The dynamical degree is the exponential rate of the volume growth. The dynamical degree is one of the essential quantity to study of rational surface mappings. For example, a birational mapping on $mathhbb{P}^2$ is birationally equivalent to a rational surface automorphism if and only if the dynamical degree is a Salem number. For any given birational mapping $f$ on $mathhbb{P}^2$, it is known that we can always construct a modification whose action on $H^{2,2}$ gives the dynamical degree of $f$.

  We will discuss how to construct such modifications and how to compute the dynamical degree of a given rational map.

 Let X be a manifold obtained by blowing up points of k-dimensioanl projective space. We say f: X ----> X is a pseudo-automorphism if for every codimension 1 variety H, both the codimensions of f(H) and f^{-1}(H) are equal to 1. In this talk , we will discuss an explicit method for constructing pseudo-automorphisms on X. The centers of blowups are chosen to lie on an algebraic curve of degree (k+1) with one singular point and are determined using the arthmetic on the curve. These pseudo-automoprhisms have dynyamical degree greater than 1. This is a joint work with Eric Bedford and Jeffery Diller.

I will introduce examples of non-normal very ample toric 3-folds.
Next I will give classes of toric 3-folds whose any ample line bundles are
normally generated. Main subject is how to prove normality of
toric 3-folds admitting surjective morphism onto the projectie line.

Let f be a rational surface automorphism with positive entropy. It is known that the entropy of f is determined by its ``orbit data'', thee positive integers and a permutation sigma in S_3. Under the assumption that there is a curve C such that the closure of f(C - I(f)) = C, one can construct an automorphism with given entropy. We will discuss possible configuration of invariant curves and the construction of an automorphism with given invariant curve. This construction can be done in any dimension (ge 2).

A Fatou set is where the dynamics of a mapping is regular. On interesting kind is a rotation domain, a Fatou component on which the automorphism induces a torus action. In this second part we will discuss a rational surface automorphism with a ``huge'' rotation domain. It is huge in the sense that it contains both a curve of fixed points and isolated fixed points and there is a global linear model for it.

외환 파생상품 시장에서 거래되고 있는 옵션들과 활용 예시를 소개하고 가격결정의 주요 요소인 volatility surface의 구축과 관련된 이슈들을 소개한다. 마지막으로 유동성이 상대적으로 낮은 cross FX rate의 volatility surface를 multi-factor stochastic volatility 모형으로 추정하는 방법을 제시한다.

 The revolution of molecular biology in the early 1980s has revealed complex network of non-linear and stochastic biochemical interactions underlying biological systems. To understand this complex system, mathematical modeling has been widely used.
 In this talk, I will introduce the typical process of applying mathematical models to biological systems including mathematical representation of biological systems, model fitting to data, analysis and simulations, and experimental validation. I will also describe our efforts to develop and integrate mathematical tools across the different steps of the modeling process.
 Finally, I will discuss the shortcomings of our present approach and how they point to the parts of current toolbox of mathematical biology that need further mathematical development.

  I will give an estimate of degree of ideals defining toric
varieties
by using the dimension of the varieties, and give a characterization
of toric varieties whose dfining ideals need elements of the highest degree.
And I also talk about higher syzigies of toric varieties.

무선통신 망 성능분석에 널리 사용되는 라이시안 분포(S. O. Rice)를 소개하고

We give a closed formula for the expression of sums of period polynomials multiplied its associated Hecke eigenform on level N with N square-free. We also show that for N=2, 3, 5 this formula completely determines the Fourier expansions all Hecke eigenforms of all weights on level N. This is joint work with Youngju Choie and Don Zagier.

The revolution of molecular biology in the early 1980s has revealed complex network of non-linear and stochastic biochemical interactions underlying biological systems. To understand this complex system, mathematical modeling has been widely used.

 In this talk, I will introduce the typical process of applying mathematical models to biological systems including mathematical representation of biological systems, model fitting to data, analysis and simulations, and experimental validation. I will also describe our efforts to develop and integrate mathematical tools across the different steps of the modeling process. Finally, I will discuss the shortcomings of our present approach and how they point to the parts of current toolbox of mathematical biology that need further mathematical development.

The theory of algebraic cycles with modulus, such as the additive higher Chow group introduced by Bloch and Esnault and the Chow group with modulus by Binda, Kerz and Saito, is an emerging branch of algebraic cycle theory. The concept "modulus" concerns how cycles behave at the boundary, expressed by a Cartier divisor. In this talk we exhibit how the contravariance (in affine smooth varieties) of these theories can be deduced from a new moving lemma with modulus. We explain what kind of difficulties are caused by the modulus condition when establishing it.

초록:  1920년대 말기에 폴란드 태생의 수학자 Stefan Bergman 박사는 지금은 Bergman kernel function이라는 이름으로 알려진 개념을발견하였다. 복소함수론의 전통 깊은 코시 적분공식처럼 복소해석함수를 재생해낼 수 있는 적분 공식을 구성하는 함수 (이런 것들을 통틀어 kernel function이라 부른다)를 발견하고, 연구를 거듭하며 이로부터 파생되는 Kaehler(캘러) 거리 텐서를 포함한 여러 개념을 구성하고 이를 통해 복소함수론을 “재구성”할 수 있을 것으로 예상하였다. 그의 착상은 여러 위대한 수학자들에 의해 발전되고 연구되어 지난90여년 간 활발히 연구되어 왔다. 이 강연에서는 이 분야의 발생에서부터 현재의 연구까지 역사와 주요 연구 결과를 살펴 보고, 최근 연구의 발전 방향 등을 소개하며 앞으로 나아갈 길을 조망해 보려 한다.

참석하고자 하시는 분은 아래 링크를 통해 사전등록을 해주시면 감사하겠습니다^^
https://goo.gl/7qB5uV

 In this talk, we consider the problem of determining which modular curves can have infinitely many K-rational points when K varies over all the number fields of fixed degree.

The Ramsey number of a graph G is the minimum integer n for which every edge coloring of the complete graph on n vertices with two colors admits a monochromatic copy of G. It was first introduced in 1930 by Ramsey, who proved that the Ramsey number of complete graphs are finite, and applied it to a problem of formal logic. This fundamental result gave birth to the subfield of Combinatorics referred to as Ramsey theory which informally can be described as the study of problems that can be grouped under the common theme that “Every large system contains a large well-organized subsystem.”

In this talk, I will review the history of Ramsey numbers of graphs and discuss recent developments.

The hydrodynamic limit theory of Guo, Papanicolaou and Varadhan suggests a concrete way of analyzing the large-scale behavior of a non-equilibrium interacting particle system. Although the hydrodynamic limit theory has been successfully applied for numerous models, the particle system can be better understood by studying the so-called empirical process. Accordingly, Quastel, Rezakhanlou and Varadhan suggested a way to achieve this by using the symmetric simple exclusion process (SSEP) as a sample model. Despite their methodology being very robust, such an analysis is difficult because of several technical obstacles. Consequently, results were only achieved for two systems: the SSEP and zero-range process (ZRP). Recently, we obtained a third result of this type in the system of locally interacting Brownian motion. This model is a kind of continuum system, whereas the two previous models are lattice systems. Our work verifies that the results of SSEP and ZRP are valid for our model as well.
  We start by introducing the standard methods and classic results of the hydrodynamic limit theory and present our results thereafter.

We will recall how the regularity of powers of ideals behave

in general. Then we will present results concerning powers of ideals of maximal minors of matrices of linear forms. Joint with WinfiredBruns and Matteo Varbaro.

Let R be a commutative Noetherian ring. It is a classical result that R is regular if and only if it has finite global dimension. In recent years, certain non-commutative rings which are modules-finite over R and has finite global dimension have become objects of intense interests. They can serve as "non-commutative desingularizations" of Spec(R) and have come up in the three-dimensional solution of the Bondal-Orlov conjecture, higher Auslander-Reiten theory and non-commutative minimal model program. Despite all that attention, these objects remain rather mysterious, for examle we do not know fully  when they exist, or what global dimensions can occur. In this talk I will describe some very recent work on these questions. Some of the work are joined with E. Faber, C. Ingalls, O. Iyama, R. Takahashi, I. Shipman and C. Vial. 

We study multigraded ideals with a radical generic initial ideal.
Our main new result is that if a multigraded ideal has a radical multigraded generic initial ideal then the same is true for every multigraded hyperplane section and for every multigradedprojection.
Connection to universal Gr"obner bases for determinantal ideals, Koszul algebras associated to subspaces configurations and to ideals associated to the multiview varieties of Aholt, Sturmfels and Thomas will be discussed. Joint work with Emanuela De Negri and Elisa Gorla.

소셜 네트워킹 서비스(SNS)의 인기와 함께 스마트폰, 웨어러블 기기와 같은 모바일 기기의 보급으로 인해 개인에 대한 다양한 정보를 수집하는 것이 가능해졌다. SNS 데이터는 온라인 상에서의 행동을, 모바일 데이터는 오프라인 상에서의 행동을 나타내기 때문에 이러한 두 가지 형태의 데이터를 결합하여 개인의 행동을 보다 더 정확하게 모델링 할 수 있다. 본 강연에서는 최근 2-3년간 SNS 및 모바일 데이터 분석과 관련하여 수행한 연구를 요약하여 발표하고자 한다. 좀 더 구체적으로는 커뮤니티 발견, 전문가 발견, 위치기반 질문 처리, 이동경로 패턴발견 등을 논하고자 한다.

Lecture 3) 8. 14(Fri)  11:00 ~ 12:10 

                  Generic syzygy schemes and classification

Abstract: A main reason for non-vanishing of linear syzygies of curves is that they lie on varieties with special geometry. We can ask for the converse: if a curve carries non-zero linear sygygies, can we build interesting varieties containing the curve out of this situation? This question was answered by Mark Green, Frank-Olaf Schreyer, Stefan Ehbauer and Hans-Christian von Bothmerwho introduced the syzygy schemes associated to syzygies and began to study their properties. In my lectures, I intend to discuss various aspects of the geometry of syzygy schemes and present some applications.

Lecture 4)8. 14(Fri) 16:00 ~ 17:10

                  Applications of syzygy schemes

Abstract: A main reason for non-vanishing of linear syzygies of curves is that they lie on varieties with special geometry. We can ask for the converse: if a curve carries non-zero linear sygygies, can we build interesting varieties containing the curve out of this situation? This question was answered by Mark Green, Frank-Olaf Schreyer, Stefan Ehbauer and Hans-Christian von Bothmerwho introduced the syzygy schemes associated to syzygies and began to study their properties. In my lectures, I intend to discuss various aspects of the geometry of syzygy schemes and present some applications.

Lecture 7) More on the geometry of border rank algorithms.

Abstract: I will introduce the problem of determining the complexity of matrix multiplication and approaches to it via algebraic geometry.
The first part of the series will only require a knowledge of linear algebra.

Lecture 2)8. 13(Thu)  16:00 ~ 17:10  

                  Strong Castelnuovo Lemma and syzygy schemes

Abstract: A main reason for non-vanishing of linear syzygies of curves is that they lie on varieties with special geometry. We can ask for the converse: if a curve carries non-zero linear sygygies, can we build interesting varieties containing the curve out of this situation? This question was answered by Mark Green, Frank-Olaf Schreyer, Stefan Ehbauer and Hans-Christian von Bothmerwho introduced the syzygy schemes associated to syzygies and began to study their properties. In my lectures, I intend to discuss various aspects of the geometry of syzygy schemes and present some applications.

Mazur and Rubin found sufficient conditions for elliptic curves to have infinitely many quadratic twists that have 2-Selmer ranks r, for any given non-negative integer r. We generalize this result to hyperelliptic curves. 

Lecture 5) Geometry of Strassen's algorithm.

Abstract: I will introduce the problem of determining the complexity of matrix multiplication and approaches to it via algebraic geometry.
The first part of the series will only require a knowledge of linear algebra.

Lecture 6) Geometry of border rank algorithms (special curves in Grassmannians)
 

Abstract: I will introduce the problem of determining the complexity of matrix multiplication and approaches to it via algebraic geometry.
The first part of the series will only require a knowledge of linear algebra.

Lecture 1) 8. 12(Wed)  17:10 ~ 18:10

                  Syzygies and Koszulcohomology

Abstract: A main reason for non-vanishing of linear syzygies of curves is that they lie on varieties with special geometry. We can ask for the converse: if a curve carries non-zero linear sygygies, can we build interesting varieties containing the curve out of this situation? This question was answered by Mark Green, Frank-Olaf Schreyer, Stefan Ehbauer and Hans-Christian von Bothmerwho introduced the syzygy schemes associated to syzygies and began to study their properties. In my lectures, I intend to discuss various aspects of the geometry of syzygy schemes and present some applications.

Lecture 3) Strassen's equations and a classical problem in linear algebra
 

Abstract: I will introduce the problem of determining the complexity of matrix multiplication and approaches to it via algebraic geometry.
The first part of the series will only require a knowledge of linear algebra.

Lecture 4) Generalizations of Strassen's equations.

Abstract: I will introduce the problem of determining the complexity of matrix multiplication and approaches to it via algebraic geometry.
The first part of the series will only require a knowledge of linear algebra.

A continuous map R^m -> R^N or C^m -> C^N is called k-regular if the images of any k points are linearly independent. Given integers m and k  a problem going back to Chebyshev and  Borsuk is to determine the minimal value of N for which such maps exist. The methods of algebraic topology provide lower bounds for N, however there are very few results on the existence of such maps for particular values m. During the talk, using the methods of algebraic geometry we will construct k-regular maps. We will relate the upper bounds on N with secant varieties and the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for k< 10, and we will provide explicit examples for k at most 5. We will also provide upper bounds for arbitrary m and k.

A continuous map R^m -> R^N or C^m -> C^N is called k-regular if the images of any k points are linearly independent. Given integers m and k  a problem going back to Chebyshev and  Borsuk is to determine the minimal value of N for which such maps exist. The methods of algebraic topology provide lower bounds for N, however there are very few results on the existence of such maps for particular values m. During the talk, using the methods of algebraic geometry we will construct k-regular maps. We will relate the upper bounds on N with secant varieties and the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for k< 10, and we will provide explicit examples for k at most 5. We will also provide upper bounds for arbitrary m and k.

Lecture 1)  Strassen's algorithm and the astounding conjecture

Abstract: I will introduce the problem of determining the complexity of matrix multiplication and approaches to it via algebraic geometry.
The first part of the series will only require a knowledge of linear algebra.

A classical question in knot theory: given a knot type, what is the minimal number of sticks needed to build a stick knot (i.e., embedded piecewise-linear circle) of that knot type? This turns out to be rather difficult, and the answer is only known for the simplest knot types. It is helpful to dualize the question and ask: given a positive integer n, what knot types is it possible to realize with n sticks? With what frequencies do the different knot types arise? And, more generally, what is the structure of the moduli space of n-stick knots? I will give a detailed description of the geometry of this moduli space, which turns out to be a toric symplectic manifold which is a symplectic reduction of a complex Grassmannian, and give some initial results on the probability of knotted hexagons and heptagons. This geometric description also leads to algorithms for sampling stick knots thus for simulating ring polymers, which are modeled by stick knots. This is joint work with Jason Cantarella, Tetsuo Deguchi, and Erica Uehara.

Lecture 2) Strassen's equations: from linear to multi-linear algebra
 

Abstract: I will introduce the problem of determining the complexity of matrix multiplication and approaches to it via algebraic geometry.
The first part of the series will only require a knowledge of linear algebra.

In 1911, Dubouis determined all positive integers that are represented by a sum of k positive squares for any k geq 4.

In this talk, we generalize Dubouis' result to the binary case.

We determine all binary forms that are represented by a sum of k nonzero squares for any k geq 5.

Deep learning is a neural network technique that gained great prominence in recent years for recognizing faces (Facebook), translating speech (Microsoft) and identifying cat videos (Google). Before deep learning, neural networks were unpopular due to overfitting, problems with local minima and difficulty in choosing appropriatehyperparameters for regularization. In fact, Sumio Watanabe and his collaborators showed that the optimalhyperparameters are dictated by the structure of singularities in the models, and neural networks in particular are highly singular models. In this talk, we discuss how deep learning overcomes these singularities using Monte Carlo methods such as contrastive divergence and minimum probability flow.

C. Simpson introduced a coarse projective moduli space of semistable sheaves with a fixed Hilbert polynomial on a smooth projective variety. When the degree of the Hilbert polynomial is one, the supports of the semistable sheaves are one-dimensional and it gives an inspiration on the study of Hilbert scheme of curves, because certain components of the moduli space can be viewed as a compactifications of an open part of the corresponding Hilbert scheme. 

In this talk, we describe the relationship between these two families over a smooth quadric threefold in a very special case, using the double line structures on it that are also called ribbons. 

This is a joint work with E. Ballico. 

 Around 1940, engineers working on communication systems encountered a new challenge: How can one preserve the integrity of digital data, where minor errors in transmission can have catastrophic effects? The resulting theories of information (Shannon 1948) and error-correcting codes (Hamming 1950) created a “marriage made in heaven” between mathematics and its applications. On the one hand emerged a profound theory that could measure information and preserve it under a variety of errors; and on the other hand the practical consequences propelled telephony, satellite communication, digital hardware and the internet. In this talk I will give a brief introduction to the history of the mathematical theory of communication and then describe some of my work in this area that focus on efficient algorithms that can deal with large amounts of error, and on communication when sender and receiver are uncertain about each other’s context.

An order-t Markov chain is a discrete process where the outcome of each trial is linearly determined by the outcome of most recent t trials. The set of outcomes can be modelled by functions from a set V to a set F. The linear influences can be described as t-linear maps. When t=1, the set of linear influences can be conveniently described as digraphs on the vertex set V. Most of our talk is concerned with a combinatorial counterpart of Markov chains, where we can only tell the difference between zero probability and positive probability. We especially focus on the Boolean case, namely F is a 2-element set. This talk is to introduce several easy-to-state combinatorial problems about discrete dynamics, which arise from the combinatorial considerations of Markov chains.

The computer has influenced all kinds of sciences, with mathematical sciences being no exception. Mathematicians have been looking for a new foundation of mathematics replacing ZFC (Zermelo-Fraenkel set theory with the axiom of choice) and category theory, both of which have been successful to a great extent. Indeed, a theory, known as Type Theory, is rising up as a powerful alternative to all these traditional foundations. In type theory, any mathematical object is represented as a type.