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.

I will explain the basic notions and methods of an algebro-geometric theory over semirings and over somewhat exotic objects called `hyperrings'. Both developments reveal previously unseen links to other theories which include tropical geometry. In particular, I will focus on the following: 1. Cech cohomology on semiring schemes, 2. Construction of hyperring schemes, 3. Hyperstructure of the underlying space of an affine algebraic group scheme.

In this talk, we will look at how congruences between Hecke eigensystems of modular forms affect the Iwasawa invariants of their anticyclotomic p-adic L-functions. It can be regarded as an application of the ideas of Greenberg-Vatsal and Emerton-Pollack-Weston on the variation of Iwasawa invariants to the anticyclotomic setting. As an application, we establish new examples of the anticyclotomic main conjecture for modular forms. At the end, we discuss a higher weight generalization of the result (joint work with F. Castella and M. Longo) and give an explicit example.

Bounds on the complex dielectric constant of a two-component material at fixed 

frequency were derived about 35 years ago independently by Milton and Bergman 

using the analytic representation formula for the effective dielectric constant as a 

function of the component dielectric constant. These bounds become tighter the more 

information is incorporated about the composite geometry, such as the volume 

fractions of the constituents and whether it is isotropic or not. These bounds were 

subsequently generalized to elasticity in works of Berryman, Gibiansky, Lakes and 

Milton, using the variational principles of Cherkaev and Gibiansky. All these bounds 

are applicable when the applied fields are time harmonic. But what happens when the 

applied fields are not time harmonic? One would like to bound for each moment in time,

 the transient response of the induced average displacement field given the applied time

 varying electric field. We obtain such bounds using the analytic method, and we find 

that they can be very tight, tighter the more information is known about the composite. 

The bounds are also applicable to the mathematically equivalent problem of antiplane 

elasticity, where one is interested in bounding the stress relaxation and creep of 

composites of two viscoelastic phases.

Here we show how the analytic properties reviewed in Lecture 1 can be used to derive bounds on the effective moduli of composites, in particular the "Bergman-Milton" bounds that were derived independently by David Bergman and myself way back in 1979. (Chapter 27 of book "Theory of Composites” by Graeme Milton). 

Lecture 1: A Landscape of Graph Polynomials.

We introduce the most prominent graph polynomials (characteristic, Laplacian, chromatic, matching, Tutte) and discuss how to compare them.

Lecture 2: Why is the Chromatic Polynomial a Polynomial?

We give an alternative proof for the fact that the chromatic polynomial is indeed a polynomial. From this we introduce generalized chromatic polynomials, and show that this actually represents the most general case; Every (reasonably defined) graph polynomial can be represented as a generalized chromatic polynomial.

Lecture 3: Hankel matrices and Graph Polynomials.

We introduce Hankel matrices of graph paramaters, which generalize Lovasz’ connection matrices. We show that many (but not all) graph polynomials have Hankel matrices of finite rank. We show how to use the Finite Rank Property to show definability/non-definability of graph parameters/polynomials in Monadic Second Order Logic.

Network and graph theory has proven useful for modelling, analysis, and solving of problems arising in mathematics, theoretical computer science, natural sciences, social sciences, and even in finance. The connectivity, interdependence, and complexity in financial markets and systems are increasing. The analysis of networks and graphs will help us understand issues and problems arising in finance and provide appropriate models. This talk is a gentle introduction to network and graph theory.

Photonic devices are emerging for an increasing variety of technological applications, ranging from sensors to solar cells. I will show how large-scale computational optimization and rigorous analytical frameworks enable rapid search through large design spaces, and spur discovery of fundamental limits to interactions between light and matter. Our simple analysis of solar-cell emissivity showed that solar cells should be good LEDs, a counterintuitive idea leveraged by a start-up company that recently set a record for single-junction photovoltaic efficiency. I will then pivot to reviewing large-scale adjoint-based optimization methods, which we used to design new solar-cell textures and super-scattering nanoparticles. Finally, our computational nanoparticle designs led to new analytical limits to the response of metals, which have applications ranging from smoke-grenade obscurance to the near-field radiative transfer of heat.

In this lecture we will review and discuss several aspects of linear (time) translation-invariant (LTI) systems. We will begin by focusing our attention on causal and passive LTI systems, their fundamental properties, and the relation- ship between causality, passivity, and energy dissipation. After we have given a discussion of such systems in the time domain, we will discuss their properties in the frequency domain (dispersion). This leads naturally to positive (real) functions and Herglotz functions. We will then review their analytic properties and how they are related to causality, passivity, dissipation, and the Kramers-Kronig relations (i.e., dispersion relations). Finally, we will introduce the notion of a transparency window for a passive LTI system and describe its consequences. Simple examples from mathematics (e.g., the resolvent of a self-adjoint opera- tor), physics, and engineering (e.g., a spring-mass-damper system or an RLC circuit with one degree-of-freedom; constitutive relations in electromagnetism) will be used to illustrate how ubiquitous such passive LTI systems are in many areas of science.

Lecture 1: A Landscape of Graph Polynomials.

We introduce the most prominent graph polynomials (characteristic, Laplacian, chromatic, matching, Tutte) and discuss how to compare them.

Lecture 2: Why is the Chromatic Polynomial a Polynomial?

We give an alternative proof for the fact that the chromatic polynomial is indeed a polynomial. From this we introduce generalized chromatic polynomials, and show that this actually represents the most general case; Every (reasonably defined) graph polynomial can be represented as a generalized chromatic polynomial.

Lecture 3: Hankel matrices and Graph Polynomials.

We introduce Hankel matrices of graph paramaters, which generalize Lovasz’ connection matrices. We show that many (but not all) graph polynomials have Hankel matrices of finite rank. We show how to use the Finite Rank Property to show definability/non-definability of graph parameters/polynomials in Monadic Second Order Logic.

Here we discuss the analytic properties of the effective (conductivity, elastic, piezoelectric, etc.) tensor of composite materials as a function of the moduli of the component moduli, and present the associated representation formulas for these functions. (Chapter 18 of book "Theory of Composites” by Graeme Milton).

Lecture 1: A Landscape of Graph Polynomials.

We introduce the most prominent graph polynomials (characteristic, Laplacian, chromatic, matching, Tutte) and discuss how to compare them.

Lecture 2: Why is the Chromatic Polynomial a Polynomial?

We give an alternative proof for the fact that the chromatic polynomial is indeed a polynomial. From this we introduce generalized chromatic polynomials, and show that this actually represents the most general case; Every (reasonably defined) graph polynomial can be represented as a generalized chromatic polynomial.

Lecture 3: Hankel matrices and Graph Polynomials.

We introduce Hankel matrices of graph paramaters, which generalize Lovasz’ connection matrices. We show that many (but not all) graph polynomials have Hankel matrices of finite rank. We show how to use the Finite Rank Property to show definability/non-definability of graph parameters/polynomials in Monadic Second Order Logic.

Let A be a commutative ring. A subset X of A^n is a polynomial

family with d parameters if it is the range of a polynomial map from A^d to

A^n. It is an old question of Skolem (1938) whether SL_2(A) is a polynomial

family, where A is the ring of integers. Only recently Vaserstein (2010)

answered Skolem's question in the affirmative. In this talk, I will discuss

my recent result proving that SL_2(A) is a polynomial family, where A is

the ring of polynomials over a finite field of q elements. This is a

function field analogue of Vaserstein's theorem.