We discuss the structure of braid groups on complexes that
is embedded in a surface using configuration spaces of complexes. We
show that the discrete configuration space of some cube complex that
is homeomorphic to a given complex is an Eilenberg-MacLane space of a
braid group on the complex.

Heron's formula relates the square of the area of a triangle to the 4-dimensional volume of a hyper-rectangle. As such, it should lend itself to a 4-dimensional proof. In this talk, I show how to use a scissors congruence proof of the Pythagorean Theorem to create a scissors congruence proof of Heron's formula. The talk will be an excursion into some interesting aspects of 4-dimensional hyper-solids. 

The generalized Dirichlet Problem in a plane region is established by Perron for the regions whose boundary has positive logarithmic capacity. Let be a region whose boundary has positive logarithlmic capacity. For a bounded continuous function on, there exists a unique harmonic function (the Perron function) on whose boundary function is n.e. (i.e., outside a set of capacity zero) on The solution is related with the Green's function on but the explicit form of the Green's function is known only for special class of regions, like disks, half plane, annulus and their conformally equivanent regions by means of conformal maps. In this talk, the Greens functions will be given by a boundary preserving Nevanlinna class function from the unit disk onto. In this way the geometric property of is shown to be related with the function theoretic property of the analytic function. I wish to make this talk accessible to the first year graduate students by starting with the Dirichlet problem on the unit disk with explicit Green's function and Possion integral for the solution.

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강연 30분전 세미나실 앞에서 다과가 있습니다.

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In this talk, I will introduce the idea of an n-dimensional foam which generalizes trivalent graphs, and the usual notion of a surface foam. Such foams can be knotted in (n+2)-dimensional space. Local pictures for the crossing points are obtained in all dimensions. There are different crossing types that are easy to parametrize. Also local crossings have signs associated to them. In all dimensions it is possible to examine quandle colorings and group-flows on n-foams. As a result, group-families of quandles, and cocycles that are associated to these can be used to distinguish different knotted foams. The subject of this talk is being developed in conjunction with Masahico Saito.

In this talk, I will give a brief introduction to Nonparametric (NP) Bayesian statistical modeling. First, I will describe some key components of Bayesian statistical inference. Then, I will begin with a statistical modeling example for which parametric modeling may have limitations and introduce the NP Bayes methodology for more flexible modeling. Focuses will be on NP Bayes approaches involving Dirichlet process (DP). I will also discuss computation-based inference procedure focusing on Markov Chain Monte Carlo (MCMC). I will conclude with a summary and some discussions of future research directions.

In this talk, we are interested in the stability and dynamic bifurcation for the two dimensional Swift-Hohenberg equation with an odd periodic condition. It is shown that an attractor bifurcates from the trivial solution as the control parameter crosses the critical value. The bifurcated attractor consists of finite number of singular points and their connecting orbits. Using the center manifold theory, we verify the nondegeneracy and the stability of the singular points.

We consider an optimal financial planning problem of an
economic agent with labor income when the agent has limited
opportunities to borrow against future labor income. The economic
agent determines his/her inter-temporal consumption, portfolio, and
contribution on annuity contract to maximize his/her utility of
lifetime consumption. We transform the agent’s inter-temporal problem
into a dual problem to derive the optimal policies. It can be shown
that constraints on the borrowing opportunities are necessary to
remove the arbitrage opportunities.

Recently, imaging techniques in science, engineering and medicine have evolved to expand our ability to visualize internal information of an object such as the human body.  In particular, there has been marked progress in electromagnetic property imaging techniques where cross-sectional image reconstructions of conductivity, permittivity and susceptibility distributions inside the human body are pursued. They will widen applications of imaging methods in medicine, biotechnology, non-destructive testing, monitoring of industrial process and others. 

This lecture focuses on mathematical modeling and analysis on electromagnetic tissue property imaging. The imaging problems can be formulated as inverse problems that are intrinsically nonlinear, and finding solutions with practical significance and value requires deep understanding of underlying physical phenomenon (Maxwell's equations) with data acquisition systems as well as implementation details of image reconstruction algorithms. We will explain strategies dealing with these complicated structures using a simple linear algebra.

Garside theory was initiated from the work of F. A. Garside on the word and conjugacy 

For centuries, researchers have attempted to grapple with the basic question of what risk is and how to measure risk. Especially, financial markets are becoming increasingly sophisticated in pricing, isolating, repackaging, and transferring risks thanks to tools such as derivatives and securitization. Therefore, financial risk management is vital to the survival of financial institutions and the stability of the financial system. At this point, financial risk management highly depends on a quantitative assessment of risk involved in a financial position. In this talk, we will discuss various issues related on financial risk analysis and management and discover the importance of advanced mathematics in those issues.

The minimal model program (MMP) refers to a series of theorems and
conjectures which arise naturally when one attempts to classify
projective varieties in terms of their pluricanonical line bundles.
The theory of multiplier ideal sheaves has played a central role in
the recent development of MMP.

A multiplier ideal sheaf is determined by a singular hermitian metric
of a line bundle. In fact, a singular hermitian metric contains more
information than its multiplier ideal sheaf. We will give an overview
of these fundamental notions and their applications in the context of
MMP.

On the other hand, in the subclass of algebraic multiplier ideal
sheaves, it is known that not every integrally closed ideal is an
algebraic multiplier ideal. We extend this statement to the full class
of analytic multiplier ideal sheaves, answering a question asked by
Lazarsfeld.

The cycle counting rook numbers, hit numbers, and q-rook numberes and q-hit numbers have been studied by many people, and Briggs and Remmel introduced the theory of p-rook and p-hit numbers which is a rook theory model of the weath product of the cyclic group C_p and the symmetric group S_n.
We extend the cycle-counting q-rook numberes and q-hit numbers to the Briggs-Remmel model. In such a settinig, we define multivariable version of the cycle-counting q-rook numbers and cycle-counting q-hit numbers where we keep track of cycles of pernutation and partial permutation of C_pwearth product with S_n according to the signs of the cycles.
This work is a joint work with Jim Haglund at University of Pennsylvania and Jeff Remmel at UCSD.

We consider the time evolution of hypersurfaces immersed in Euclidean space with the speed of the square root of the scalar curvature times a positive conformal factor.  This is an example of the geometric flow deforming the immersions which are similar to the mean curvature flow and the Gauss curvature flow.  The main ingredient for the convergence and the existence is the pinching estimate modifying that by B. Andrews.  In dimension two, a monotone quantity is obtained from the divergence structure for the Gauss curvature.  This is a joint work with Lami Kim and Kiahm Lee.

For a group G with a finite presentation and a subgroup H of G, the Reidemeister-Schreier method enables us to find a presentation of H. By applying the Reidemeister-Schreier method to right-angled Artin groups, Bell(2011) obtained a result concerning a kind of subgroups of a given right-angled Artin group. 

Following Bell's idea, we apply the Reidemeister-Schreier method to right-angled Coxeter groups. 

By modifying the definition of moments of ranks and cranks, we study the odd moments of ranks and cranks. In particular, we prove the inequality between the first crank moment ‾M₁(n) and the first rank moment ‾N₁(n):

We also study new counting function ospt(n) which is equal to ‾M₁(n) – ‾N₁(n). We will also discuss higher order moments of ranks and cranks.
This is a joint work with G. E. Andrews and S. H. Chan.

 

For a given representation of a knot group, the twisted Alexander
polynomial(TAP) is obtained by taking the Reidemeister torsion of a certain
chain complex associated to the representation. In recent years TAPs have
been successfully used for finding many topological properties of knots
such as fiberedness, knot genus, mutation, knot concordance, and more. In
this talk, we introduce a new approach for the problem of detecting
fiberedness of knots using SL(2,C)-character varieties and TAPs. We

conjecture that TAPs associated to SL(2,C)-character varieties determine if
a knot is fiebered, and give evidence for the conjecture.

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

A general goal of noncommutative geometry (in the sense of Alain Connes) is to translate the main tools of differential geometry into the Hilbert space formalism of quantum mechanics by taking advantage of the familiar duality between spaces and algebras. In this setting noncommutative spaces are only represented through noncommutative algebras that play formally the role of algebras of functions on these (ghost) noncommutative spaces. As a result, this allows us to deal with a variety of geometric problems whose noncommutative nature prevent us from using tools of classical differential geometry. In particular, the Atiyah-Singer index theorem untilmately holds in the setting of noncommutative geometry. 

The talk will be an overview of the subject with a special emphasis on quantum space-time and diffeomorphism invariant geometry. In particular, if time is permitted,  it is planned to allude to recent projects in biholomorphism invariant geometry of complex domains and contactomorphism invariant geometry of contact manifolds. 

We show that the higher-order multilinear analogue  of the fractional integral operator due to L. Grafakos has the endpoint weak type boundedness.
Furthermore, we discuss that  the method about theorems of the multilinear fractional integral operator is related to the research of multilinear (trilinear)  Hilbert transform.

The Bergman Tau function is a holomorphic function defined over Teichm"uller spaces. This satisfies modular property with repsect to the mapping class group. In this talk, we will explain an infinite product expression of the Bergman Tau function. This can be considered as a generalization of the Dedekind eta function to higher genus case. The complex valued Chern-Simons functional will be introduced for this infinite product expression. We will also explain some corollaries of this result about the eta invariant and a Polyakov type formula. 

Weakly separated set families were first studied by Leclerc and Zelevinsky in the context of quantum flag variety. Two quantum Plücker coordinates quasi-commute whenever their indexing sets are weakly separated. It was conjectured that maximal such families always have the same size. Similar question was asked by Scott when she studied quantum Grassmannian. These conjectures were independently proved by Danilov-Karzanov-Koshevoy and Oh-Postnikov-Speyer using some planar graphs and by the author using truncation. In this talk, definitions and motivations for the weakly separated set families will be explained, including Oh-Postnikov-Speyer’s point of view on the subject. The proof of the purity conjecture using truncation will be provided, and related questions will be discussed.

With availability of high frequency financial data, recently growing numbers of related mathematical methods have been developed. Among those we explain the methods based on quadratic variation process of return series with applications to measurement, forecast and pricing.

We introduce critical surfaces, which are topological index two surfaces.

In this talk, we briefly introduce the theory of degree and its application to some elliptic PDEs.
Under certain symmetry of the system concerned, most natural deformation of PDEs
fall down to ODEs.   One can use equivariant degree formula to the system in this case.
We demonstrate the detailed analysis and gives some recent results.