Department of

Mathematics


Seminar Calendar
for events the day of Thursday, April 5, 2018.

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Thursday, April 5, 2018

11:00 am in 241 Altgeld Hall,Thursday, April 5, 2018

Large gaps in sieved sets

Kevin Ford (UIUC Math)

Abstract: For each prime $p\le x$, remove from the set of integers a set $I_p$ of residue classed modulo $p$, and let $S$ be the set of remaining integers. As long as $I_p$ has average 1, we are able to improve on the trivial bound of $\gg x$, and show that for some positive constant c, there are gaps in the set $S$ of size $x(\log x)^c$ as long as $x$ is large enough. As a corollary, we show that any irreducible polynomial $f$, when evaluated at the integers up to $X$, has a string of $\gg (\log X)(\log\log X)^c$ consecutive composite values, for some positive $c$ (depending only on the degree of $f$). Another corollary is that for any polynomial $f$, there is a number $G$ so that for any $k\ge G$, there are infinitely many values of $n$ for which none of the values $f(n+1),\ldots,f(n+k)$ are coprime to all the others. For $f(n)=n$, this was proved by Erdos in 1935, and currently it is known only for linear, quadratic and cubic polynomials. This is joint work with Sergei Konyagin, James Maynard, Carl Pomerance and Terence Tao.

2:00 pm in 243 Altgeld Hall,Thursday, April 5, 2018

Lipschitz differentiability and rigidity for convex-cocompact actions on rank-one symmetric spaces

Guy C. David (Ball State University)

Abstract: We discuss a recent theorem of the speaker and Kyle Kinneberg concerning rigidity for convex-cocompact actions on non-compact rank-one symmetric spaces, which generalizes a result of Bonk and Kleiner from real hyperbolic space. A key part of the proof concerns analysis on some non-Euclidean metric spaces (Cheeger's "Lipschitz differentiability spaces" and Carnot groups), and this will be the main focus of the talk.

4:00 pm in 245 Altgeld Hall,Thursday, April 5, 2018

Ideals in L(L_p)

William B. Johnson (Texas A&M University)

Abstract: I'll discuss the Banach algebra structure of the spaces of bounded linear operators on l_p and L_p := L_p(0, 1). The main new results are
1. The only non trivial closed ideal in L(L_p), 1 ≤  p < ∞ , that has a left approximate identity is the ideal of compact operators (joint with N. C. Phillips and G. Schechtman).
2. There are in nitely many; in fact, a continuum; of closed ideals in L(L_1) (joint with G. Pisier and G. Schechtman).
The second result answers a question from the 1978 book of A. Pietsch, "Operator ideals".