[Turkmath:7845] Yeditepe Universitesi Matematik Bolumu Seminerleri: 12 Agustos 2011/Yediitepe University , Department of Mathematics, Departmental Seminar onAugust 12, 2011

[Itır Mogultay]

Degerli Liste Uyeleri;

Yeditepe Universitesi Matematik Bolumu Seminerleri kapsaminda 12 Agustos Cuma gunu Oklahoma Universitesi'nden Prof. Dr. "Murad Ozaydin" konusmaci olacaktir.  Konusma ile ilgili detayli bilgi asagidadir:

Gun: 12 Agustos, Cuma, 2pm-3pm.
Yer: Yeditepe Universitesi Matematik Bolumu Seminer Odasi.
Konusmaci: Prof. Murad Ozaydin

Baslik ve ozet:

"Three Coins for Sylvester"

How many sums are not payable using 5p and 12p coins? What's the largest such sum?
If mcnuggets come in boxes of 6, 9 and 20, which sizes can not be obtained? What's the largest unobtainable size?
In a certain game one can score 7, 11, 15 or 19 points. How many total points are impossible? What's the largest one?

In the puzzles above we are asked to find the genus and the conductor of a numerical monoid (i.e., a cofinite subset of non-negative integers containing 0, closed under addition). These problems in combinatorial number theory are intimately connected with homological and commutative algebra (graded resolutions, Gorenstein rings), algebraic geometry (monomial curves, Arf rings), Convexity (lattice points in polyhedra), Fourier analysis (Dedekind sums), complexity theory, transformation groups (stable genus increment), etc. Several recent books (by Alfonsin-Ramirez, Beck & Robins, Barvinok, Rosales & Garcia-Sanchez) treat this topic, but there are still more questions than answers.

The question of computing the genus (the size of the complement of the numerical monoid) for two (relatively prime) generators a and b was solved by J. J. Sylvester in the 1880's: genus=1/2(a-1)(b-1). The answer for three generators can not be an algebraic function (of the generators) and it took over a century more to figure it out. With four or more generators little is known except for very special cases (like arithmetic or geometric progressions). Finding the conductor (the largest element of the complement) of the numerical monoid is known as the linear Diophantine Frobenius problem, again the best results beyond three generators (in the general case) are about polynomial-time algorithms.

The Hilbert series (of the numerical monoid), expressed as a "short" rational function is an effective method for computing both the genus and the conductor. The talk will mostly be about explaining, finding and using the Hilbert series, illustrated with examples. I will try to sketch an elementary proof (that could have been given in Sylvester's time) of the solution for three coins.


Saygilarimla;
ITIR MOGULTAY.

___________________________________________________________________________

Dear Group Members;

On August 12th, 2011, our seminar speaker at the Department of Mathematics, Yeditepe University,  will be  Prof. Dr. " Murad Ozaydin" from University of Oklahoma. The title and the abstract of his talk are given below:

Date: August 12, 2pm-3pm
Place: Seminar room, Department of Mathematics, Yeditepe University
Speaker: Prof. Murad Ozaydin

Title and abstract:

"Three Coins for Sylvester"

How many sums are not payable using 5p and 12p coins? What's the largest such sum?
If mcnuggets come in boxes of 6, 9 and 20, which sizes can not be obtained? What's the largest unobtainable size?
In a certain game one can score 7, 11, 15 or 19 points. How many total points are impossible? What's the largest one?

In the puzzles above we are asked to find the genus and the conductor of a numerical monoid (i.e., a cofinite subset of non-negative integers containing 0, closed under addition). These problems in combinatorial number theory are intimately connected with homological and commutative algebra (graded resolutions, Gorenstein rings), algebraic geometry (monomial curves, Arf rings), Convexity (lattice points in polyhedra), Fourier analysis (Dedekind sums), complexity theory, transformation groups (stable genus increment), etc. Several recent books (by Alfonsin-Ramirez, Beck & Robins, Barvinok, Rosales & Garcia-Sanchez) treat this topic, but there are still more questions than answers.

The question of computing the genus (the size of the complement of the numerical monoid) for two (relatively prime) generators a and b was solved by J. J. Sylvester in the 1880's: genus=1/2(a-1)(b-1). The answer for three generators can not be an algebraic function (of the generators) and it took over a century more to figure it out. With four or more generators little is known except for very special cases (like arithmetic or geometric progressions). Finding the conductor (the largest element of the complement) of the numerical monoid is known as the linear Diophantine Frobenius problem, again the best results beyond three generators (in the general case) are about polynomial-time algorithms.

The Hilbert series (of the numerical monoid), expressed as a "short" rational function is an effective method for computing both the genus and the conductor. The talk will mostly be about explaining, finding and using the Hilbert series, illustrated with examples. I will try to sketch an elementary proof (that could have been given in Sylvester's time) of the solution for three coins.


Sincerely;
ITIR MOGULTAY.




___________________________________________________

Itır Mogultay
Assistant Professor
Department of Mathematics
Yeditepe University, Turkey

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