Microscopic properties of eigenvalues of normal random matrices change drastically in a narrow belt around the edge of the spectrum. I present an elementary method to prove Borodin and Sinclair's theorem on the scaling limit of correlation kernels for the soft-edge Ginibre ensemble. This method gives new result for the hard-edge Ginibre ensemble. After a discussion of the general properties of this scaling limit, I state a universality conjecture and provide arguments to support it. This is a joint work with Y. Ameur and N. Makarov.

임의의 수학적인 구조가 있을때, 그 대칭성들의 집합은 군을 이룹니다.

역으로, 임의로 주어진 군을 연구하는 데에는 그에 대응하는 특별한 그래프가 효과적으로 쓰일 수 있습니다.

본 강연에서는 그래프처럼 미적분이 정의되지 않는 공간에서 기하학적인 성질들을 찾아내는 방법을 소개하고, 이 접근이 어떻게 무한군의 성질들을 밝히는 데에 쓰일 수 있는지 소개합니다.

A nonnegative harmonic function defined on an open ball in
\R^d(d\ge 2)can be represented as an integral over the Euclidean
boundary of the ball. For the general domain, it is represented as an
integral over the Martin boundary. We consider the problem of
identification of Martin boundary and Euclidean boundary.

We will construct the primitive generators of the ray class
fields over imaginary quadratic fields by using the singular values of
suitable powers of Siegel functions. We investigate the algorithm for
finding all Galois conjugates of singular values obtained by Gee and
Stevenhagen and Shimura's reciprocity law. And then, by comparing the
absolute values of all Galois conjugates of given singular value, we
construct the ray class invariants of a given imaginary quadratic
field K.

Furthermore, we obtain the normal bases for class fields by using the
singular values of suitable powers of Siegel functions.

We introduce a new class of rate one half codes, called complementary information set codes. A binary linear code of length 2n and dimension n is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune functions of use in the security of hardware implementations of  cryptographic primitives. In this talk, we give optimal or best known CIS codes of length <132. We  derive general constructions based on cyclic codes, double circulant codes, strongly regular graphs, and doubly regular tournaments. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths up to 12 by the building up construction. This is a joint work with Claude Carlet, Philippe Gaborit, and Patrick Sole.

http://mathsci.kaist.ac.kr/~manifold/Arithmetics.html

We survey the theory of non-abelian Galois representations and its applications, as developed

out of the anabelian programme of Grothendieck from the 1980's.

The Internet has evolved into a vast heterogeneous system, comprised of independent selfish users who value the benefits they derive from the network much more than the efficiency of the network as a whole. Independent selfish user behavior results in a non-cooperative distributed network environment that increasingly undermines many important congestion control schemes (e.g., TCP and CSMA/CA) relying on voluntary participation. Considering that selfish users may modify the current standard congestion protocol and exploit knowledge of the cooperative behavior of others, we have abandoned the paradigm of assumed-cooperative users in the networks and considered networks with only selfish users who pursue only their own benefit.

In this seminar, we will focus on transmission rate control algorithms with selfish users in wireless networks. This seminar covers non-cooperative games among selfish users in random access networks, specifically networks using simple slotted ALOHA or IEEE 802.11 DCF. A transmission rate control algorithm in ALOHA networks and a generalization of the rate control algorithm will be discussed. A transmission rate control algorithm for a variant of IEEE 802.11 DCF will be proposed to consider selfish user behaviors. Existence and uniqueness of equilibrium points of the proposed algorithm will be explored. Convergence properties of the equilibrium points will be studied via Lyapunov stability theory.

Multiple hovering UAVs equipped with signal capturing sensors are used for target tracking due to economic efficiency. Comparing the received signals reveals the differences in the distances to the target from different UAVs, and the statistical analysis of the data allows for the estimation of the target location. The estimation process can be simplified by employing geometric approximations suitable for practical applications. Minimizing an objective function with an Lagrange addivity constant solves the problem of finding the maximum likelihood estimator for the target location. A sensitivity analysis of this process provides a useful suggestions as to which flight formation is more efficient than others. (Joint work with prof. Sung-Ho Kim, ADD project)

This will be a series of roughly 6-8 lectures. The first half will be on well-known and classical material on equivariant cohomology and Schubert calculus, while I will concentrate on my recent joint work with Tymoczko, Bayegan, and Dewitt in the second half of the lecture series.

We discuss initial boundary value problems in the regime of classical solutions to the Vlasov-Poisson system with large data. We also talk on the exponential time decay rate of smooth solutions of small amplitude to the Vlasov-Poisson-Fokker-Planck equations to Maxwellian.

Hessenberg varieties are a class of subvarieties of the flag variety which appear in many areas, e.g. in geometric representation theory. In order to generalize Schubert calculus to Hessenberg varieties, a first step is to construct computationally convenient module bases for the (equivariant) cohomology rings of Hessenberg varieties analogous to the famous Schubert classes which are a basis for the cohomology of flag varieties. Goresky-Kottwitz-MacPherson ("GKM") theory gives a concrete combinatorial description of the equivariant cohomology of spaces with torus action which satisfy certain conditions (usually called the GKM conditions). We propose a framework for approaching the problem of constructing module bases for Hessenberg varieties which uses GKM theory. The main conceptual challenge in this context is that conventional GKM theory requires a `sufficiently large-dimensional torus' action (to be made precise in the talk), while Hessenberg varieties generally have only a circle action. To resolve this, we define the notion of GKM-compatible subspaces of GKM spaces and give applications in some special cases of Hessenberg varieties. The talk will be intended for a wide audience, and in particular I will begin with a conceptual sketch of the main ideas in Schubert calculus and of classical GKM theory.

This is mainly joint work with Tymoczko; time permitting, I will mention joint work with Bayegan, and also with Dewitt.

This will be a series of roughly 6-8 lectures. The first half will be on well-known and classical material on equivariant cohomology and Schubert calculus, while I will concentrate on my recent joint work with Tymoczko, Bayegan, and Dewitt in the second half of the lecture series.

This will be a series of roughly 6-8 lectures. The first half will be on well-known and classical material on equivariant cohomology and Schubert calculus, while I will concentrate on my recent joint work with Tymoczko, Bayegan, and Dewitt in the second half of the lecture series.

Kashaev volume conjecture suggests that the limit of the colored Jones polynomial of a hyperbolic knot gives the volume and the Chern-Simons invariant of the knot complement. It is one of the most important problems in quantum topology because it shows many non-trivial relations between many areas including knot theory, hyperbolic geometry, quantum group, extended Bloch group, and more.

In this talk, we survey many aspects of the volume conjecture. Especially, we focus on how the extended Bloch group plays a crucial role in proving the optimistic limit version of the volume conjecture.

In signal processing/communications, an analogue (or continuous)
signal is represented by its discrete counterpart which is called
samples of the analog signal. Since it is inevitable in practice that
some of the samples are missing during transfer, not only engineers
but also mathematicians have been trying to circumvent this problem
with various kind of approaches. For band-limited signals, It is well
known that any finitely many missing samples can be recovered from the
remaining known samples when the signal is oversampled at a rate
higher than the minimum Nyquist rate. In this talk, we consider a
similar problem of recovering missing samples for signals in shift
invariant spaces.

We study an optimal portfolio and consumption choice problem of a family that combines life insurance for parents who receive deterministic labor income until the fixed time T. We consider utility functions of parents and children separately and assume that parents have an uncertain lifetime. If parents die before time T, children have no labor income and they choose the optimal consumption and portfolio with remaining wealth and life insurance benefit. The object of the family is to maximize the weighted average of utility of parents and that of children. We obtain analytic solutions for the value function and the optimal policies, and then analyze how the changes of the weight of the parents’ utility function and other factors affect the optimal policies.

Given an even subset T of the vertices of an undirected graph, a T-join is a subgraph in which the subset of vertices with odd degree is exactly T. Given edge weights, the weighted T-join problem is to find a T-join of minimum weight. With nonnegative edge weights, the problem can be reduced to finding a minimum weight perfect matching on the metric completion of the vertices in T.
Given an undirected graph with nonnegative edge weights but no specific T, the Max-Min Weighted T-join problem is to find an even cardinality vertex subset T such that the minimum weight T-join for this set is maximum. The unweighted case of the problem when all weights are either unit or infinity has been well characterized by a decomposition of the underlying graph into factor critical and matching-covered bipartite subgraphs (Frank1993). We consider the weighted version which is NP-hard even on a cycle. After showing a simple exact solution on trees, we present a 2/3-approximation algorithm for the general case. Our algorithm is based on a natural cut packing upper bound obtained using an LP relaxation and uncrossing, and relating it to the T-join problem using duality.
This is a joint work with R. Ravi.

This will be a series of roughly 6-8 lectures. The first half will be on well-known and classical material on equivariant cohomology and Schubert calculus, while I will concentrate on my recent joint work with Tymoczko, Bayegan, and Dewitt in the second half of the lecture series.

Submodular functions are discrete analogues of convex functions.

Examples include cut capacity functions, matroid rank functions,

and entropy functions. Submodular functions can be minimized in

polynomial time, which provides a fairly general framework of

efficiently solvable combinatorial optimization problems.

In contrast, the maximization problems are NP-hard and several

approximation algorithms have been developed so far.

In this talk, I will review the above results in submodular

optimization and present recent approximation algorithms for

combinatorial optimization problems described in terms of

submodular functions.

This will be a series of roughly 6-8 lectures. The first half will be on well-known and classical material on equivariant cohomology and Schubert calculus, while I will concentrate on my recent joint work with Tymoczko, Bayegan, and Dewitt in the second half of the lecture series.

In the very first lecture, I intend to give a very introductory overview lecture on Schubert calculus, concentrating on the most classical case: that of the (cohomology of the) Grassmannian of k-planes in complex n-space. This lecture will be elementary, requiring (mostly) only undergraduate material (mainly linear algebra), but I hope to give the broad overview of a beautiful and elegant subject where algebra, combinatorics, and geometry come together. It will serve as a motivation for the entire lecture series.


In the next lecture(s), I intend to give a broad overview of the theory of equivariant cohomology and, in particular, the so-called `GKM' (Goresky-Kottwitz-MacPherson) theory which gives, under suitable conditions, a combinatorial description of the equivariant cohomology in terms of the data of the equivariant 1-skeleton of a space equipped with a group action. I will also discuss some of the Morse-theoretic interpretations of aspects of GKM theory, and in particular take some time to discuss how GKM theory provides a good technology for constructing computationally convenient module bases for equivariant cohomology.

In a following lecture(s) I will review the basic geometry of flag varieties associated to compact Lie groups (or complex reductive algebraic groups), including the description of the Schubert varieties sitting inside the flag varieties. I will also briefly recall the corresponding objects in equivariant cohomology, namely the Schubert classes. I will briefly describe some of the known formulas for computing products of certain of these classes. Finally, I will explain some Bruhat-order combinatorics and the Sara Billey formula for computing localizations of Schubert classes at T-fixed points, and how this is another method for a computation `in principle' of the structure constants in the equivariant cohomology ring.


In the next lecture(s) I will describe in some detail my recent work with Julianna Tymoczko and Darius Bayegan, in which we generalize classical Schubert calculus to a certain class of subvarieties of flag varieties, namely, the Peterson varieties. We exploit the GKM theory on the ambient flag variety in order to derive a computationally convenient module basis for the S^1-equivariant cohomology ring of Peterson varieties. I will explain in detail how this is done, and also explain some consequences of our approach: namely, Tymoczko and I derive a manifestly-positive and manifestly-integral Monk formula for the structure constants, and Bayegan and I similarly derive a Giambelli formula.

In the following lecture(s) I will explain how to generalize the techniques in my work with Tymoczko-Bayegan on Peterson varieties. In particular I explain the key new concept of `GKM-compatible subspaces' of an ambient GKM space. The crucial idea is that, although a subspace X of a GKM space Y may not be itself GKM, the inclusion map of X into Y (and associated map on equivariant cohomology) may have good enough properties that we can deduce properties about the equivariant geometry of X through the GKM theory on Y. Motivated by the notion of GKM-compatibility, Tymoczko and I also introduce a new combinatorial game called `poset pinball', which in some situations allow us to explicitly build computationally convenient module bases analogous to the Schubert classes in Schubert calculus.

In the last lecture(s) I will explain concrete examples and applications of the general techniques of GKM-compatibility and poset pinball introduced above, which are contained in my joint work with Tymoczko, Bayegan, and Dewitt. For example, we give an explicit construction of an equivariant lift of the classical Springer action on the cohomology of subregular Springer varieties of type A.


For further reading:

Kleiman-Laksov, Schubert Calculus. The American Mathematical Monthly, Vol. 79, No. 10, (Dec., 1972), pp. 1061-1082.

Allen Knutson, The symplectic and algebraic geometry of Horn's problem.  http://arxiv.org/abs/math/9911088

Julianna Tymoczko. An introduction to equivariant cohomology andhomology, following Goresky, Kottwitz, and MacPherson. Snowbird lectures in algebraic geometry. AMS.

Julianna Tymoczko. Permutation actions on equivariant cohomology of flag varieties. Toric Topology conference proceedings. AMS.

Tara Holm. Act globally, compute locally: group actions, fixed points, and localization. Toric Topology conference proceedings. AMS.

Fulton, Young tableaux. Cambridge University Press.

Megumi Harada and Julianna Tymoczko, A positive Monk formula in the S^1-equivariant cohomology of type A Peterson varieties. Proc. London Math. Soc. (2011) doi: 10.1112/plms/pdq038.

Megumi Harada and Julianna Tymoczko, Poset pinball, GKM-compatible subspaces, and Hessenberg varieties. http://arxiv.org/abs/1007.2750

Darius Bayegan and Megumi Harada, A Giambelli formula for the $S^1$-equivariant cohomology of type A Peterson varieties.
http://arxiv.org/abs/1012.4053

Darius Bayegan and Megumi Harada, Poset pinball, the dimension pair algorithm, and type A regular nilpotent Hessenberg varieties.
http://arxiv.org/abs/1012.4054

Barry Dewitt and Megumi Harada, Poset pinball, highest forms, and (n-2,2) Springer varieties. http://arxiv.org/abs/1012.5265

This will be a series of roughly 6-8 lectures. The first half will be on well-known and classical material on equivariant cohomology and Schubert calculus, while I will concentrate on my recent joint work with Tymoczko, Bayegan, and Dewitt in the second half of the lecture series.

 In this seminar, I will talk about the properties and the classification of embeddings of homogeneous spaces, especially the case of affine normal embeddings of reductive groups. We might guess that as in the case of toric varieties, some specific subset of one-parameter subgroups may contribute to the classification of affine embeddings of general reductive group. To check this, we review the theory of affine normal SL(2)-embeddings, and prove that the classification cannot be solved entirely based on one-parameter subgroups. We can also observe that even though this set does not give a complete answer to the classification problem, but still contains useful information about varieties. If time permitted, I will also give examples of GL(2)-embeddings which had not previously been constructed in detail, which might be helpful in understanding the general classification of affine normal G-embeddings.

 In this talk, we establish an oscillation estimate of nonnegative harmonic functions for a large class of integro-differential
operators. Such operators are
the infinitesimal generators of pure-jump subordinate Brownian motion. As an application, we give a probabilistic proof of relative Fatou  theorem for harmonic functions
for the integro-differential operators in bounded $\kappa$-fat open set.   That is, if $u$ is a positive harmonic function in  a bounded $\kappa$-fat open set $D$ and $h$ is a positive  harmonic function in $D$ vanishing on $D^c$, then the non-tangential limit
 of  $u/h$ exists almost everywhere with respect to the Martin-representing measure of $h$.
Under the gaugeability assumption, relative Fatou theorem is true for operators obtained from the generator of pure-jump subordinate Brownian motion in bounded $\kappa$-fat open set $D$ through non-local Feynman-Kac transforms.

Many problems that are intractable on general graphs become linear time solvable on graphs of bounded treewidth. The constant factor however of such algorithms is exponential or worse in the treewidth of the tree decomposition that is used. In this talk,  a number of known and some new results are surveyed. In particular, it is shown how speed improvements can be obtained using convolutions, and how a recent technique called "cut-and-count" can be used to obtain fast probabilistic algorithms for problems like Hamiltonian Circuit.

In this talk, we investigate the structure of highest weight
modules over the quantum queer superalgebra $U_q(q(n))$ and develop
the crystal basis theory for $U_q(q(n))$. We define the notion of
crystal bases and prove the tensor product rule for
$U_q(q(n))$-modules in the category $O_{int}^{\ge 0}$. Moreover, we
give an explicit combinatorial realization of the crystal $B(\lambda)$
for an irreducible highest weight $U_q(q(n))$-module $V(\lambda)$ in
terms of semistandard decomposition tableaux.

The Rogers-Ramanujan identities have many natural and
significant generalizations. The generalization we will present in
this talk was first studied by D. Bressoud, by considering the
partitions he named as ”footed partition”. Bressound then made a
conjecture that two sets of partitions under certain constraints are
equinumerous. The validity of the conjecture in the first two cases implies
exactly the partition-theoretical interpretation for the Rogers-Ramanujan
identities. We give a nearly bijective proof of the conjecture, and we
provide examples to demonstrate the bijection as well.

http://mathsci.kaist.ac.kr/~manifold/Arithmetics.html

In order to understand complex geometry, it can be useful
to construct "pathological" examples. Nonkähler or nonprojective
compact complexe manifolds are such examples. Hopf (in 1948) and
Calabi and Eckmann (1953) constructed complex structures on product of
spheres with odd dimension which are not kähler. In this talk, we will
present a generalization due to Bosio of these manifolds. These
manifolds are called LVMB manifolds are their interest lies in the
very combinatorial nature. In particular, we will show that LVMB
manifolds are homeomorphic to quotients of important objects in toric
topology, namely moment-angle complexes (MAC).We will also show the
relationship between LVMB manifolds and toric (algebraic) varieties.

A posteriori error estimation produces qualitative and/or quantitative information on the numerical errors and now plays a fundamental role in adaptive mesh refinement for efficient implementation of numerical methods. Recently, there has been growing interest in the type of a posteriori error estimation which provides a fully computable upper bound on the acutal error. In this talk we will give an overview of some recent results on this issue for primal and mixed finite element methods.

This is an elementary introduction to Langlands program, focusing
on simple examples of GL(2) and modular curves. We will start with
the definition of modular forms and explain their relation to automorphic
representations of GL(2). After defining L-groups and automorphic
L-functions, we will state the Langlands' principle of functoriality and
explain for the cases of Jacquet-Langlands correspondence and
base change. One of main tools used to prove these cases of functoriality
is the trace formula. We will explain this trace formula (or rather simplified
version of it) and another application of it to show that the Hasse-Weil zeta
function of modular curves can be expressed as a product of automorphic
L-functions.

This is an elementary introduction to Langlands program, focusing
on simple examples of GL(2) and modular curves. We will start with
the definition of modular forms and explain their relation to automorphic
representations of GL(2). After defining L-groups and automorphic
L-functions, we will state the Langlands' principle of functoriality and
explain for the cases of Jacquet-Langlands correspondence and
base change. One of main tools used to prove these cases of functoriality
is the trace formula. We will explain this trace formula (or rather simplified
version of it) and another application of it to show that the Hasse-Weil zeta
function of modular curves can be expressed as a product of automorphic
L-functions.

We generalize two bijections due to Garsia and Gessel to compute the generating functions of the two vector statistics (des_G, maj, ℓ_G, col) and (des_G, ides_G, maj, imaj, col, icol) over the wreath product of a symmetric group by a cyclic group. Here des_G, ℓ_G, maj, col, ides_G, imaj_G, and icol denote the number of descents, length, major index, color weight, inverse descents, inverse major index, and inverse color weight, respectively. Our main formulas generalize and unify several known identities due to Brenti, Carlitz, Chow-Gessel, Garsia-Gessel, and Reiner on various distributions of statistics over Coxeter groups of type A and B.

In order to understand complex geometry, it can be useful
to construct "pathological" examples. Nonkähler or nonprojective
compact complexe manifolds are such examples. Hopf (in 1948) and
Calabi and Eckmann (1953) constructed complex structures on product of
spheres with odd dimension which are not kähler. In this talk, we will
present a generalization due to Bosio of these manifolds. These
manifolds are called LVMB manifolds are their interest lies in the
very combinatorial nature. In particular, we will show that LVMB
manifolds are homeomorphic to quotients of important objects in toric
topology, namely moment-angle complexes (MAC).We will also show the
relationship between LVMB manifolds and toric (algebraic) varieties.

http://mathsci.kaist.ac.kr/~manifold/Arithmetics.html

In 1979, Paul Yang showed that the bidisc does not admit any complete K ̈ahler metric with its bisectional curvature bounded between any two negative constants. His proof works for any dimensional polydiscs as well as the Hermitian symmetric domains with rank not less than 2. This method works also for product of complete K ̈ahler manifols, as shown by H. Seshadri and F.Zheng. Still, whether there are examples falling into this category that are neither homogeneous nor product is poorly understood. In this talk, I will present new examples of inhomogeneous bounded domains that cannot admit complete Kahler metrics with their bisectional curvature bounded between any prescribed negative constants.

Transportation and network polytopes are classical objects in operations research. In this talk, we focus on recent advances on the diameters of several classes of transportation polytopes, motivated by the efficiency of the simplex algorithm. In particular, we discuss results on 2-way transportation polytopes, including a recent result of Stougie and report on joint work with Bruhn-Fujimoto and Pilaud, concerning 2-way transportation polytopes with a certain support structure. We also present a bound on 3-way transportation polytopes in joint work with De Loera, Onn, and Santos. To conclude, we discuss avenues for future work on transportation polytopes and their diameters.

We survey our recent results on rigidity vs flexibility of representations in semisimple Lie groups and group actions on character variety.

인터넷의 가장 중요한 문제중 하나인 혼잡제어 문제를 바라보는 수학적인 시각에 대해서 얘기합니다. 협조(cooperation)와 경쟁(competition)의 논리에서 어떻게 인터넷을 제어할 수 있는가에 따라서, 바라보는 문제의 다름에 대해서 논하고, 최적화 이론과 게임 이론과 같은 수학적 도구가 어떻게 인터넷을 발전시키는데 공헌을 하는지에 대해서 얘기합니다.

※5시부터 시작전 다과회(피자제공)가 있습니다.

In coherent imaging systems, such as synthetic aperture radar
(SAR), the observed images are contaminated by multiplicative noise. Due
to edge preserving feature of the total variation (TV), variational
models with TV regularization have attracted much interest in removing
multiplicative noise. However, the fidelity term of the variational
model, based on maximum a posteriori estimation, is not convex and so it
is usually difficult to find a global solution. In this talk, we
introduce how to relax the nonconvexity of the variational model.
Algorithm based on the augmented Lagrangian function can be applied to
solve our proposed model. But this algorithm requires to solve a
subproblem, which does not have closed form solution, at each iteration.
Hence we adapt the linearized proximal alternating minimization
algorithm (LPAMA) which does not require any inner iterations. Also, the
proposed method is very simple and highly parallelizable and thus it is
efficient to remove multiplicative noise in huge SAR images.
models with TV regularization have attracted much interest in removing
multiplicative noise. However, the fidelity term of the variational
model, based on maximum a posteriori estimation, is not convex and so it
is usually difficult to find a global solution. In this talk, we
introduce how to relax the nonconvexity of the variational model.
Algorithm based on the augmented Lagrangian function can be applied to
solve our proposed model. But this algorithm requires to solve a
subproblem, which does not have closed form solution, at each iteration.
Hence we adapt the linearized proximal alternating minimization
algorithm (LPAMA) which does not require any inner iterations. Also, the
proposed method is very simple and highly parallelizable and thus it is
efficient to remove multiplicative noise in huge SAR images.

Toric varieties are in one-to-one correspondence with rational fans.
This establishes a useful bridge between algebraic geometry and convex
geometry. Algebraic geometers can translate many problems into
statements which are more amenable to computations. Conversely, and
even more remarkably, some combinatorial questions can be successfully
reformulated in algebraic-geometric terms. I will describe a model of
toric varieties associated to nonrational fans, with supporting
evidence that the rich interplay between algebraic and convex
geometries carries over. This is a joint work with F. Battaglia.

Groups, Graphs and Geometry

- a gentle introduction to circle packing

평면이나 구면 상에 겹치지 않는 원판들이 있을 때, 그 접점 관계(tangency relation)를 나타내는 그래프는 “동전 그래프”라 불립니다. 이 강연에서 우리는 Koebe (1936) – Andreev (1970) – Thurston (1978)에 의해 밝혀진 다음 정리의 증명을 살펴 봅니다.
정리. 임의의 평면 그래프는 동전 그래프이다.
학부 수준의 복소 해석학만으로 증명할 수 있는 이 아름다운 정리는 다면체에 맞는 원형 철망을 씌우는 문제, 다각형의 예각 삼각형화를 찾는 문제, 3차원 다양체를 기하화하는 문제 등과 연관됩니다. 이는 기하학적 군론, 그래프 이론, 위상수학을 어우르는 주제입니다.

 Text : Basic Analytic Number Theory 

        by A. Karatsuba

If one draws the Hasse diagram for the face lattice of a polytope, this may appear to be the 1-skeleton of some other polytope. In 1971 Lindström asked whether the intervals of a polytope’s face lattice ordered by containment can be realized as the face lattice of another polytope. If so, this would be the polytope appearing the Hasse diagram. In this talk, I will give necessary and sufficient conditions for the intervals of a polytope’s face lattice to be realizable, and I will provide a counter example showing that not every polytope satisfies these conditions.

 복수측정벡터 문제는 0이 아닌 성분이 공통된 위치에 분포하는 성긴 벡터들을 복원하는 방법을 다루는 문제이며, 도착방향 추정문제, 산란광단층촬영, 역산란 문제 등 여러 실용적인 문제에 응용될 수 있다. 2000년대 중반부터 신호처리와 정보이론 분야 등에서 주목을 받기 시작한 압축감지 기법은, 성긴 속성을 갖는 벡터들을 샤논 샘플링 비율보다 상당히 낮은 샘플링 비율로도 높은 확률로 정확히 복원할 수 있다는 점을 보여주고 있으며, 이로 인해 복수측정벡터 문제에 압축감지 기법을 응용하는 일련의 연구들이 이어져 왔다.
본 세미나에서는 먼저 압축감지 기법의 개념과 주요한 결과들을 소개하고, 복수측정벡터 문제에 대한 압축감지 기법의 연구결과에 대해 살피도록 한다. 또한, 압축감지 기법을 이용한 복수측정벡터 문제에 대한 알고리즘은 측정벡터의 개수가 늘어날수록 성능향상이 둔화되어, 일정 갯수 이상의 측정벡터에 대해서는 배열신호처리 기반의 복수신호분리(MUSIC) 알고리즘보다 성능이 더 나쁜 것을 볼 수 있는데, 이러한 문제점을 해결하기 위해 연구한 최근의 결과에 대해서도 살펴보고자 하며, 이는 기존의 압축감지 알고리즘 혹은 배열신호처리 알고리즘을 개별적으로 복수측정벡터 문제에 적용할 때 생기는 한계를 극복할 수 있게 해 준다.

In this talk, we will explain a Hitchin system in an elementary way. That is, we will review basics in symplectic geometry such as a moment map, the Marsden-Weinstein quotient, etc in a finite dimensional setting. After reviewing these materials and related elementary examples, we will sketch out how theses can be generalized
to the comparable ones in an infinite dimensional setting, i.e., in a gauge theoretic way.