פעילויות השבוע

טרנספורם רדון הוא כלי מתמטי בעל חשיבות רבה ברפואה ובהנדסה. נציג את הטרנספורם ואת המתמטיקה מאחורי נוסחת ההיפוך. אחר כך נפנה לשימוש בסימטריות, חבורות והצגות שלהן כדי ללכסן את הטרנספורם ולהראות כי הוא הפיך. בדרך נפגוש את הלמה של Schur את הלפלסיאן וגם גזירה שבורה של פונקציות.

Period Problem is one of the most popular interesting problems in recent years, such as the Gan-Gross-Prasad conjectures. In this talk, we mainly focus on the local period problems, so called the relative Langlands programs. Given a quadratic local field extension E/F and a quasi-split reductive group G defined over F with associated quadratic character $\chi_G$, let $\pi$ be an irreducible admissible representation of G(E). Assume the Langlands-Vogan conjecture. Dipendra Prasad uses the enhanced L-parameter of $\pi$ to give a precise description for the multiplicity $\dim Hom_{G(F)}(\pi,\chi_G)$ if the L-packet $\Pi_\pi$ contains a generic representation. Then we can verify this conjecture if G=GSp(4).

An observation by Jens Marklof shows that the primitive rational points of a fixed denominator along the periodic unipotent orbit of volume equal to the square of the denominator equidistribute inside a proper submanifold of the modular surface. This concentration as well as the equidistribution are intimately related to classical questions of number theoretic origin. We discuss the distribution of the primitive rational points along periodic orbits of intermediate size. In this case, we can show joint equidistribution with polynomial rate in the modular surface and in the torus. We also deduce simultaneous equidistribution of primitive rational points in the modular surface and of modular hyperbolas in the two-torus under certain congruence conditions. This is joint work with M. Einsiedler and N. Shah.