In 1983 Dan Voiculescu used a family of unitary matrices, now
known as "Voiculescu's Unitaries," to provide the first counter-example to
an old conjecture of Halmos regarding "almost commuting" matrices. Later,
Ruy Exel and Terrry Loring used "Voiculescu's Unitaries" in an elementary
and elegant proof to provide another counter-example on "almost commuting"
matrices. In this talk, we present two new counter-examples using
"Voiculescu's Unitaries." The talk should be accessible to anyone with
knowledge of basic real analysis and linear algebra.

The Gross-Kudla-Schoen modified diagonal cycle on the triple product of the modular curve with itself provides a wealth of arithmetic information about modular forms, including derivatives of complex L-functions, special values of p-adic L-functions, and points on elliptic curves, known as Chow-Heegner points. In this talk, I will discuss a formula expressing the p-adic logarithm of a Chow-Heegner point in terms of the coefficients of the ordinary projection of certain p-adic modular forms.

We prove a B-SD conjecture for elliptic curves (for the p^infinity Selmer groups with arbitrary rank) a la Mazur-Tate and Darmon in anti-cyclotomic setting, for certain primes p. This is done, among other things, by proving a conjecture of Kolyvagin in 1991 on p-indivisibility of (derived) Heegner points over ring class fields. Some applications follow, for example, the p-part of the refined B-SD conjecture in the rank one case.

Mandatory for 298 Grad Seminar students who have been a TA for less than 2 quarters at UCI

Mandatory for 298 Grad Seminar students who have been a TA for less than 2 quarters at UCI