Department of

Mathematics


Seminar Calendar
for events the day of Thursday, November 14, 2013.

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Thursday, November 14, 2013

11:00 am in 241 Altgeld Hall,Thursday, November 14, 2013

The Siegel zero and zeros of other L -functions

Yitang Zhang (Univ. New Hampshire)

Abstract: We provide a possible method to eliminate the Siegel zero of $L(s,\chi)$, or, equivalently, to establish an e ffective lower bound for $L(1,\chi)$, where $\chi$ is a real primitive character $(\mod D)$. There are two major ingredients. The fi rst is to relate the lower bound for $L(1,\chi)$ to the distribution of zeros of $L(s,\chi)L(s,\chi\psi)$, with $\psi$ belonging to a large set of primitive characters $\Psi$, in a region $\Omega$. It is shown that if $$L(1,\chi) < (\log D)^{-B}$$, where $B$ is a large constant, then for most $\psi \in \Psi$, not only all the zeros of $L(s,\chi)L(s,\chi\psi)$ in $\Omega$ are critical and simple, but also all the gaps between consecutive zeros are near to the average gap. The second is, with the aim of deriving a contradiction from the gap assertion, to reduce the problem to evaluating a discrete mean. Eventually, the problem is reduced to proving a lower bound for the norm of a self adjoint operator on a Hilbert space.

1:00 pm in 347 Altgeld Hall,Thursday, November 14, 2013

A non-injective skinning map with a critical point

Jonah Gaster (UIC Math)

Abstract: Following Thurston, certain classes of 3-manifolds yield holomorphic maps on the Teichm uller spaces of their boundary components. Inspired by numerical evidence of Kent and Dumas, we present a negative result about these maps. Namely, we construct a path of deformations of the hyperbolic structure on a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry to conclude that the skinning map sends a specied path to itself, and use estimates on extremal length functions to show non-monotonicity and the existence of a critical point. Time permitting, we will indicate some surprising unexplained symmetry that comes out of our calculations.

2:00 pm in 243 Altgeld Hall,Thursday, November 14, 2013

Some results on completely p-summing maps

Alejandro Chavez-Dominguez (University of Texas at Austin)

Abstract: The theory of p-summing operators is one of the cornerstones of modern Banach space theory, having given rise to a wide array of results not only within the realm of Banach spaces but also reaching into other areas. In the context of operator spaces, which can be viewed as a noncommutative version of Banach spaces, one possible replacement for p-summing operators is the completely p-summing maps of Pisier. In this talk we present several results on completely p-summing maps, including: 1) A composition theorem: if 1/r = 1/p+1/q, then the composition of a completely p-summing map and a completely q-summing one is completely r-summing. 2) An operator space version of the Chevet-Saphar tensor products, which are in operator space trace duality with the completely p-summing maps. 3) An operator space version of the following result of Tomczak-Jaegermann: if X is an n-dimensional Banach space such that all its k-dimensional subspaces are C-isomorphic to a Hilbert space, then X itself is C(2n/k)^{1/2}-isomorphic to a Hilbert space.

3:00 pm in 243 Altgeld Hall,Thursday, November 14, 2013

Quasi-Gorensteiness of Extended Rees Algebras

Youngsu Kim (Purdue University)

Abstract: A Noetherian local ring having a canonical module is called quasi-Gorenstein if it is isomorphic to the canonical module. A quasi-Gorenstein ring is Gorenstein if it is Cohen-Macaulay. There are rings which are quasi-Gorenstein, but not Gorenstein. We show that for some classes of extended Rees algebras, the quasi-Gorensteinness implies Gorensteinness.

4:00 pm in 314 Altgeld Hall,Thursday, November 14, 2013

Bounded gaps between primes

Yitang Zhang (University of New Hampshire)

Abstract: The proof of the existence of infinitely many pairs of primes whose gaps are bounded by a constant consists of two aspects: The GPY sieve is applied for the main term and a stronger version of the Bombieri-Vinogradov theorem is applied to bound the error term. In this talk, we first briefly recall the history, then we describe the new idea which applies to the main and error terms equally, and eventually leads to a solution to the problem. We also discuss potential application of the method to other problems.