# Graduate Colloquium Fall 2016

McHenry Library Room 4130

Refreshments served at 3:30 in room 4161

For further information, please contact Gabriel Martins or call 831-459-2969

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**September 28, 2016**

**October 5, 2016**

**October 12, 2016**

**October 19, 2016**

**October 26, 2016**

**November 2, 2016**

**"On the h-cobordism theorem and the Poincare conjecture"**

**Alvin Jin, University of California, Santa Cruz**

The Poincare conjecture asserts that every simply-connected, closed 3-manifold is homeomorphic to the 3-sphere. In 2000, it was named one of the seven Millennium Prize Problems. Grigori Perelman proved this in 2002 using Ricci flow. We won't go into the proof of this conjecture, but we will look at the generalized version: Let M be a smooth compact n-manifold homotopy-equivalent to the n-sphere. Then M is homeomorphic to the n-sphere. It turns out that for n greater than or equal to 5, the h-cobordism theorem, proved by Stephen Smale for n greater than or equal to 7 in 1960 (and extended to n greater than or equal to 5), almost gives an immediate proof of the generalized conjecture in dimensions 5 or greater.

**November 9, 2016**

**November 16, 2016 *SPECIAL COLLOQUIUM***

**Noetherian Ring Seminar**

**Talk 1: 2:45-3:30pm**

**Giang Le, MSRI**

*"Action dimension of groups"*

The action dimension of a discrete group G is the minimum dimension of a contractible manifold which admites a proper G-action. This was first defined in a paper by M. Bestvina, M. Kapovich and B. Kleiner. In this talk, I will provide basic definition, examples and introduce one method for calculating the action dimension. If time permits, I will talk about the action dimension of Artin groups. The main result is that the action dimension of an Artin group with the nerve L of dimension n for n 6= 2 is less than or equal to (2n + 1) if the Artin group satisfies the K(π, 1)-Conjecture and the top cohomology group of L with Z-coefficients is trivial. For n = 2, we need one more condition on L to get the same inequality; that is the fundamental group of L is generated by r elements where r is the rank of H1(L, Z).

**Talk 2: 5:00-5:45pm**

**Pei Wang, MRSI**

**"***Introduction to Elementary Theory of Free Groups"*

Around 1945, Alfred Tarski asked whether the free groups on two or more generators have the same elementary theory. It remained open for 60 years until a positive answer was given by two groups of authors Z. Sela and O. Kharlampovich & A. Myasnikov.We will discuss basic ingredients: limit groups and real trees, in Sela’s approach. Time permitting, we will construct the Makanin-Razborov diagram.

**November 23, 2016**

**November 30, 2016**

**December 7, 2016**

**September 28, 2016****October 5, 2016****October 12, 2016****October 19, 2016****October 26, 2016****November 2, 2016**

**"On the h-cobordism theorem and the Poincare conjecture"**

**Alvin Jin, University of California, Santa Cruz**

The Poincare conjecture asserts that every simply-connected, closed 3-manifold is homeomorphic to the 3-sphere. In 2000, it was named one of the seven Millennium Prize Problems. Grigori Perelman proved this in 2002 using Ricci flow. We won't go into the proof of this conjecture, but we will look at the generalized version: Let M be a smooth compact n-manifold homotopy-equivalent to the n-sphere. Then M is homeomorphic to the n-sphere. It turns out that for n greater than or equal to 5, the h-cobordism theorem, proved by Stephen Smale for n greater than or equal to 7 in 1960 (and extended to n greater than or equal to 5), almost gives an immediate proof of the generalized conjecture in dimensions 5 or greater.

**November 9, 2016****November 16, 2016 *SPECIAL COLLOQUIUM***

**Noetherian Ring Seminar**

**Talk 1: 2:45-3:30pm**

**Giang Le, MSRI**

*"Action dimension of groups"*

The action dimension of a discrete group G is the minimum dimension of a contractible manifold which admites a proper G-action. This was first defined in a paper by M. Bestvina, M. Kapovich and B. Kleiner. In this talk, I will provide basic definition, examples and introduce one method for calculating the action dimension. If time permits, I will talk about the action dimension of Artin groups. The main result is that the action dimension of an Artin group with the nerve L of dimension n for n 6= 2 is less than or equal to (2n + 1) if the Artin group satisfies the K(π, 1)-Conjecture and the top cohomology group of L with Z-coefficients is trivial. For n = 2, we need one more condition on L to get the same inequality; that is the fundamental group of L is generated by r elements where r is the rank of H1(L, Z).

**Talk 2: 5:00-5:45pm**

**Pei Wang, MRSI**

**" Introduction to Elementary Theory of Free Groups"
**

Around 1945, Alfred Tarski asked whether the free groups on two or more generators have the same elementary theory. It remained open for 60 years until a positive answer was given by two groups of authors Z. Sela and O. Kharlampovich & A. Myasnikov.We will discuss basic ingredients: limit groups and real trees, in Sela’s approach. Time permitting, we will construct the Makanin-Razborov diagram.

**November 23, 2016****November 30, 2016****December 7, 2016**