Activities This Week

Consider a morphism of germs of Noetherian schemes, f: (X,x)-> (Y,y). When is it ‘stable’ under perturbations by higher order terms? I.e. when can such a perturbation be undone by a group action, e.g. by the local coordinate changes. This question has been extensively studied for real/complex analytic (or C^k) maps (k^n,o)-> (k^m,o). The idea is to reduce the orbit study, Gf, to the study of the tangent space, T_G f. The classical methods used vector field integration and infinite dimensional Lie groups, thus obstructing extensions to the zero/positive characteristic. During the last years we have developed a purely algebraic approach to this problem, extending the results to arbitrary characteristic. The key tool is the ‘Lie-type pair’. This is a group G, its would-be tangent space T_G, and certain maps between G, T_G, approximating the classical exponential/logarithm.

(joint work with G. Belitskii, A.F. Boix, G.M. Greuel.)

Long periodic geodesics on the unit tangent bundle of the modular surface are not necessarily equidistributed. However, there is a natural way to group finitely many geodesics together so that the so-obtained unions do equidistribute. This theorem (in this instance going back to Duke ‘88) is very well studied nowadays. In the talk, we discuss a dynamical approach due to Linnik through what is nowadays called Linnik’s basic lemma (providing in particular an entropy lower bound). We present here a new proof of Linnik’s basic lemma based on geometric invariant theory. This is joint work with Pengyu Yang.

In many data-driven applications, the data follows some geometric structure, and the goal is to recover this structure. In many cases, the observed data is noisy and the recovery task is even more challenging. A common assumption is that the data lies on a low dimensional manifold. Estimating a manifold from noisy samples has proven to be a challenging task. Indeed, even after decades of research, there was no (computationally tractable) algorithm that accurately estimates a manifold from noisy samples with a constant level of noise.

In this talk, we will present a method that estimates a manifold and its tangent. Moreover, we establish convergence rates, which are essentially as good as existing convergence rates for function estimation.

נציג את מושג הצביעה הקלאסית של גרף פשוט ונכליל אותו להיפרגרף (כלומר לקבוצה של קודקודים ואוסף כלשהו של תת-קבוצות שלה). ישנן כמה הכללות אפשריות, אנחנו נתמקד בהכללה הבאה: נאמר שצביעה של הקודקודים היא חסרת קונפליקטים אם בכל אחת מקבוצות האוסף יש לפחות קודקוד אחד עם צבע שלא ניתן לאף קודקוד אחר בקבוצה. נראה את הקשר בין המושג הזה לבעיות של השמות תדרים באנטנות סלולריות.

ניתן לראות פרומו חסר כל ערך מתמטי בסרטון