פעילויות השבוע

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

The general structure of absolute Galois groups of fields as profinite groups is still a mystery. Among the very few known properties of such groups are several “Intersection Theorems”, describing subgroups in standard filtrations of absolute Galois groups as the intersection of all normal open subgroups with quotient in a prescribed list of finite groups. These theorems are based on deep cohomological properties of absolute Galois groups. We will present a general “Transfer Theorem” for profinite groups, which explains what lies behind these intersection theorems.

The Vertex Cover Problem is the optimization problem of finding a vertex cover V_c of minimal cardinality in a given graph. It is a classic NP-hard problem, and various algorithms have been suggested for it. In this talk, we will start with a basic algorithm for solving the problem. Using a probabilistic idea, we use it to develop an improved algorithm. The algorithm is greedy; at each step it adds to the cover a vertex such that the expected cover size, if we continue randomly after this step, is minimal. We will study the new algorithm theoretically and empirically, and present simulations that compare its performance to that of some algorithms of a similar nature.