Let V be a subvariety of the complex projective space. The amoeba of V is the set of all real vectors log|x| where x runs over all points of V in the complex torus. The asymptotic behavior of the amoeba is given by a polyhedral fan called the Bergman fan of V. We use the tropical geometry of the Bergman fan to prove the log-concavity conjecture of Rota and Welsh over any field. This work is joint with Eric Katz and is based on arXiv:1104.2519.

We will use the primary result of the last lecture to illustrate the rich structure of right-angles Artin subgroups of right-angled Artin groups.  The talk will culminate in a complete classification of right-angled Artin subgroups of two-dimensional right-angled Artin groups.  This talk will concern joint work with Sang-hyun Kim.

Simple, distributed and iterative algorithms, popularly known as the
message passing algorithms, have emerged as the choice of the
architecture for engineered networks as well as canonical behavioral
model for societal and biological networks. Despite their simplicity,
message passing algorithms have been surprisingly effective. In this
talk, I will present a new framework to design such algorithms in the
context of communication networks. Two fundamental natures required to
understand to architect communication networks well are `interference'
(i.e. simultaneous transmissions may not be possible under certain
combinatorial constraints) and `routing' (i.e. transmissions may lead
to creation of demand at another transmissions). We developed two
principles for designing message passing algorithms utilizing these
communication natures effectively. I will present our first principle
for 'interference' primarily focusing on medium access in wireless
networks. Our second principle for 'routing' will be described in the
context of networks of data switches.
The first (interference) part of this talk is a joint work with
Devavrat Shah (MIT), Prasad Tetali (Georgia Tech) and the second
(routing) part is with Ton Dieker (Georgia Tech).

I will discuss in detail how right-angled Artin groups occur as subgroups of mapping class groups and delve into some of the hyperbolic geometry details of the proof.

I will give an overview of the general problem of understanding subgroups of mapping class groups and right-angled Artin groups, and I will draw connections to other important questions in geometry and topology, such as the virtually fibered conjecture.

Transition probabilities associated with a solution to an Ito stochastic
differential equation satisfy a partial differential equation called a
Fokker-Planck or Kolmogorov equation (FPK for short). A similar
connection holds for appropriately scaled limits of continuous time
random walks. Namely, transition probabilities of the limit processes of
continuous time random walks satisfy time-fractional order partial
differential equation. This connection represents the simplest case of
the interrelation between a time-changed stochastic processes and their
associated FPK equations. In the talk a wide class of stochastic
differential equations driven by specially constructed semimartingale
driving processes and their associated fractional order FPK equations
will be discussed.

Since investor risk aversion determines the premium required for bearing risk, a comparison thereof provides evidence of the different structure of risk premium across markets. This paper estimates and compares the degree of risk aversion of three actively traded options markets: the S&P 500, Nikkei 225, and KOSPI 200 options markets. The estimated risk aversions is found to follow S&P 500, Nikkei 225, and KOSPI 200 options in descending order, implying that S&P 500 investors require more compensation than other investors for bearing the same risk. To prove this empirically, we examine the effect of risk aversion on volatility risk premium, using delta-hedged gains. Since more risk-averse investors are willing to pay higher premiums for bearing volatility risk, greater risk averseness can result in a severe negative volatility risk premium, which is usually understood as hedging demands against the underlying asset’s downward movement. Our findings support the argument that S&P 500 investors with higher risk aversion pay more premiums for hedging volatility risk.  

The hard-core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics, a multicast model of calls in communication networks, and as a weighted independent set problem in combinatorics, probability and theoretical computer science.
In this model, each independent set I in a graph G is weighted proportionally to λ^|I|, for a positive real parameter λ. For large λ, computing the partition function (namely, the normalizing constant which makes the weighting a probability distribution on a finite graph) on graphs of maximum degree Δ≥3, is a well known computationally challenging problem. More concretely, let λ_c(T_Δ) denote the critical value for the so-called uniqueness threshold of the hard-core model on the infinite Δ-regular tree; recent breakthrough results of Dror Weitz (2006) and Allan Sly (2010) have identified λ_c(T_Δ) as a threshold where the hardness of estimating the above partition function undergoes a computational transition.
We focus on the well-studied particular case of the square lattice Z², and provide a new lower bound for the uniqueness threshold, in particular taking it well above λ_c(T₄). Our technique refines and builds on the tree of self-avoiding walks approach of Weitz, resulting in a new technical sufficient criterion (of wider applicability) for establishing strong spatial mixing (and hence uniqueness) for the hard-core model. Our new criterion achieves better bounds on strong spatial mixing when the graph has extra structure, improving upon what can be achieved by just using the maximum degree. Applying our technique to Z² we prove that strong spatial mixing holds for all λ<2.3882, improving upon the work of Weitz that held for λ<27/16=1.6875. Our results imply a fully-polynomial deterministicapproximation algorithm for estimating the partition function, as well as rapid mixing of the associated Glauber dynamics to sample from the hard-core distribution.
This is joint work with Ricardo Restrepo, Jinwoo Shin, Prasad Tetali, and Linji Yang. A preprint is available from the arXiv at: arxiv:1105.0914

By comparing liquidity and price discovery effect, the market microstructure literature including Chakravarty et al. (2004) and Easley et al. (1998) insists that in-the-money options (ITMs) are informationally inferior to out-of-the-money options (OTMs). However, such an argument is at odds with the anecdotal point that ITMs may be more effective for hedging future volatility risk. ITMs are not only driven by institutional investors who are considered as informed traders, but also can provide significant hedging benefits such that a hedging with ITMs needs fewer options and requires less frequent rebalancing. To clear this suspicion, we compare the implied risk-neutral densities, the implied risk aversions and the volatility forecasting performances. Contrary to the anecdotal evidence, our findings show the inferiority of ITMs on forecasting future volatilities, even after adjusting the risk attitude of investors, thereby supporting the argument of the extant market microstructure literature.

Constant mean curvature surfaces come up in the study of lipid bilayers
[5], soap films, soap bubble clusters [1, 11], protein folding problem [4], Chaplygin
gas, black holes [2], etc. Fascinating applications of constant mean curvature surfaces
have also appeared in harmonic map theory, Perelman’s proof of the Poincar´e
conjecture [8], positive mass theorem [10], isolated physical systems [6] and Penrose
inequality in general relativity [2].
Calabi [3, 9] proved an interesting duality between minimal surface equation in
Euclidean space R3 and maximal surface equation in Lorentz space L3. In this talk,
we exploit the classical Poincar´e Lemma to construct the twin correspondence [7] for
constant mean curvature equations in Riemannian and Lorentzian Bianchi–Cartan–
Vranceanu spaces. The twin correspondence is an extension of Calabi’s duality.

It is very well known that every graph on n vertices and m edges admits a bipartition of size at least m/2. This bound can be improved to m/2 + (n-1)/4 for connected graphs, and m/2 + n/6 for graphs without isolated vertices, as proved by Edwards, and Erdős, Gyárfás, and Kohayakawa, respectively. A bisection of a graph is a bipartition in which the size of the two parts differ by at most 1. We prove that graphs with maximum degree o(n) in fact contain a bisection which asymptotically achieves the above bounds. These results follow from a more general theorem, which can also be used to answer several questions and conjectures of Bollobás and Scott on judicious bisections of graphs.
Joint work with Po-Shen Loh (CMU) and Benny Sudakov (UCLA)

이자율 관련 파생상품의 가격결정 모델을 구축하기 위해서는 이자율을 수학적으로 모델링해야한다.

 이 강의에서는 확률과정을 통해 이자율을 모델링하는 다양한 방법을 설명한다.

이자율 파생상품은 은행 및 증권사의 FICC 등에서 가장 많이 거래하는 상품이다.

이 강의에서는 이자율 파생상품의 종류 및 가격결정 모델에 대해 설명한다.

Lascoux-Leclerc-Thibon conjectured the simple modules over Hecke akgebras
are controled by the fundamental representation of affine quantum groups.
This is proved by Ariki in a more generalized form.
Recently, Kovanov-Lauda and Rouquier introduced a new algebra
which is generalization of Hecke algebras,
and they conjectured that this algebra categorifies highest modules
of quantum groups. This conjecture is solved by myself and
Seok-Jin Kang (arXiv:1102.4677). In this talk, I will explain them.

There are three ways to define an orbifold: orbifolds
charts, groupoids, stacks. I will explain these notions in smooth
category by working out a few basic examples. It will be one of the
goals of the talk to give some understanding of the notion of stacks,
which is an important language to study moduli problems and also an
orbifold as a presentation-free intrinsic object. If time allows, I
will get to the definitions of group actions on orbifolds and the
(equivariant) cohomology over integer coefficients.

critical  exponent for the Sobolev embedding  of H 1 (Ω) into Lq (Ω).  We discuss some

existence  results  for the problem  above.

A set A of integers is a Sidon set if all the sums a₁+a₂, with a₁≤a₂and a₁, a₂∈A, are distinct. In the 1940s, Chowla, Erdős and Turán showed that the maximum possible size of a Sidon set contained in [n]={0,1,…,n-1} is √n (1+o(1)). We study Sidon sets contained in sparse random sets of integers, replacing the ‘dense environment’ [n] by a sparse, random subset R of [n].

Let R=[n]_m be a uniformly chosen, random m-element subset of [n]. Let F([n]_m)=max {|S| : S⊆[n]_m Sidon}. An abridged version of our results states as follows. Fix a constant 0≤a≤1 and suppose m=m(n)=(1+o(1))n^a. Then there is a constant b=b(a) for which F([n]_m)=n^b+o(1) almost surely. The function b=b(a) is a continuous, piecewise linear function of a, not differentiable at two points: a=1/3 and a=2/3; between those two points, the function b=b(a) is constant. This is joint work with Yoshiharu Kohayakawa and Vojtech Rödl.