Activities This Week

We introduce a new class of locally compact groups, namely the strongly compactly covered groups, which are the Hausdorff topological groups G such that every element of G is contained in a compact open normal subgroup of G. For continuous endomorphisms ϕ:G→G of these groups we compute the algebraic entropy and study its properties. Also an Addition Theorem is available under suitable conditions.

This is joint work with Anna Giordano Bruno and Daniele Toller.

בקונגרס הבינלאומי במתמטיקה שהתקיים בשנת 1900 הציג דוד הילברט 23 בעיות פתוחות בכל התחומים של מתמטיקה. חלקן נפתרו (או נפתרו חלקית) וחלקן פתוחות עד היום.

הבעיה השלישית ברשימה התייחסה לקיום של פאונים בעלי נפחים זהים אך כאלה שלא ניתן לפרקם למספר סופי של תת פאונים חופפים. בעיה זו נפתרה תוך שנתיים ע”י תלמידו של הילברט – מקס דן. בהרצאה נדבר על הבעיה המקורית, על פתרונה ועל בעיות קשורות.

In 1981, Sansuc obtained a formula for Tamagawa numbers of reductive groups over number fields, modulo some then unknown results on the arithmetic of simply connected groups which have since been proven, particularly Weil’s conjecture on Tamagawa numbers over number fields. One easily deduces that this same formula holds for all linear algebraic groups over number fields. Sansuc’s method still works to treat reductive groups in the function field setting, thanks to the recent resolution of Weil’s conjecture in the function field setting by Lurie and Gaitsgory. However, due to the imperfection of function fields, the reductive case is very far from the general one; indeed, Sansuc’s formula does not hold for all linear algebraic groups over function fields. We give a modification of Sansuc’s formula that recaptures it in the number field case and also gives a correct answer for pseudo-reductive groups over function fields. The commutative case (which is essential even for the general pseudo-reductive case) is a corollary of a vast generalization of the Poitou-Tate nine-term exact sequence, from finite group schemes to arbitrary affine commutative group schemes of finite type. Unfortunately, there appears to be no simple formula in general for Tamagawa numbers of linear algebraic groups over function fields beyond the commutative and pseudo-reductive cases. Time permitting, we may discuss some examples of non-commutative unipotent groups over function fields whose Tamagawa numbers (and relatedly, Tate-Shafarevich sets) exhibit various types of pathological behavior.

We introduce the notion of stationary actions in the context of C-algebras, and prove a new characterization of C-simplicity in terms of unique stationarity. This ergodic theoretical characterization provides an intrinsic understanding for the relation between C-simplicity and the unique trace property, and provides a framework in which C-simplicity and random walks interact. Joint work with Mehrdad Kalantar.