Activities This Week

We propose a procedure (the first of its kind) for computing a fully data-dependent interval that traps the mixing time t_mix of a finite reversible ergodic Markov chain at a prescribed confidence level. The interval is computed from a single finite-length sample path from the Markov chain, and does not require the knowledge of any parameters of the chain. This stands in contrast to previous approaches, which either only provide point estimates, or require a reset mechanism, or additional prior knowledge.

The interval is constructed around the relaxation time t_relax, which is strongly related to the mixing time, and the width of the interval converges to zero roughly at a sqrt{n} rate, where n is the length of the sample path. Upper and lower bounds are given on the number of samples required to achieve constant-factor multiplicative accuracy. The lower bounds indicate that, unless further restrictions are placed on the chain, no procedure can achieve this accuracy level before seeing each state at least \Omega(t_relax) times on the average. Finally, future directions of research are identified.

Join work with Daniel Hsu and Csaba Szepesvári

More than 40 years ago, Jensen constructed a free Souslin tree of height $\omega_1$. We show how to construct a free $\kappa$-Souslin tree, where $\kappa$ is an arbitrary regular uncountable cardinal. This is joint work with Assaf Rinot.

We describe the Calabi-Yau problem on complex manifolds and its analog in non-archimedean geometry. On complex manifolds the Calabi-Yau problem asks locally for solutions of a PDE of Monge-Ampère type. The complex Calabi-Yau problem was posed by Calabi in 1954 and solved by Yau in 1978. After a presentation of the complex case we give a brief introduction to non-archimedean geometry and report on the recent progress in the non-archimedean case.

מערכות דינמיות זהו תחום רחב מאוד ובעל שימושים רבים במתמטיקה. ישנן בעיות רבות שהיו פתוחות לאורך שנים ולבסוף באו על פתרונן בעקבות ניסוח דינמי שקול, ובאמצעות כלים מדינמיקה. בהרצאה זו נדון על קצה המזלג במהי מערכת דינמית, ומהן השאלות ששואלים שם. אחר כך נראה כיצד ניתן להשתמש ברעיונות אלה על מנת לייצר פרקטלים.