem is polynomial-time solvable whenever p{0,1} and, even when k=3, it is NP-complete whenever p2.
We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree Δ(G) of the input graph G and give two FPT-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FPT parameterized by Δ(G). Second, we show that b-Coloring is FPT parameterized by Δ(G)+k(G), where k(G) denotes the number of vertices of degree at least k.
This is joint work with Paloma T. Lima.

Host: 엄상일     미정     2019-05-13 09:30:25

An equitable tree-k-coloring of a graph is a vertex coloring using kdistinct colors such that every color class (i.e, the set of vertices in a common color) induces a forest and the sizes of any two color classes differ by at most one. The minimum integer k such that a graph G is equitably tree-k-colorable is the equitable vertex arboricity of G, denoted by vaeq(G). A graph that is equitably tree-k-colorable may admits no equitable tree-k-coloring for some k>k. For example, the complete bipartite graph K9,9 has an equitable tree-2-coloring but is not equitably tree-3-colorable. In view of this a new chromatic parameter so-called the equitable vertex arborable threshold is introduced. Precisely, it is the minimum integer k such that G has an equitable tree-k-coloring for any integer kk, and is denoted by vaeq(G). The concepts of the equitable vertex arboricity and the equitable vertex arborable threshold were introduced by J.-L. Wu, X. Zhang and H. Li in 2013. In 2016, X. Zhang also introduced the list analogue of the equitable tree-k-coloring. There are many interesting conjectures on the equitable (list) tree-colorings, one of which, for example, conjectures that every graph with maximum degree at most Δ is equitably tree-k-colorable for any integer k(Δ+1)/2, i.e, vaeq(G)(Δ+1)/2. In this talk, I review the recent progresses on the studies of the equitable tree-colorings from theoretical results to practical algorithms, and also share some interesting problems for further research.

Host: 엄상일     미정     2019-05-09 09:18:00

(This is a reading seminar for graduate students.) We defined Quillen's higher algebraic $K$-theory and examined its basic properties in previous talks. By the localization theorem and the dévissage theorem, the codimension filtration on $operatorname{Coh}(X)$ for a finite dimensional noetherian scheme $X$ gives the Brown-Gersten-Quillen spectral sequence from page 1. If $X$ is a regular algebraic scheme, then the second page of this spectral sequence is given by $E_2^{p,-q}=H^p_{Zar}(X, G_{q})$ and $E_2^{p,-p}=CH^p(X)$, where $G_{q}$ denotes the (Zariski) sheafification of $Umapsto G_p(U)$. To prove this, we employ Quillen's geometric presentation lemma. This is the third and last part of Quillen's algebraic $K$-theory.

Host: 박진현     Contact: 박진현 (2734)     한국어     2019-04-23 20:27:06

Let be a field. The monodromy group of a rational function $r(X) = f(X)/g(X) in K(X)$, i.e., the Galois group of $f(X− tg(X)$ over $K(t)$, is an important object of study in problems from number theory, geometry, arithmetic dynamics, etc.

Classifying which finite groups occur as monodromy groups has been of great interest, since this knowledge helps reducing many arithmetic problems to pure group theory. The celebrated Guralnick-Thompson conjecture (1990; eventually proved by Frohardt and Magaard) asserts that apart from alternating and cyclic groups, only finitely many simple groups occur as composition factors of monodromy groups of rational functions over (so-called "geometric" monodromy groups). In the case of functionally indecomposable $r(X)$, later work by Neftin, Zieve and others classified not only the "exceptional" groups, but actually the rational functions with exceptional monodromy group, assuming sufficiently large degree. In joint work in progress with Mueller, Neftin and Zieve, we reach a similar result for "arithmetic" monodromy groups. That is, we extend the above classification to arbitrary fields of characteristic zero. As a consequence, we also prove a generalization of the Guralnick-Thompson conjecture for arbitrary fields.

Host: Bo-Hae Im     영어     2019-03-11 19:14:36

In this talk, we study the Schrödinger-Lohe (S-L) model as a phenomenological model for describing possible quantum synchronization phenomenon. In order to reflect real-world phenomenon, we employ several dynamical properties such as general network structure and interaction frustration, and present sufficient frameworks leading to complete and practical synchronizations. First, we begin with an all-to-all network topology to show that the complete synchronization occurs for generic initial data. Moreover, uniform stability of standing wave solutions is established using the complete synchronization results. Next, we consider the general network structure to observe that several emergent behaviors arise, for instance, repulsive behavior, bi-polar synchronized state and periodic orbits. On the other hand, we employ frustration functions into our model and provide sufficient conditions leading to complete and practical synchronizations and periodic trajectories.

Host: 변재형     미정     2019-05-07 09:40:36
In this talk, I will review some known results on multiplicity of autormorphic representations of GL_N, and I will announce my result on it.

Host: 백상훈     한국어     2019-02-25 15:45:28

Let NN be the set of natural numbers. A set ANA⊂N is called a Sidon set if the sums a1+a2a1+a2 , with a1,a2Sa1,a2∈S and a1a2a1≤a2 , are distinct, or equivalently, if |(x+w)(y+z)|1|(x+w)−(y+z)|≥1 for every x,y,z,wSx,y,z,w∈S with x<yz<wx<y≤z<w . We define strong Sidon sets as follows:

For a constant αα with 0α<10≤α<1 , a set SNS⊂N is called an αα -strong Sidon set if |(x+w)(y+z)|wα|(x+w)−(y+z)|≥wα for every x,y,z,wSx,y,z,w∈S with x<yz<wx<y≤z<w .

The motivation of strong Sidon sets is that a strong Sidon set generates many Sidon sets by altering each element a bit. This infers that a dense strong Sidon set will guarantee a dense Sidon set contained in a sparse random subset of NN .

In this talk, we are interested in how dense a strong Sidon set can be. This is joint work with Yoshiharu Kohayakawa, Carlos Gustavo Moreira and Vojtěch Rödl.

Host: 엄상일     미정     2019-04-30 09:22:38

S. Kamada introduced chart diagrams to describe two-dimensional braids in four-dimensional space, which (roughly speaking) are to classical braids what Cerf diagrams are to Morse functions. In this talk we recall chart diagrams, and discuss their application in defining Vassiliev invariants and approaching problems in linkhomotopy of 2-spheres in the 4-sphere.

Host: 백형렬     미정     2019-05-07 09:45:27

Magombedze and Mulder (2013) studied the gene regulatory system of Mycobacterium Tuberculosis (Mtb) by partitioning this into three subsystems based on putative gene function and role in dormancy/latency development. Each subsystem, in the form of S-system, is represented by an embedded chemical reaction network (CRN), defined by a species subset and a reaction subset induced by the set of digraph vertices of the subsystem. Based on the network decomposition theory initiated by Feinberg in 1987, we have introduced the concept of incidence-independent and developed the theory of C- and C*-decompositions including their structure theorems in terms of linkage classes. With the S-system CRN N of Magombedze and Mulder's Mtb model, its reaction set partition induced decomposition of subnetworks that are not CRNs of S-system but constitute independent decomposition of N. We have also constructed a new S-system CRN for which the embedded networks are C*-decomposition. We have shown that subnetworks of N and the embedded networks (subnetworks of N*) are digraph homomorphisms. Lastly, we attempted to explore modularity in the context of CRN.

Host: 김재경 교수     영어     2019-03-25 10:57:21

Chemical reaction network theory (CRNT) is an area of applied mathematics that attempts to model the behavior of real world chemical systems. CRNT has become a tool to study complex biology independent of rate parameters, that is, certain behaviors of networks are examined by analyzing their structures only. In this talk, preliminary CRNT concepts will be presented. We focus on the existence of complex balanced equilibria for weakly reversible reaction networks with power law kinetics elaborating on the so called “Weak Reversibility Theorems”. We also discuss some particular applications of our theoretical results.

Host: 김재경 교수     영어     2019-03-25 11:00:37

A reaction network is a graphical configuration that describes an interaction between species (molecules). If the abundances of the network system is small, then the randomness inherent in the molecular interactions is important to the system dynamics, and the abundances are modeled stochastically as a jump by jump fashion continuous-time Markov chain. One of challenging issues facing researchers who study biological systems is the often extraordinarily complicated structure of their interaction networks. Thus, how to characterize network structures that induce characteristic behaviors of the system dynamics is one of the major open questions in this literature. In this talk, I will provide an analytic approach to find a class of reaction networks whose associated Markov process has a stationary distribution. Moreover I will also talk about the convergence rate for the process to its stationary distribution with the mixing time.

Host: 김재경 교수     영어     2019-03-25 11:03:23

We survey some results in random matrix theory and their universal nature. For instance, consider the largest eigenvalue of a randomly chosen Hermitian matrix. This random variable converges to a certain distribution as the dimension becomes large. It was proved by many different researchers over the last twenty years that this distribution also describes many different models in probability which do not have an apparent connection to matrices. The examples include Coulomb gas, random tilings of a hexagon, random growth models, and directed polymers among others. We will discuss this fascinating university aspect of random matrix theory through several examples.

Host: 이지운     미정     2019-04-24 10:46:29

This talk is about the spectra of non-Hermitian random matrix models. Their asymptotic analysis reveals remarkable high dimensional phenomena, which are, among other aspects, related to entropy maximization and free probability theory. We  will present some of the general phenomena and methodologies, and we will focus mostly on the circular law phenomenon. We will present several works, notably in collaboration with Charles Bordenave and Pietro Caputo. We will also present briefly some few open problems.

Host: 폴정     영어     2019-04-09 08:54:19

The real Ginibre ensemble consists of square real matrices whose entries are i.i.d. standard normal random variables. In the large dimension limit, the empirical distribution converges to the circular law in a disk. This is same as the complex Ginibre ensemble. However, unlike the complex version, there is a positive probability that there are real eigenvalues. The law of the largest real eigenvalue was studied by Rider, Sinclair and Poplavskyi, Tribe, Zaboronski. Building on their work, we will show that the limiting distribution of the largest real eigenvalue admits a closed form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov-Shabat system which is related to the nonlinear Schr¨odinger equation. The results of this talk are based on a recent joint with Thomas Bothner (King's College London).

Host: 폴정     영어     2019-04-26 10:42:34

Describing the eigenvalue distribution of the sum of two general Hermitian matrices is basic question going back to Weyl. If the matrices have high dimensionality and are in general position in the sense that one of them is conjugated by a random Haar unitary matrix, the eigenvalue distribution of the sum is given by the free additive convolution of the respective spectral distributions. This result was obtained by Voiculescu on the macroscopic scale. In this talk, I show that it holds on the microscopic scale all the way down to the eigenvalue spacing in the bulk and at the regular edges. This shows a remarkable rigidity phenomenon for the eigenvalues. Joint work with Z.G. Bao and L. Erdos.

Host: 폴정     영어     2019-04-24 10:50:49

S. Kamada introduced chart diagrams to describe two-dimensional braids in four-dimensional space, which (roughly speaking) are to classical braids what Cerf diagrams are to Morse functions. In this talk we recall chart diagrams, and discuss their application in defining Vassiliev invariants and approaching problems in linkhomotopy of 2-spheres in the 4-sphere.

Host: 백형렬     영어     2019-04-23 13:19:09

The geometry and cohomology of modular curves play an important role in constructing the Langlands correspondence for GL(2,mathbb{Q}). Similarly, the tower of algebraic varieties called Shimura varieties plays an important role in the Langlands programme for reductive groups over number fields. It is then a natural question to ask if there is a ‘p-adic local analogue’ of modular curves and Shimura varieties, which is now commonly referred to as 'local Shimura varieties' or ‘Rapoport—Zink spaces'.

In this survey talk, I will try to motivate the concept of local Shimura varieties starting from the review of modular curves and introducing classical examples of local Shimura varieties (that have been known even before the general concept of local Shimura varieties emerged). If time permits, I will introduce my contribution and work in progress on constructing more general classes of local Shimura varieties.

Host: Bo-Hae Im     미정     2019-04-22 14:39:46

Circle graphs are the intersection graphs of chords on a circle; vertices correspond to chords, and two vertices are adjacent if their chords intersect. We prove that every circle graph with clique number k has chromatic number at most 4k2. Joint with James Davies.

Host: 엄상일     미정     2019-04-19 09:33:00

Recently, the classification of isoparametric hypersurfaces in spheres has been completed. Therefrom, various new research projects in geometry have been initiated. The study of minimal lagrangian submanifolds via isoparametric hypersurfaces is one of the most active projects a la mode. In this talk, we have an introduction to the study of Isoparametric hypersurfaces and minimal Lagrangian submanfolds, and discuss the relationship between them.

Host: 백상훈     미정     2019-02-25 15:44:17

The current focus of our research is to reveal fundamental design principles of the biological clock mechanism and pathogenesis of circadian disorders. We develop unique mouse models to simulate human diseases that can cause circadian disorders directly and indirectly, and unravel how the diseases compromise circadian rhythms including wake-sleep cycles, the most salient feature of circadian rhythms in animals, at the molecular level. In the first half of my talk, I will discuss how cytoplasmic congestion, normally associated with metabolic diseases and aging, can disrupt the clock mechanism and how sleep disorders in patients with these diseases can be treated at the core clock level. In the second half of my talk, I will discuss arguably the most upstream event in rhythm generation. We recently found that phosphorylation of the circadian pacemaker protein, PER is regulated by a strong Hill-type reaction, the basis of robustness in many signaling pathways. Phosphorylation kinetics of PER increases proportionally to concentration of PER: the more PER is generated, the faster PER phosphorylates, similar to O2 binding to hemoglobin. Since PER phosphorylation is considered the time-generating step, we believe this is the most upstream event and simple mutations in Per genes (SNPs) may cause sleep disorders in many humans.

Host: 김재경 교수     영어     2019-03-25 10:24:10

The circadian clock is an autonomous molecular mechanism that controls biochemical, physiological, and behavioral processes with a periodicity of 24 h in living organisms and can be entrained by environmental cues. The clock is sustained by a coordinated interplay of positive and negative transcriptional-translational feedback loops driven by circadian factors, a core group of proteins that either possess intrinsic transcriptional activity or modulate gene expression. We previously reported that the circadian factor PERIOD 2 (PER2) forms a stable complex with the tumor suppressor and checkpoint protein p53. The PER2:p53 complex undergoes time-of-day–dependent nuclear-cytoplasmic shuttling, thus generating an asymmetric distribution of each protein in different cellular compartments. In unstressed cells, PER2 mediates p53’s stability by binding to its C-terminal domain and preventing p53 from being ubiquitylated at sites targeted by the RING finger–containing E3 ligase and oncoprotein mouse double minute 2 homolog (MDM2). We found that PER2, p53, and MDM2 co-exist as a trimeric and stable complex in the nuclear compartment, although p53 is released from the complex to become transcriptionally active after cells experience genotoxic stimuli. More recently, we found that PER2 could also act as a bona fide substrate for MDM2 in the absence of p53. Indeed, PER2 was efficiently ubiquitylated in vitro and in cells at numerous sites by MDM2 in a process that was independent of PER2 phosphorylation. Accordingly, PER2’s half-life is critically influenced by the abundance and enzymatic activity of MDM2, as shown in cells in which MDM2 expression was either enhanced or silenced and its catalytic activity was pharmacologically inhibited. As a consequence, direct manipulation of MDM2 expression influenced period length by reducing PER2 stability. Our results uncover previously unknown regulatory players that likely impact our view of how other mechanisms crosstalk and modulate the clock itself. Furthermore, it exposes an uncharacterized druggable node that is often found to be deregulated during tumorigenesis.

Host: 김재경 교수     영어     2019-03-25 10:53:23

Grothendieck's $K_0$-group has an exact sequence $K_0(Z)to K_0(X)to K_0(X-Z)to 0$ for a closed immersion $Zto X$ of regular noetherian schemes, whose kernel of the leftmost map is usually nontrivial. To prolong this sequence functorially, Quillen defined higher algebraic $K$-groups as the higher homotopy groups of some mysterious category associated to an exact category. For this, we introduce simplicial sets, $Q$-construction of an exact category, higher $K$-groups of schemes, and their remarkable theorems, part of which extends the previously mentioned exact sequence for $K_0$ and relates higher K-theory with Chow groups. This is the second half of Quillen's algebraic $K$-theory.

Host: 박진현     Contact: 박진현 (2734)     한국어     2019-03-20 13:42:08

In X-ray computed tomography (CT), deep learning techniques have shown great potential for reduction of various artifacts (e.g., noise arising from low-dose, streaking artifacts due to sparse view). Most of these approaches learn the relationship between artifact images and artifact-free (ground-truth) images. However, paired training data are not generally available in clinical practice. In this talk, I will introduce simulation-based and unpaired learning methods, which can be used to circumvent such issue.

Host: 이창옥     미정     2019-04-09 09:58:14

A Boolean function is a function from the set Q of binary vectors of length n (i.e., the binary n-dimensional hypercube) to F2={0,1}. It has several applications to complexity theory, digital circuits, coding theory, and cryptography.
In this talk we give a connection between Boolean functions and Artificial Neural Network. We describe how to represent Boolean functions by Artificial Neural Network including linear and polynomial threshold units and sigmoid units. For example, even though a linear threshold function cannot realize XOR, a polynomial threshold function can do it. We also give currently open problems related to the number of (Boolean) linear threshold functions and polynomial threshold functions.

Host: 엄상일     미정     2019-04-11 09:24:29

Many modern applications such as machine learning require solving large-dimensional optimization problems. First-order methods are widely used to solve such problems, since their computational cost per iteration mildly depends on the problem dimension. However, they suffer from relatively slow convergence rates, and this talk will discuss recent progress on the acceleration of first-order methods, particularly using the close relationship between convex optimization methods and maximally monotone operators.

Host: 백상훈     영어     2019-02-25 15:43:09

Let $X$ be an abelian variety of dimension $g$ over a field $k$. In general, the group $textrm{Aut}_k(X)$ of automorphisms of $X$ over $k$ is not finite. But if we fix a polarization $mathcal{L}$ on $X$, then the group $textrm{Aut}_k(X,mathcal{L})$ of automorphisms of the polarized abelian variety $(X,mathcal{L})$ over $k$ is known to be finite. Then it is natural to ask which finite groups can be realized as the full automorphism group of a polarized abelian variety over $k.$

In this talk, we give a classification of such finite groups for the case when $k$ is a finite field and $g$ is a prime number. If $g=2,$ then we need a notion of maximality in a certain sense, and for $g geq 3,$ we achieve a rather complete list without conveying maximality.

Host: Bo-Hae Im     한국어 (필요한 경우 영어 가능) ( )     2019-02-18 12:43:14

Finite element discretization of solutions with respect to simplicial/cubical meshes has been studied for decades, resulting in a clear understanding of both  the relevant mathematics and computational engineering challenges. Recently, there has been both a desire and need for an equivalent body of research regarding discretization with respect to generic polygonal/polytopal meshes. General meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and  non-matching interfaces. Such a flexibility represents a powerful tool towards the efficient solution of problems with complex inclusions as in geophysical applications or posed on very complicated or possibly deformable geometries as encountered in basin and reservoir simulations, in fluid-structure interaction, crack propagation or contact problems.

In this talk,  a new computational paradigm for discretizing PDEs is presented via staggered Galerkin approach on general meshes. First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal meshes for elliptic problems are proposed.  The method can be flexibly applied to rough grids such as highly distorted  meshes. Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes.

We derive a simple residual-type error estimator.  Numerical results indicate that optimal convergence can be achieved for both the potential  and vector variables, and the singularity
can be well-captured by the proposed error estimator.  Then, some applications to diffusion equations, Stokes equations, and linear elasticity equations are considered.
Finally, we comment on the possibility of extending this approach to high-order polynomial approximations on general meshes.
Host: 김재경     영어     2019-02-25 15:42:05

Goldman parametrizes the $mathrm{PSL}_3(mathbb{R})$-Hitchin component of a closed oriented hyperbolic surface of genus $g$ by $16g-16$ parameters. Among them, $10g- 10$ coordinates are canonical. We prove that the $mathrm{PSL}_3(mathbb{R})$-Hitchin component equipped with the Atiyah-Bott-Goldman symplectic form admits a global Darboux coordinate system such that the half of its coordinates are canonical Goldman coordinates. To this end, we establish a version of the action-angle principle and show that the Hitchin component can be decomposed into a product of smaller Hitchin components.

Host: 백형렬     영어     2019-03-29 09:06:35

Grothendieck's $K_0$-group has an exact sequence $K_0(Z)to K_0(X)to K_0(X-Z)to 0$ for a closed immersion $Zto X$ of regular noetherian schemes, whose kernel of the leftmost map is usually nontrivial. To prolong this sequence functorially, Quillen defined higher algebraic $K$-groups as the higher homotopy groups of some mysterious category associated to an exact category. For this, we introduce simplicial sets, $Q$-construction of an exact category, higher $K$-groups of schemes, and their remarkable theorems, part of which extends the previously mentioned exact sequence for $K_0$ and relates higher K-theory with Chow groups. This is the first half of Quillen's algebraic $K$-theory.