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UNCG Clock Talk
Chris Ratigan
Tufts University
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Clock Problem
A clock has minute hand and hour hand which are distinct, butindistinguishable. Most of the time, you can tell what time it is.Question: How many times can you not tell what time it is?
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12:00
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6:00 vs. 12:30
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Ambiguous?
Fact 1 A time is ambiguous if interchanging hands creates a newtime.Fact 2 The Minute hand goes around 12 times the speed of thehour hand.
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Related Rates?
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Solution
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12:00
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Solution
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Epilogue
Image from ”Survey of Graph Embeddings Into Compact Surfaces” on Researchgate
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Thank You!
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Short presentations of An and Sn
Peter Huxford
Supervisor: Professor Eamonn O’Brien
The University of Auckland,New Zealand
June 28, 2019
Peter Huxford Short presentations of An and Sn June 28, 2019 1 / 6
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Length of a Presentation
What measurements do we care about?
Number of generatorsNumber of relationsTotal length of relations
IWord length (view relators as words)
IBit-length (can use exponents written as binary strings)
Bit-length corresponds closely to computational cost for evaluation.
Example
Dn = hr , s | rn = s2= (rs)2 = 1i
2 generators. 2 relations. Word length: O(n). Bit-length: O(log n).
Peter Huxford Short presentations of An and Sn June 28, 2019 3 / 6
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Length of a Presentation
What measurements do we care about?Number of generators
Number of relationsTotal length of relations
IWord length (view relators as words)
IBit-length (can use exponents written as binary strings)
Bit-length corresponds closely to computational cost for evaluation.
Example
Dn = hr , s | rn = s2= (rs)2 = 1i
2 generators. 2 relations. Word length: O(n). Bit-length: O(log n).
Peter Huxford Short presentations of An and Sn June 28, 2019 3 / 6
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Length of a Presentation
What measurements do we care about?Number of generatorsNumber of relations
Total length of relationsI
Word length (view relators as words)
IBit-length (can use exponents written as binary strings)
Bit-length corresponds closely to computational cost for evaluation.
Example
Dn = hr , s | rn = s2= (rs)2 = 1i
2 generators. 2 relations. Word length: O(n). Bit-length: O(log n).
Peter Huxford Short presentations of An and Sn June 28, 2019 3 / 6
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Length of a Presentation
What measurements do we care about?Number of generatorsNumber of relationsTotal length of relations
IWord length (view relators as words)
IBit-length (can use exponents written as binary strings)
Bit-length corresponds closely to computational cost for evaluation.
Example
Dn = hr , s | rn = s2= (rs)2 = 1i
2 generators. 2 relations. Word length: O(n). Bit-length: O(log n).
Peter Huxford Short presentations of An and Sn June 28, 2019 3 / 6
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Length of a Presentation
What measurements do we care about?Number of generatorsNumber of relationsTotal length of relations
IWord length (view relators as words)
IBit-length (can use exponents written as binary strings)
Bit-length corresponds closely to computational cost for evaluation.
Example
Dn = hr , s | rn = s2= (rs)2 = 1i
2 generators. 2 relations. Word length: O(n). Bit-length: O(log n).
Peter Huxford Short presentations of An and Sn June 28, 2019 3 / 6
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Length of a Presentation
What measurements do we care about?Number of generatorsNumber of relationsTotal length of relations
IWord length (view relators as words)
IBit-length (can use exponents written as binary strings)
Bit-length corresponds closely to computational cost for evaluation.
Example
Dn = hr , s | rn = s2= (rs)2 = 1i
2 generators. 2 relations. Word length: O(n). Bit-length: O(log n).
Peter Huxford Short presentations of An and Sn June 28, 2019 3 / 6
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Length of a Presentation
What measurements do we care about?Number of generatorsNumber of relationsTotal length of relations
IWord length (view relators as words)
IBit-length (can use exponents written as binary strings)
Bit-length corresponds closely to computational cost for evaluation.
Example
Dn = hr , s | rn = s2= (rs)2 = 1i
2 generators. 2 relations. Word length: O(n). Bit-length: O(log n).
Peter Huxford Short presentations of An and Sn June 28, 2019 3 / 6
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Length of a Presentation
What measurements do we care about?Number of generatorsNumber of relationsTotal length of relations
IWord length (view relators as words)
IBit-length (can use exponents written as binary strings)
Bit-length corresponds closely to computational cost for evaluation.
Example
Dn = hr , s | rn = s2= (rs)2 = 1i
2 generators. 2 relations. Word length: O(n). Bit-length: O(log n).
Peter Huxford Short presentations of An and Sn June 28, 2019 3 / 6
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The symmetric group is a Coxeter group.
Sn = hs1, . . . , sn�1 | s2i = 1, (si�1si )
3= 1,
(si sj)2= 1 if |i � j | � 2i.
s1 s2 sn�2 sn�1
(1, 2) (2, 3) (n � 2, n � 1) (n � 1, n)
This presentation is due to Moore (1897).
It has O(n) generators, O(n2)
relations, and bit-length O(n2).
There is a presentation of An+2 due to Carmichael (1923) with similarmeasurements, on the generators (i , n + 1, n + 2) for i = 1, . . . , n.
Peter Huxford Short presentations of An and Sn June 28, 2019 4 / 6
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The symmetric group is a Coxeter group.
Sn = hs1, . . . , sn�1 | s2i = 1, (si�1si )
3= 1,
(si sj)2= 1 if |i � j | � 2i.
s1 s2 sn�2 sn�1
(1, 2) (2, 3) (n � 2, n � 1) (n � 1, n)
This presentation is due to Moore (1897). It has O(n) generators, O(n2)
relations, and bit-length O(n2).
There is a presentation of An+2 due to Carmichael (1923) with similarmeasurements, on the generators (i , n + 1, n + 2) for i = 1, . . . , n.
Peter Huxford Short presentations of An and Sn June 28, 2019 4 / 6
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The symmetric group is a Coxeter group.
Sn = hs1, . . . , sn�1 | s2i = 1, (si�1si )
3= 1,
(si sj)2= 1 if |i � j | � 2i.
s1 s2 sn�2 sn�1
(1, 2) (2, 3) (n � 2, n � 1) (n � 1, n)
This presentation is due to Moore (1897). It has O(n) generators, O(n2)
relations, and bit-length O(n2).
There is a presentation of An+2 due to Carmichael (1923) with similarmeasurements, on the generators (i , n + 1, n + 2) for i = 1, . . . , n.
Peter Huxford Short presentations of An and Sn June 28, 2019 4 / 6
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Recent Improvements
The best bit-length possible has been achieved.
Theorem (Bray, Conder, Leedham-Green, O’Brien, 2011)An and Sn have presentations with a uniformly bounded number of
generators and relations, and bit-length O(log n).
A stronger result has also been shown.
Theorem (Guralnick, Kantor, Kassabov, Lubotzky, 2011)An and Sn have 3-generator 7-relator presentations of bit-length O(log n).
Peter Huxford Short presentations of An and Sn June 28, 2019 5 / 6
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Recent Improvements
The best bit-length possible has been achieved.
Theorem (Bray, Conder, Leedham-Green, O’Brien, 2011)An and Sn have presentations with a uniformly bounded number of
generators and relations, and bit-length O(log n).
A stronger result has also been shown.
Theorem (Guralnick, Kantor, Kassabov, Lubotzky, 2011)An and Sn have 3-generator 7-relator presentations of bit-length O(log n).
Peter Huxford Short presentations of An and Sn June 28, 2019 5 / 6
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What I have done
There are errors in (GKKL, 2011) regarding the presentations of An and Sn.
These are now fixed.
See my honours dissertation (2019) and my GitHub for supporting code.https://github.com/pjhuxford/short-presentations
Peter Huxford Short presentations of An and Sn June 28, 2019 6 / 6
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Lightning Talk–UNCG Computational Aspects of
Buildings Summer School 2019
Richard Mandel
Stevens Institute of Technology
June 28, 2019
1 / 16
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About me
Originally from Philadelphia, PA
Grew up partly in Melbourne, Australia.
Also lived in UK, Romania and Israel as a child.
2 / 16
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About me
Originally from Philadelphia, PA
Grew up partly in Melbourne, Australia.
Also lived in UK, Romania and Israel as a child.
3 / 16
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About me
Originally from Philadelphia, PA
Grew up partly in Melbourne, Australia.
Also lived in UK, Romania and Israel as a child.
4 / 16
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About me
BS in Mathematics from Temple University, Philadelphia; completedin 2015...
...but started in 2005(!)
MS in Mathematics from City College of New York (2016)
PhD student at Stevens Institute of Technology (Hoboken, NJ) sincefall 2017
5 / 16
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About me
BS in Mathematics from Temple University, Philadelphia; completedin 2015...
...but started in 2005(!)
MS in Mathematics from City College of New York (2016)
PhD student at Stevens Institute of Technology (Hoboken, NJ) sincefall 2017
6 / 16
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About me
BS in Mathematics from Temple University, Philadelphia; completedin 2015...
...but started in 2005(!)
MS in Mathematics from City College of New York (2016)
PhD student at Stevens Institute of Technology (Hoboken, NJ) sincefall 2017
7 / 16
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About me
BS in Mathematics from Temple University, Philadelphia; completedin 2015...
...but started in 2005(!)
MS in Mathematics from City College of New York (2016)
PhD student at Stevens Institute of Technology (Hoboken, NJ) sincefall 2017
8 / 16
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Current research (with Alexander Ushakov)
Baumslag-Solitar groups are groups with presentation
BS(m, n) = ha, t��ta
mt
�1 = a
ni.
The Diophantine Problem (DP) for spherical equations over BS(m, n)is the following decision problem. Given a spherical equation W :
w
�1
1
c
1
w
1
· · ·w�1
k ckwk = 1,
with unknowns wi and constants ci , is it decidable whether or not Whas a solution?
9 / 16
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Current research (with Alexander Ushakov)
Baumslag-Solitar groups are groups with presentation
BS(m, n) = ha, t��ta
mt
�1 = a
ni.
The Diophantine Problem (DP) for spherical equations over BS(m, n)is the following decision problem. Given a spherical equation W :
w
�1
1
c
1
w
1
· · ·w�1
k ckwk = 1,
with unknowns wi and constants ci , is it decidable whether or not Whas a solution?
10 / 16
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Current research
Easy cases:We have BS(1, 1) ⇠= Z⇥ Z and BS(1,�1) ⇠= Zo Z; in these cases itcan be shown that the problem is decidable in polynomial time.
Hard cases:For |mn| > 1, we wish to prove that the Diophantine Problem forspherical equations over BS(m, n) is NP-complete.
NP-hardness can be proved by exhibiting a reduction of the3-partition problem.
Remains to show that DP2NP.
11 / 16
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Current research
Easy cases:We have BS(1, 1) ⇠= Z⇥ Z and BS(1,�1) ⇠= Zo Z; in these cases itcan be shown that the problem is decidable in polynomial time.
Hard cases:For |mn| > 1, we wish to prove that the Diophantine Problem forspherical equations over BS(m, n) is NP-complete.
NP-hardness can be proved by exhibiting a reduction of the3-partition problem.
Remains to show that DP2NP.
12 / 16
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Current research
Easy cases:We have BS(1, 1) ⇠= Z⇥ Z and BS(1,�1) ⇠= Zo Z; in these cases itcan be shown that the problem is decidable in polynomial time.
Hard cases:For |mn| > 1, we wish to prove that the Diophantine Problem forspherical equations over BS(m, n) is NP-complete.
NP-hardness can be proved by exhibiting a reduction of the3-partition problem.
Remains to show that DP2NP.
13 / 16
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Current research
Easy cases:We have BS(1, 1) ⇠= Z⇥ Z and BS(1,�1) ⇠= Zo Z; in these cases itcan be shown that the problem is decidable in polynomial time.
Hard cases:For |mn| > 1, we wish to prove that the Diophantine Problem forspherical equations over BS(m, n) is NP-complete.
NP-hardness can be proved by exhibiting a reduction of the3-partition problem.
Remains to show that DP2NP.
14 / 16
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Current research
We hope to generalize methods to a wider class of groups (e.g.generalized Baumslag-Solitar groups).
15 / 16
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Thank you!
Thank you!
16 / 16