Mathematics Department - Logic Seminar - Fall 2016

# Logic Seminar - Fall 2016

### Organizer(s)

Rebecca L Coulson, Gregory Cherlin

### Website

http://www.math.rutgers.edu/~rlg131/logicseminar.html

## Sunday, December 18th

### "Proving Structural Ramsey Theorems, II"

Time: 5:00 PM
Location: Hill 705
Abstract: POSTPONED FROM NOV. 28TH

Abstract: Continued.....this pair of talks will cover recent results of Hubicka and Nesetril that provide a greatly simplified framework for proving structural Ramsey theorems. The first talk will contain a brief review of the subject, state the first theorem of Hubicka and Nesetril, use it to prove certain classes are Ramsey, and point out some deficiencies. The second talk will cover the framework of closures introduced to remedy these deficiencies and then use the corresponding theorem to prove even more classes are Ramsey.

## Monday, December 5th

### "A local Ramsey theory for block sequences"

Time: 5:00 PM
Location: Hill 705
Abstract: Gowers proved an approximate Ramsey theorem for analytic partitions of the space of block sequences in a Banach space. Exact, discretized, versions of this result were later given by Rosendal. We isolate the combinatorial properties of the space of block sequences which enable these constructions and prove that they can be carried out within certain subfamilies, analogous to selective coideals or "happy families" and the role they place in the Ramsey theory of $mathbb{N}$. Under large cardinal hypotheses, these results are extended to partitions in $mathbf{L}(mathbb{R})$.

## Monday, November 28th

### "Proving Structural Ramsey Theorems, II"

Time: 5:00 PM
Location: Hill 705
Abstract: Continued.....this pair of talks will cover recent results of Hubicka and Nesetril that provide a greatly simplified framework for proving structural Ramsey theorems. The first talk will contain a brief review of the subject, state the first theorem of Hubicka and Nesetril, use it to prove certain classes are Ramsey, and point out some deficiencies.

The second talk will cover the framework of closures introduced to remedy these deficiencies and then use the corresponding theorem to prove even more classes are Ramsey.

## Monday, November 21st

### "Proving Structural Ramsey Theorems"

Time: 5:00 PM
Location: Hill 705
Abstract: This pair of talks will cover recent results of Hubicka and Nesetril that provide a greatly simplified framework for proving structural Ramsey theorems. The first talk will contain a brief review of the subject, state the first theorem of Hubicka and Nesetril, use it to prove certain classes are Ramsey, and point out some deficiencies. The second talk will cover the framework of closures introduced to remedy these deficiencies and then use the corresponding theorem to prove even more classes are Ramsey.

## Monday, November 7th

### " Central extensions in groups of finite Morley rank"

Time: 5:00 PM
Location: Hill 705
Abstract: We discuss the proof of the following theorem: Theorem. A group of finite Morley rank which is a perfect central extension of an algebraic group is itself algebraic.

The proof relies on Steinberg's theory of central extensions of Chevalley groups and related results in the Milnor K-theory of fields. The published proof glosses over a couple of important points, which we will try to address more explicitly. The result was first proved with Altinel and Borovik in a special case, then in the generality stated with Altinel.

## Monday, October 31st

### "Lattices of Elementary Substructures"

Time: 5:00 PM
Location: Hill 705
Abstract: Given any model of Peano Arithmetic, the collection of its elementary substructures forms a lattice under inclusion. The lattice problem for models of PA asks which lattices can be represented as substructure lattices of some model of PA. This question dates back to Gaifman's work on minimal types, which showed that the lattice 2 (the two element chain) can be represented as a substructure lattice. Since then, there have been many important contributions to this problem, including by Paris, Wilkie, Mills and Schmerl, though it remains open. The study of this question involves knowledge of models of PA as well as some nontrivial lattice theory and combinatorics. In this talk I will survey some of the major results and give a flavor of some of the techniques used in constructing models with prescribed substructure lattices. If there is time, I hope to describe how I have applied these techniques to construct a model of PA whose substructure lattice includes an infinite descending chain.

## Monday, October 24th

### "Random graphs: a stronger logic, but with the zero one law , V"

Time: 5:00 PM
Location: Hill 705
Abstract: THIS IS THE LAST TALK IN THE SERIES.......This is based on paper [1077] We like to find a logic stronger than first order such that: on the one hand it satisfies the 0-1 law, e.g. for the random graph $cG_{n,1/2}$ and on the other hand there is a formula $varphi(x)$ such that for no first order $psi(x)$ do we have: for every random enough $G_{n,1/2}$ are the formulas $varphi(x),psi(x)$ equivalent in it. We do it adding a quantifier on graph $bQ_{old t}$, i.e. have a class of finite graphs closed under isomorphisms and being able to say that if $(varphi_0(x,ar c),varphi_1(x_0,x_1,ar c))$ a pair of formulas with parameter define a graph in $cG_{n,1/2}$, hen , we can form a formula $psi(ar y)$ such $psi(ar c)$ says that the graph belongs $K_{ar{old t}}$. Presently we do it for random enough $ar{old t}$. In later versions we shall do it for $K_{old t} = {H:H$ a non-2-weak graph with number of cliques with $log log(|H|)$ nodes$}$ is one of $1,2,ldots lfloor sqrt{loglog(|H|)} floor$ modulo $lfloor loglog(|H|) floor$.

## Monday, October 17th - CANCELLED

### "Random graphs: a stronger logic, but with the zero one law , IV"

Time: 5:00 PM
Location: Hill 705
Abstract: and so....THE TALK CONTINUES.......This is based on paper [1077] We like to find a logic stronger than first order such that: on the one hand it satisfies the 0-1 law, e.g. for the random graph $cG_{n,1/2}$ and on the other hand there is a formula $varphi(x)$ such that for no first order $psi(x)$ do we have: for every random enough $G_{n,1/2}$ are the formulas $varphi(x),psi(x)$ equivalent in it. We do it adding a quantifier on graph $bQ_{old t}$, i.e. have a class of finite graphs closed under isomorphisms and being able to say that if $(varphi_0(x,ar c),varphi_1(x_0,x_1,ar c))$ a pair of formulas with parameter define a graph in $cG_{n,1/2}$, hen , we can form a formula $psi(ar y)$ such $psi(ar c)$ says that the graph belongs $K_{ar{old t}}$. Presently we do it for random enough $ar{old t}$. In later versions we shall do it for $K_{old t} = {H:H$ a non-2-weak graph with number of cliques with $log log(|H|)$ nodes$}$ is one of $1,2,ldots lfloor sqrt{loglog(|H|)} floor$ modulo $lfloor loglog(|H|) floor$.

## Monday, October 10th

### "Random graphs: a stronger logic, but with the zero one law , III"

Time: 5:00 PM
Location: Hill 705
Abstract: THE TALK CONTINUES.......This is based on paper [1077] We like to find a logic stronger than first order such that: on the one hand it satisfies the 0-1 law, e.g. for the random graph $cG_{n,1/2}$ and on the other hand there is a formula $varphi(x)$ such that for no first order $psi(x)$ do we have: for every random enough $G_{n,1/2}$ are the formulas $varphi(x),psi(x)$ equivalent in it. We do it adding a quantifier on graph $bQ_{old t}$, i.e. have a class of finite graphs closed under isomorphisms and being able to say that if $(varphi_0(x,ar c),varphi_1(x_0,x_1,ar c))$ a pair of formulas with parameter define a graph in $cG_{n,1/2}$, hen , we can form a formula $psi(ar y)$ such $psi(ar c)$ says that the graph belongs $K_{ar{old t}}$. Presently we do it for random enough $ar{old t}$. In later versions we shall do it for $K_{old t} = {H:H$ a non-2-weak graph with number of cliques with $log log(|H|)$ nodes$}$ is one of $1,2,ldots lfloor sqrt{loglog(|H|)} floor$ modulo $lfloor loglog(|H|) floor$.

## Monday, October 3rd - CANCELLED

### "Random graphs: a stronger logic, but with the zero one law , III"

Time: 5:00 PM
Location: Hill 705
Abstract: WILL CONTINUE NEXT WEEK!!!! THE TALK CONTINUES.......This is based on paper [1077] We like to find a logic stronger than first order such that: on the one hand it satisfies the 0-1 law, e.g. for the random graph $cG_{n,1/2}$ and on the other hand there is a formula $varphi(x)$ such that for no first order $psi(x)$ do we have: for every random enough $G_{n,1/2}$ are the formulas $varphi(x),psi(x)$ equivalent in it. We do it adding a quantifier on graph $bQ_{old t}$, i.e. have a class of finite graphs closed under isomorphisms and being able to say that if $(varphi_0(x,ar c),varphi_1(x_0,x_1,ar c))$ a pair of formulas with parameter define a graph in $cG_{n,1/2}$, hen , we can form a formula $psi(ar y)$ such $psi(ar c)$ says that the graph belongs $K_{ar{old t}}$. Presently we do it for random enough $ar{old t}$. In later versions we shall do it for $K_{old t} = {H:H$ a non-2-weak graph with number of cliques with $log log(|H|)$ nodes$}$ is one of $1,2,ldots lfloor sqrt{loglog(|H|)} floor$ modulo $lfloor loglog(|H|) floor$.

## Monday, September 26th

### "Random graphs: a stronger logic, but with the zero one law , II"

Time: 5:00 PM
Location: Hill 705
Abstract: THE TALK CONTINUES.......This is based on paper [1077] We like to find a logic stronger than first order such that: on the one hand it satisfies the 0-1 law, e.g. for the random graph $cG_{n,1/2}$ and on the other hand there is a formula $varphi(x)$ such that for no first order $psi(x)$ do we have: for every random enough $G_{n,1/2}$ are the formulas $varphi(x),psi(x)$ equivalent in it. We do it adding a quantifier on graph $bQ_{old t}$, i.e. have a class of finite graphs closed under isomorphisms and being able to say that if $(varphi_0(x,ar c),varphi_1(x_0,x_1,ar c))$ a pair of formulas with parameter define a graph in $cG_{n,1/2}$, hen , we can form a formula $psi(ar y)$ such $psi(ar c)$ says that the graph belongs $K_{ar{old t}}$. Presently we do it for random enough $ar{old t}$. In later versions we shall do it for $K_{old t} = {H:H$ a non-2-weak graph with number of cliques with $log log(|H|)$ nodes$}$ is one of $1,2,ldots lfloor sqrt{loglog(|H|)} floor$ modulo $lfloor loglog(|H|) floor$.

## Monday, September 19th

### "Random graphs: a stronger logic, but with the zero one law , I"

Time: 5:00 PM
Location: Hill 705
Abstract: This is based on paper [1077]

We like to find a logic stronger than first order such that: on the one hand it satisfies the 0-1 law, e.g. for the random graph $cG_{n,1/2}$ and on the other hand there is a formula $varphi(x)$ such that for no first order $psi(x)$ do we have: for every random enough $G_{n,1/2}$ are the formulas $varphi(x),psi(x)$ equivalent in it. We do it adding a quantifier on graph $bQ_{old t}$, i.e. have a class of finite graphs closed under isomorphisms and being able to say that if $(varphi_0(x,ar c),varphi_1(x_0,x_1,ar c))$ a pair of formulas with parameter define a graph in $cG_{n,1/2}$, hen , we can form a formula $psi(ar y)$ such $psi(ar c)$ says that the graph belongs $K_{ar{old t}}$. Presently we do it for random enough $ar{old t}$. In later versions we shall do it for $K_{old t} = {H:H$ a non-2-weak graph with number of cliques with $log log(|H|)$ nodes$}$ is one of $1,2,ldots lfloor sqrt{loglog(|H|)} floor$ modulo $lfloor loglog(|H|) floor$.

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