Activities This Week

The space Hom(\Gamma,G) of homomorphisms from a finitely-generated group \Gamma to a complex semisimple algebraic group G is known as the G-representation variety of \Gamma. We study this space when G is fixed and \Gamma is a random group in the few-relators model. That is, \Gamma is generated by k elements subject to r random relations of length L, where k and r are fixed and L tends to infinity.

More precisely, we study the subvariety Z of Hom(\Gamma,G), consisting of all homomorphisms whose images are Zariski dense in G. We give an explicit formula for the dimension of Z, valid with probability tending to 1, and study the Galois action on its geometric components. In particular, we show that in the case of deficiency 1 (i.e., k-r=1), the Zariski-dense G-representations of a typical \Gamma enjoy Galois rigidity.

Our methods assume the Generalized Riemann Hypothesis and exploit mixing of random walks and spectral gap estimates on finite groups.

Based on a joint work with E. Breuillard and P. Varju.

כשלייבניץ פיתח את האנליזה, הוא ניסח את התורה שלו באמצעות “גדלים אינפינטסימליים”, שמשקפים היטב את האינטואיציה מאחורי מושגים כמו רציפות, גזירות ואינטגרציה, אולם הוא וממשיכיו לא הצליחו להגדיר אובייקטים כאלה בצורה מספיק מדויקת, ולכן הגישה הזאת נזנחה, לטובת ההגדרות המוכרות לנו כיום. בהרצאה אשתדל להסביר איך אפשר בכל זאת להעמיד את הגישה של לייבניץ על בסיס מדויק, וגם איך אפשר לחלק פך שמן אחד (או מגש פיצה) למספר גדול מאד של חלקים, כל זאת באמצעות שימוש מושכל בכלים של לוגיקה מסדר ראשון (אותם נסביר תוך כדי ההרצאה)

We give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, we find that the central value of the L-function of the relevant motive is non-vanishing, consistent with the conjectures of Beilinson and Bloch. We speculate on a possible explanation for the existence of these torsion Ceresa classes, based on some computations with cyclic Fermat quotients. This is joint work with Ari Shnidman.

Discrepancy is a measure of equidistribution for sequences of points. A bounded remainder set is a set with bounded discrepancy, that is, the number of times it is visited differs by the expected time only by a constant. We discuss dynamical, symbolic, and spectral approaches to the study of bounded remainder sets for Kronecker sequences. We consider in particular discrepancy in the setting of symbolic dynamics and we discuss the existence of bounded remainder sets for some families of zero entropy subshifts. Note that bounded discrepancy has also to do with the notion of bounded displacement to a lattice in the context of Delone sets. We focus on the case of Pisot parameters for toral translations and then show how to construct symbolic codings in terms of multidimensional continued fraction algorithms.
This is joint work with W. Steiner and J. Thuswaldner.