In these lectures we shall introduce motives and present results in Jannsen's paper, which say that the "conjectural" category of motives is semisimple abelian iff the adequate equivalence relation taken is numerical equivalence. We shall also explain what is still "conjectural" About this.

Koszul algebras are the algebras over which the resolution of the residue class field is given entirely by linear matrices. This series of talks will be a survey on results obtained about Koszul algebras since they were introduced by Priddy in 1970. In the first talk, We shall see lots of examples of Koszul algebras, and discuss several characterizations of Koszul algebras.

Some convergence results in probability measures are used to prove approximations to the factorial of natural numbers and the Weierstrass Approximation Theorem.

In this talk I will introduce generalities of the interaction between problems in Diophantine arithmetic and dynamics of flows on homogeneous spaces, and set the tone for subsequent lectures.

We consider the one dimensional compressible Navier-Stokes system near a constant steady state with the periodic boundary conditions. The linearized system around the constant steady state is a hyperbolic-parabolic coupled system. We discuss some of the properties of the linearized system and its spectrum. Next we study some controllability results of the system.

The class of multiplicative error models are particularly suited to model nonnegative time series such as financial durations, realized volatility, and squared returns. Threshold models are also known to play an important role in time series analysis. In this talk we shall present a lack-of-fit test for fitting a two-phase threshold model to the conditional mean function in a multiplicative error model. The proposed testing procedure can also be applied to a class of autoregressive conditional heteroscedastic threshold models. A simulation study shows some superiority of the proposed test over some commonly used existing tests. We shall illustrate the testing procedure by some data examples.

We consider the question of variable selection in complex models. This is often a difficult problem due to the inherent nonlinearity of the models and the resulting non-conjugacy in their Bayesian analysis. Bayesian variable selection in time-to-event models often utilize cross-validated predictive model selection criteria which can be relatively easy to estimate for a given model. However, the performances of these criteria are not well-studied in large-scale variable selection problems and, evaluation of these criteria for each model under consideration can be difficult to infeasible. An alternative criterion is based on the highest posterior model but its implementation is difficult in non-conjugate lifetime models. In this presentation, we compare the performances of these different criteria in complex lifetime data models including models with limited failure. We also propose an efficient variable selection method and illustrate its performance in simulation studies and real example.

Two-phase sampling design is a common practice in many medical studies with rare disorders. Generally, the first-phase classification is fallible but relatively cheap, while the accurate second-phase state of-the-art medical diagnosis is complex and rather expensive to perform. When constructed efficiently it offers great potential for higher true case detection as well as for higher precision. In this talk, we consider epidemiological studies with two-phase sampling design. However, instead of a single two-phase study we consider a scenario where a series of two-phase studies are done in longitudinal fashion. Efficient and simultaneous estimation of prevalence as well incidence rate are being considered at multiple time points from a sampling design perspective. Simulation study is presented to measure accuracy of the proposed estimation technique under many different circumstances. Finally, proposed method is applied to a population of elderly adults for the prognosis of major depressive disorder.

The Rost nilpotence principle is an important tool in the study of motivic decompositions of smooth projective varieties over a field. We will introduce this principle and briefly survey the cases in which it is known to hold. We will then outline a new approach to the question using etale motivic cohomology, which helps us to give a simpler and more conceptual proof of Rost nilpotence for surfaces, generalize the known results over one-dimensional bases and sheds more light on the situation in higher dimensions. The talk is based on joint work with Andreas Rosenschon.

Since time immemorial, people have been trying to understand the rational number solutions of systems of homogeneous polynomial equations with integer coefficients (called a Diophantine system). It is more convenient to think of them as rational points on associated projective varieties X, which we wll take to be smooth. This talk will introduce the various questions of this topic, and briefly review the reasonably well understood one-dimensional situation. But then the focus will be on dimension 2, and some progress for those covered by the unit ball will be discussed. The talk will end with a program (joint with Mladen Dimitrov) to establish an analogue of a result of Mazur.