Since a risk premium is determined by the riskiness of an asset and the attitude of investors to risk, one of them or both of them could predict the future return variations. As a proxy for the risk attitude of investors, I adopt the risk aversion and investigate the relationship with return predictability. More specifically, I propose the new method to estimate the timevarying risk aversion and examine its predictability for the future excess returns of the S&P
500 index. The results show that the risk aversion has a predictive power for future index returns even in a short horizon and does not lose the significance in the presence of conventional forecasting variables including dividend yield, short rate, and variance premium.
 Additionally, I test its Sharpe ratio predictability to guarantee the accuracy of the risk
aversion measure. The results regarding the Sharpe ratio prediction support the relevance of
the new measure for capturing the time-varying risk aversion.

A transversal matroid is a collection of objects that encodes maximal matchings in a bipartite graph. Generalized permutohedra is a class of polytopes obtained by deforming the permutohedron. We introduce a nice bijection that allows one to view transversal matroids as set of lattice points inside a generalized permutohedron. As a corollary, we solve a special case of the 30-year old conjecture by Stanley on matroids and pure O-sequences. The talk will be elementary and purely combinatorial.

The aim of these lectures is to provide an accessible understanding of time series models, heteroskedastic models and pure jump Levy models which are very useful in the study of financial mathematics. In Lecture 1, we first discuss the concept of stationary time series and then focus to discuss the basics of time series regression models and the ARCH and GARCH models. In Lecture 2, we first start with analyzing a very simple example of a pure jump process, viz. a compensated Poisson process which offers significant insight into the structure of pure jump Levy process. We next discuss several important results about Levy processes, such as Levy measures, the Levy-Ito decomposition, the Levy-Khintchine formula and infinite divisible distributions. We finally discuss how the asset price process can be modeled using Levy processes.

The aim of these lectures is to provide an accessible understanding of time series models, heteroskedastic models and pure jump Levy models which are very useful in the study of financial mathematics. In Lecture 1, we first discuss the concept of stationary time series and then focus to discuss the basics of time series regression models and the ARCH and GARCH models. In Lecture 2, we first start with analyzing a very simple example of a pure jump process, viz. a compensated Poisson process which offers significant insight into the structure of pure jump Levy process. We next discuss several important results about Levy processes, such as Levy measures, the Levy-Ito decomposition, the Levy-Khintchine formula and infinite divisible distributions. We finally discuss how the asset price process can be modeled using Levy processes.

In the course of proving the strong perfect graph theorem, Chudnovsky, Robertson, Seymour, and Thomas showed that every perfect graph either belongs to one of five basic classes or admits one of several decompositions. Four of the basic classes are closed under taking induced subgraphs (and have known forbidden subgraph characterizations), while the fifth one, consisting of double-split graphs, is not. A graph is doubled if it is an induced subgraph of a double-split graph. We find the forbidden induced subgraph characterization of doubled graphs; it contains 44 graphs.
This is joint work with Boris Alexeev, and Alexandra Fradkin.

To be announced

This work is concerned with an optimal selling rule for a large
position of stock in a market. Selling a large block of stock in a short
period typically depresses the market, which would result in a poor filling
price. In addition, the large selling intensity makes the regime more
likely to be poor state in the market. In this paper, regime switching
and depressing terms associated with selling intensity are considered on
a set of geometric Brownian models to capture movements of underlying
asset. We also consider the liquidation strategy to sell much smaller
number of shares in a long period. The goal is to maximize the overall
return under state constraints. The corresponding value function with
the selling strategy is shown to be a unique viscosity solution to the
associated HJB equations. Optimal liquidation rules are characterized
by a finite difference method. A numerical example is given to illustrate
the result.

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

The polygonality in free groups concerns the existence of a van Kampen diagram on a closed surface with certain restrictions. By formulating polygonality in terms of corresponding Whitehead graphs, we prove that the double of a rank-two free group is one-ended if and only if it contains a hyperbolic surface group. For the free groups of higher ranks, we resolve the case when the amalgamating list of words have a regular Whitehead graph; a key ingredient is Edmonds and Seymour's characterizations of perfect matching polytopes. This is partially joint work with Sang-il Oum and with Henry Wilton.

Statistical tests are introduced for distinguishing between short-range dependent
time series with changes in mean, and long-range dependent time series, with the former
making the null hypothesis. The tests are based on estimation of long- (or short-) range
dependence parameter after removing changes in mean from the series. The focus is on
GPH (Geweke and Porter-Hudak (1983)) and local Whittle estimation methods in the
spectral domain. Theoretical properties of the resulting estimators are established when
testing for zero or one break, and small sample properties of the tests are examined in
simulations. The tests are applied to several real data sets.
The introduced tests also improve on the BHKS (Berkes, Horv¶ath, Kokoszka and
Shao (2006)) test which is the only other available test for the considered problem. It
is argued that the BHKS test su®ers from low power against long-range dependence
alternatives and that this happens because the BHKS test statistic involves estimation
of the sum of the time series covariances. The BHKS test could be readily improved
by considering its R/S-like regression version which estimates long-range dependence
parameter and which does not involve the sum of covariances. Yet better alternatives
are to use more powerful estimation methods (such as GPH or local Whittle) and lead
to the tests introduced here.

In 2006 at MSRI, nine tropical geometers and combinatorialists met and announced the list of ten key open problems in (algebraic and combinatorial side of) tropical geometry. Axiomatization of tropical oriented matroids was one of them. After the work of Develin and Sturmfels, tropical oriented matroids were conjectured to be in bijection with subdivisions of product of simplices as well as with tropical pseudohyperplane arrangements. Ardila and Develin defined tropical oriented matroid, and showed one direction that tropical oriented matroids encode subdivision of product of simplices. Recently, in joint work with Oh, we proved that every triangulation of product of simplices encodes a tropical oriented matroid.

In this talk, I will give a survey on this topic, and discuss this well known conjecture. I will also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes.

The Helly number of a collection of sets is the size of its largest inclusionwise minimal subfamily with empty intersection. The precise conditions that lead to bounded Helly numbers have been studied since the 1920′s, when Helly showed that the Helly number of any collection of compact convex sets in R^d has Helly number at most d+1.

I will discuss a proof that any collection of subsets of R^d where the intersection of any subfamily consists of at most r connected components, each of which is contractible, has Helly number at most r(d+1). I will show how this implies, in a unified manner, quantitative bounds for several Helly-type theorems in geometric transversal theory.

Our main ingredients are a new variant of the nerve, a “homological nerve theorem” for this structure and an extension of a projection theorem of Kalai and Meshulam.

This is joint work with Eric Colin de Verdiere and Gregory Ginot.

Hilbert schemes of points on surfaces are the crepant resolutions of the symmetric products of surfaces. It can be studied via the orbifold theory or as moduli spaces of stable sheaves on surfaces. I will give a brief review of the study of the cohomology ring of the Hilbert schemes of points on surfaces by the method of vertex operator algebras. I will also discuss the extremal Gromov-Witten invariants of the Hilbert schemes. The former is a joint work with Z.B. Qin and W. Q. Wang. The latter is a joint work with J. Li.

Since the work of Gottsche on Betti numbers of Hilbert schemes of points on surfaces and Nakajima and Grojnowski's independent discovery of Heisenberg algebra actions on the cohomology ring of the Hilbert schemes, there have been a lot of research on the cohomology ring structure via the infinite dimensional Lie algebras. In this talk, I will explain how some infinite dimensional Lie algebras come to play a fundamental role in the seemly totally unrelated research area of topology and geometry of Hilbert schemes.

There are, so far, two classes of moduli examples which naturally carry symmetric obstruction theories: Moduli of stable objects in the abelian category of a) coherent sheaves on a Calabi-Yau threefold; and b) representations of a quiver with relations given by a superpotential. In this talk, we present one more such class motivated by the work of Diaconescu: Moduli of stable objects in the abelian category of coherent twisted quiver sheaves on a fixed projective smooth curve. It will arise as a curve counting on a holomorphic symplectic quotient described by a quiver and as an application of quasimap construction. The quasimap construction is joint with I. Ciocan-Fontanine and D. Maulik. When the symplectic quotients are the Hilbert schemes of points on the plane, the study was carried by D.E. Diaconescu.

Numerous empirical studies have shown that Geometric Levy Models (GLM) are able to fit extremely well daily returns of financial assets. On the other hand, in spite of their importance and popularity nowadays, few works have considered intraday data. In this paper, we fill this gap by analyzing the ability of two popular GLM (the Variance Gamma and Normal Inverse Gaussian) to fit the statistical features of intraday data at different sampling frequencies. Using high-frequency transaction data of the equity market, we find out that the estimator of the volatility parameter is quite stable at a wide range of intraday frequencies, in a sharp contrast to the estimator of the kurtosis parameter, which is more sensitive to market microstructure. By taking into account a microstructure noise component and then characterizing its effect in the estimation results, we put forward a heuristic method to determine suitable sampling frequencies for which the Levy model could serve as a good probabilistic approximation of the underlying price process. We also assess the performance of the two most favored estimation methods, the Method of Moments Estimators (MME) and the Maximum Likelihood Estimators (MLE), when dealing with high- frequency data sampled from a GLM. We show that neither high-frequency sampling nor MLE will result in a significant reduction of the estimation error of the volatility parameter. On the contrary, the estimation error of the parameter controlling the kurtosis of log returns can be significantly reduced by using MLE or intraday data. The theoretical infill asymptotic behavior of the bias and standard error of the MME are also provided with and without a microstructure noise component.

A moment angle complex is realized as a subspace of a polydisk (D^2)^m and the natural action of a torus (S^1)^m on (D^2)^m leaves the moment angle complex invariant. The Buchstaber invariant $s(K)$ is defined to be the maximum integer for which there is a subtorus of dimension $s(K)$ acting freely on the moment-angle complex associated with a finite simplicial complex $K$. Analogously, its real version $s_{\mathbb{R}}(K)$ can also be defined by using the real moment-angle complex instead of the moment-angle complex. The goal of this talk is to provide some properties of the (real) Buchstaber invariants $s_{\mathbb{R}}(K)$ of a seleton of a simplex in two ways.(Y.Fukukawa, M.Masuda)

We construct a basis the space of  weakly holomorphic Drinfeld modular forms.

and observe some arithmetic properties of coefficients of the basis elements.

Precisely we find that the basis elements satisty some generating function and find some duality between coefficients of the basis elements.

In this talk I will give an overview of contact topology and an important invariant of contact manifolds, namely contact homology. Contact homology is an invariant in the spirit of Floer homology: its chain complex is freely generated by so-called periodic Reeb orbits, which are closed orbits of a distinguished vector field. The differential is defined by counting certain holomorphic curves. Contact homology and its generalization, symplectic field theory, are able to detect the non-existence of fillings. I will describe how the algebraic structure of these theories can be used to show such a property.

To be announced

A Bayesian network is a graphical model that encodes probabilistic relationships among variables of interest. In this seminar I will talk about the definitions and properties of Bayesian networks. I will also describe Bayesian statistical methods for using data to estimate Bayesian network parameters. I will describe methods for learning both the parameters and structure of a Bayesian network. In addition, I will talk about the applications of the Bayesian network methods to classification tasks. If time permits, I will describe recent studies based on log-linear models for learning Bayesian network structures.

We will focus on characterizing operators up to unitary equivalence or similarity. We will use concepts and ideas fom complex geometry and harmonic analysis.