Algebra, Number Theory and Algebraic Geometry Seminar

Department of Mathematical Sciences
Sharif University of Technology

30 June 2018 (9 Tyr 1397 ) at 10:00am-11:00am and 11:30am-12:30 in Math217.
Roozbeh Hazrat (University of Western Sydney)
The curious case of Leavitt algebras

Abstract: We discuss the two relations that imposed on free rings with 2n generators which give us a very interesting ring now called a Leavitt algebra. We then describe its generalisation to weighted Leavitt algebras.

1 June 2016 (12 Khordad 1395 ) at 2:00pm in Math317.
Rahim Zaare-Nahandi (University of Tehran)
Cellular resolutions, resolutions supported on a combinatorial entity

Abstract: Around early 1960’s Irving Kaplansky proposed the problem of fining a minimal free resolution (in a closed form) of an arbitrary monomial ideal M in the polynomial ring S = k[x1 ,...,xn ] where k is a field. The difficulty of the problem is reflected in the fact that, by the Stanley-Reisner correspondence, the homology of an arbitrary simplicial complex can be encoded in the multigraded Betty numbers of M. Furthermore, algebraic geometry is basically the study of polynomial ideals in S. A suitable computational commutative algebraic method such as Gröbner Bases Theory, faithfully translates properties of a polynomial ideals into those of monomial ideals. The idea to describe the entire resolution of a monomial ideal by a single simplicial complex was first initiated by Bayer, Peeva and Sturmfels [1]. This was extended by others using some accessible entities such as cell complexes and Morse theory. The aim of this talk is to review recent results on this topic and provide a minimal free resolution supported on a polytopal cell complex.
[1] D. Bayer, I. Peeva and B. Sturmfels, Monomial resolutions, Math. Res. Lett. 5 (1998) 31–46.

25 May 2016 (5 Khordad 1395 ) at 2:00pm in Math317.
Hossein Zakeri (Kharazmi University)
On the applications of modules of generalized fractions in commutative algebra

Abstract: In this talk, I recall briefly the main ingredients in the construction of modules of generalized fractions. Then, using certain sequences of elements of a commutative ring, I provide evidences that modules of generalized fractions are naturally connected with various basic concepts and ideas of commutative algebra such as the monomial conjecture, Cohen-Macaulay and co- Cohen-Macaulay modules, Gorenstein rings and local cohomology.

18 May 2016 (29 Ordibehesht 1395 ) at 2:00pm in Math317.
Masoud Ariannejad (University of Zanjan)
Some topics in non-commutative algebra


11 May 2016 (22 Ordibehesht 1395 ) at 2:00pm in Math317.
Raoufeh Manaviyat (Payame Noor University of Yazd)
On the idempotent graph of a ring

Abstract: The idempotent graph of a ring R, denoted by I(R), is a graph whose vertices are all non-trivial idempotents of R and two distinct vertices x and y are adjacent if and only if xy = yx = 0. We show if D is a division ring, then the clique number of I(Mn(D)) (n ≥ 2) is n and for any commutative Artinian ring R the clique number and the chromatic number of I(R) are equal to the number of maximal ideals of R. Also, we prove that for every left Noetherian ring R, the clique number of I(R) is finite. For every division ring D, we also determine an independent set of I(Mn(D)) with maximum size and prove that if F is an infinite field, then the domination number of I(Mn(F)) is infinite. Moreover, it is shown that diam(I(Mn(D))) = 4, for all natural number n ≥ 4 and diam(I(M3(D))) = 5. We also provide some classes of rings whose idempotent graphs are connected.

4 May 2016 (15 Ordibehesht 1395 ) at 2:00pm in Math317.
Mehran Motiee (Babol University of Technology)
Brauer group of fields

Abstract: The history of Brauer groups began in 1929 when R. Brauer (1902-1977) proved that the set of all isomorphism classes of finite dimensional division algebras over a common center F can be furnished with the structure of a torsion abelian group. This group is nowadays called the Brauer group of F and turns out to be an important invariant of F which is closely related to Galois Cohomology via the crossed products. In this talk, we attempt to give a general exposition of the basic algebraic aspects and the explicit arithmetic of Brauer groups.

27 April 2016 (8 Ordibehesht 1395 ) at 2:00pm in Math317.
Rashid Zaare-Nahandi (Institute for Advanced Studies in Basic Sciences)
Monomial ideals with linear resolution on fields with nonzero characteristics
ایدآلهای تکجمله ای با تحلیل آزاد خطی روی مشخصه غیر صفر

خلاصه: ایدآلهای تکجمله ای با تحلیل آزاد خطی از لحاظ جبری و هندسی اهمیت دارند و رده بندی آنها سالهاست که در حال انجام بوده و هنوز به اتمام نرسیده است. خطی بودن تحلیل آزاد به مشخصه میدان زمینه بستگی دارد. تاکنون تحقیقات کمی در مورد میدانهای با مشخصه غیر صفر انجام شده است. در این سخنرانی به این مطلب پرداخته و دسته ای از ایدآلها که روی هر میدانی تحلیل خطی دارند معرفی می شود. همچنین به ازای هر عدد اول داده شده ایدآلهایی معرفی می شوند که روی همه میدانها به جز میدان با مشخصه آن عدد اول تحلیل خطی دارند.

20 April 2016 (1 Ordibehesht 1395 ) at 2:00pm in Math317.
Mohammad Reza Salarian (Kharazmi University)
Finite simple groups with Sylow 3-subgroups of order 9

Abstract: One of the interesting questions in finite group theory is to identify a group by the structure of its subgroup. In this talk our target group is the smallest simple sporadic group. We identify this group by the structure of the normalizer of its Sylow 3-subgroup.

2 March 2016 (12 Esfand 1394 ) at 2:00pm in Math317.
Hadi Seyedinejad (University of Kashan)
Extending Auslander’s criterion for flatness in the geometric category

Abstract: Auslander’s classical criterion for flatness states that a finitely generated module over a regular local ring is flat if and only if the n-fold tensor power of the module is torsion-free, where n is the dimension of the base ring. There are two restrictions here: finiteness of the module and regularity of the base ring. Efforts have been made to generalize this criterion. We give a review of some past results with an emphasis on the geometric setting of the problem. Finally, we present our result on extending this criterion to the case of singular bases and explain how geometric ideas helped us to reach the solution.

24 February 2016 (5 Esfand 1394 ) at 2:00pm in Math317.
Shahram Khazaei (Sharif University of Technology)
Number-theoretic hardness assumptions and applications in cryptography

Abstract: I will review some well-known and generally-believed number-theoretic hardness assumptions. These assumptions are based on problems that no efficient (i.e., polynomial time) algorithms have so far been discovered for them. In particular, I will introduce factoring, RSA, discrete logarithm, Decisional Diffie-Hellman (DDH), quadratic residuosity and decisional composite residuosity assumptions. We will then see how secure public-key cryptosystems can be constructed based on some of these assumptions.

16 December 2015 (25 Azar 1394 ) at 2:00pm in Math317.
Kamran Divaani-Aazar (University of Alzahra & IPM)
On weakly Laskerian rings and modules

Abstract: Let R be a commutative ring with identity. An R-module M is said to be weakly Laskerian if each quotient module of M has finitely many associated prime ideals. The class of weakly Laskerian R-modules obviously includes all Laskerian modules. It is large enough to contain all Noetherian and Artinian R-modules. When R is Noetherian, many authors have examined the class of weakly Laskerian R-modules in conjunction with local cohomology. In this talk, we investigate some ring-theoretic properties of weakly Laskerian modules over commutative (not necessarily Noetherian) rings. We show that weakly Laskerian rings behave similar to Noetherian ones in some situations. However, we present some examples to illustrate strange behavior of weakly Laskerian rings in some other situations.

25 November 2015 (4 Azar 1394 ) at 2:00pm in Math317.
Amir Jafari (Sharif University of Technology)
Polylogarithms and Algebraic K-Theory

Abstract: The purpose of this talk is to give a conjectural relation between the functional equation of polylogarithms and the construction of algebraic K-theory of fields. These were initiated by Zagier in the early 90s and were extended and proved in special cases by Goncharov. In the process of explaining these conjectures it will be tried to convey the philosophy of motives.

18 November 2015 (27 Aban 1394 ) at 2:00pm in Math317.
Zohreh Mostaghim (Iran University of Science and Technology)
The structure of finite groups according to the conjugacy class sizes

Abstract: It is well known that there is a strong relation between the structure of a group and the sizes of its conjugacy classes. In this talk we explain some research problems on classification of finite groups according to the conjugacy class sizes. Also we study some graphs related to conjugacy class sizes of finite groups.

11 November 2015 (20 Aban 1394 ) at 2:00pm in Math317.
Mohammad Mahdavi-Hezavehi (Sharif University of Technology)
Is a division algebra of prime index cyclic?

Abstract: Let D be an F-central division algebra of index n. D is called cyclic if there exists a maximal subfield K ⊆ D such that K/F is Galois and its Galois group Gal(K/F) is a cyclic group. Cyclic division algebras of index p, a prime, investigated first by Wedderburn in 1920’s. He showed that if p = 2,3, then D is cyclic. But for p ≥ 5 the problem remains unsettled. Here we present some developments in that direction.

4 November 2015 (13 Aban 1394 ) at 2:00pm in Math317.
Mahmood Behboodi (Isfahan University of Technology & IPM-Isfahan)
پیرامون بررسی و حل دو مسئله باز قدیمی از کوته و کاپلانسکی در نظریه حلقه و کاربردهایی از آنها

خلاصه: از قدیم موضوع رده بندی و مشخص سازی تحت یکریختی گروه ها و حلقه های متناهی مورد توجه ریاضیدانان بوده و هست. کارهای گوس ریاضیدان آلمانی روی فرم های درجه دوم (Quadratic form) منشا و پیدایش قدیمی ترین قضیه در نظریه گروه ها یعنی قضیه اساسی گروه های آبلی متناهی بوده است. این قضیه توسط کرونکر و استیکلبرگر (Kronecker, Stickelberger) دیگر ریاضیدانان معروف آلمانی به صورت کامل و روشن اثبات شد، این قضیه می گوید: "هر گروه آبلی متناهی حاصل جمع تعداد متناهی گروه دوری است." تعمیمی از این قضیه در نظریه حلقه ها به این صورت است که "هر مدول متناهی تولید روی یک حوزه ایدال اصلی حاصل جمع تعداد متناهی مدول دوری است." در سال ۱۹۳۵ میلادی کوته (Gottfried Köthe) ریاضیدان مشهور آلمانی با الهام گرفتن از این قضیه این سوال را مطرح کرد که "کدام حلقه ها هستند که هر مدول روی آنها حاصل جمع مستقیم مدول های دوری است؟" بعد از پانزده سال در سال ۱۹۵۱ کوهن (Cohen) و کاپلانسکی (Kaplansky) مسئله کوته را برای حلقه های جابجایی حل کردند. هم اکنون بیش از هشتاد سال است که مسئله کوته در حالت ناجابجایی حل نشده باقی مانده است.

همچنین در سال ۱۹۴۹ میلادی کاپلانسکی (I. Kaplansky) ریاضیدان مشهور کانادایی با الهام از قضیه اساسی گروه های آبلی متناهی این سوال را مطرح کرد که "کدام حلقه ها هستند که هر مدول متناهی تولید روی آنها حاصل جمع مستقیم مدول های دوری است؟" در دهه هفتاد میلادی مسئله کاپلانسکی یکی از مهمترین مسائل جبردانان بود و بالاخره این مسئله در طی پانزده سال توسط افرادی شامل (I. Kaplansky, D. T. Gill, J.P. Lafon, W. Brandal, T. S. Shores, R. Wiegand, S. Wiegand, P. Vamos) برای حلقه های جابجایی کاملاً حل شد و ساختار چنین حلقه هایی مشخص گردیدند. ولی مسئله کاپلانسکی هم مانند مسئله کوته برای حلقه هایی ناجابجایی هنوز حل نشده باقی مانده است.

در این سخنرانی به بررسی این دو مسئله مهم و همچنین حل آنها در رده های وسیعی از حلقه ها می پردازیم. سپس چندین مسئله جدید مرتبط با این مسائل را معرفی کرده و به حل برخی از آنها اشاره خواهیم کرد. در پایان کاربردهای مهمی از این نوع مسائل را ارائه خواهیم کرد که مهمترین آنها رده بندی و مشخصه سازی حلقه های متناهی از یک مرتبه داده شده چون n می باشد.

28 October 2015 (6 Aban 1394 ) at 2:00pm in Math317.
Hossein Sabzrou (University of Tehran)
Castelnuovo-Mumford regularity and multigraded modules of nested type

Abstract: We will introduce the notion of multigraded modules of nested type as a generalization of series of objects appeared in commutative algebra and algebraic geometry including generic initial ideals (and modules), and monomial ideals of nested type. More precisely, let R := 𝕜[x1,...,xn] denote the polynomial ring in n indeterminates over an arbitrary field 𝕜. Let M be a finitely generated multigraded, i.e. ℤn-graded, R-module. We say that the multigraded module M is of nested type if all associated primes of M are all of the form (x1 ,...,xi ) for various i. We will see that multigraded modules of nested type arise naturally when we want to compute some important invariants (such as the Castelnuovo-Mumford regularity) of a general ℤ-graded R-module M.
  This is based on a joint work with Massoud Tousi [ST].
[ST] H. Sabzrou and M. Tousi, Multigraded modules of nested type, J. Commut. Alg., to appear

21 October 2015 (29 Mehr 1394 ) at 2:00pm in Math317.
Mohsen Asgharzadeh (IPM)
Geometrification of solid algebras

Abstract: This is an expository talk on solid algebras. This notion introduced by Mel Hochster. We talk on geometrical properties of solid algebras, based on the works by János Kollár and Holger Brenner.

27 May 2015 (6 Khordad 1394 ) at 2:00pm in Math221.
Mohammad Taghi Dibaei (Kharazmi University & IPM)
Linkage of Modules

Abstract: In a Noetherian ring R, two ideals 𝔞 and 𝔟 are said to be linked if 𝔞=0:𝔟 and 𝔟=0:𝔞. This notion has been generalized to modules in the literature. For a linked module, over a commutative semiperfect Noetherian ring R, the connections of its invariants reduced grade, Gorenstein dimension and depth are studied. I will discuss about the paper:
(Joint with Arash Sadeghi) Linkage of finite Gorenstein dimension modules, Journal of Algebra 376 (2013) 261–278.

20 May 2015 (30 Ordibehesht 1394 ) at 2:00pm in Math317.
Ali Rajaei (Tarbiat Modares University)
Some Pólya Fields

Abstract: "The study of integer valued polynomials (polynomials taking algebraic integers to algebraic integers) dates back to Polya who was interested in entire functions with integer values at integers. He restricted the problem to the polynomials rather than general entire functions and gave a characterization in the case of the rationals and their quadratic extensions. Around the same time Ostrowski proved some general results for finite extensions of the rationals and sixty five years later Zantema proved powerful results for cyclic Galois extensions of the rationals. Recently Bhargava gave a generalization of the factorial function using his concept of P-orderings and recast the theory in a new light. In this talk we recall Zantema's notion of a "Polya field" and note that the problem of characterizing Polya fields of a given degree is wide open, despite its solution in the case of the rationals and their cyclic extensions, then we prove some results for the first non-cyclic case of the problem, namely biquadratic extensions of the rationals. This is joint work with Bahar Heidaryan."

13 May 2015 (23 Ordibehesht 1394 ) at 2:00pm in Math221.
Amir Jafari (Sharif University of Technology)
Hilbert's Nullstellensatz: history and several proofs

Abstract: This is mostly an expository talk about a theorem that is more than 100 years old. Hilbert Nullstellensatz is at the heart of connection between geometry of locus of zeros and the algebra of polynomials. We will present a historical overview of this theorem and collect several proofs from different perspectives.

6 May 2015 (16 Ordibehesht 1394 ) at 2:00pm in Math221.
Reza Taleb (Shahid Beheshti University)
Algebraic K-groups and L-functions

Abstract: In this expository talk after a brief introduction to the concepts of L-functions and algebraic K-theory groups of the ring of integers in a number field, we will describe some deep relations between these analytic and algebraic objects. More precisely, we will present the formulation and results regarding the Lichtenbaum conjecture, as a generalization of Dirichlet's analytic class number formula, and -- if time permits -- regarding the Coates-Sinnott conjecture, as a generalization of the classical Stickelberger's theorem.

29 April 2015 (9 Ordibehesht 1394 ) at 2:00pm in Math221.
Farhad Rahmati (Amirkabir University of Technology)
Algebraic Study of Singular Points of Algebraic Schemes

22 April 2015 (2 Ordibehesht 1394 ) at 2:00pm in Math317.
Saeed Tafazolian (Amirkabir University of Technology)
کاربردهای نظریه نقاط وایراشتراس در دسته بندی خم های جبری

خلاصه: در این سخنرانی ابتدا به بیان و بررسی خواص نقاط وایراشتراس روی یک خم جبری در مشخصه مثبت خواهیم پرداخت. سپس کاربردهایی از آن در دسته بندی خم های جبری با نقاط گویا زیاد روی میدان های متناهی را ارایه می کنیم.

15 April 2015 (26 Farvardin 1394 ) at 2:00pm in Math221.
Mohammad Reza Darafsheh (University of Tehran)
Character theory of Finite Groups: Problems and Conjectures

Abstract: Ordinary character theory of finite groups has played important role in the abstract theory of finite groups since its invention by G. Frobenius and pioneering researches of I. Schur, W. Burnside and R. Brauer in early twentieth century. Because of its rich content and useful applications in science related disciplines, fundamental structural theorems about finite groups were proved using character theory. But, due to further developments of the subject new conjectures and challenging problems came to existence where still resist complete solution and group theorists try to find new ideas to settle these. In this talk our aim is to explain a few of these conjectures and problems which are in the center of research on character theory of finite groups.

18 April 2012 (30 Farvardin 1391 ) at 3:00pm in Math403.
Saeed Tafazolian (IASBS & IPM)
Curves defined over finite fields

Abstract: By a curve we mean a smooth geometrically irreducible projective curve. Explicit curves (i.e., curves given by explicit equations) over finite fields with many rational points with respect to their genera have attracted a lot of attention, after Goppa discovered that they can be used to construct good linear error-correcting codes. For the number of Fq-rational points on the curve C of genus g(C) over Fq we have the following bound

#C(Fq) ≤ 1 + q + 2 q g(C),
which is well-known as the Hasse-Weil bound. This is a deep result due to Hasse for elliptic curves, and for general curves is due to A. Weil. When the cardinality of the finite field is square, a curve C over Fq2 is called maximal if it attains the Hasse-Weil bound, i.e., if we have the equality
#C(Fq2) = 1 + q2+ 2qg(C).
We introduce some geometric properties of curves with many rational points to classify certain maximal curves.

29 February 2012 (10 Esfand 1390 ) at 2:00pm in Seminar room (4th floor, Dept. of Industrial Eng.).
Siamak Yassemi (University of Tehran & IPM)
Combinatorial aspects of Commutative algebra

Abstract: In 1975 a new trend of commutative algebra arose with the work by Richard Stanley who used the theory of Cohen-Macaulay rings to prove affirmatively the upper bound conjecture for spheres. It then turned out that commutative algebra supplies basic methods in the algebraic study of combinatorics on convex polytopes and simplicial complexes. Stanley was the first who used in a systematic way concepts and technique from commutative algebra to study simplicial complexes by considering the Hilbert function of Stanley-Reisner rings, whose defining ideals are generated by square-free monomials. Since then, the study of square-free monomial ideals from both the algebraic and combinatorial point of view is one of the most exciting topics in commutative algebra.

In this talk we present a survey on some research on combinatorial commutative algebra.

4 January 2012 (14 Dey 1390 ) at 2:00pm in Math203.
Peyman Nasehpour (Universität Osnabrück)
Gaussian Elements and Zero Divisors of Semigroup Modules

Abstract: Let M be an R-module and S a semigroup. Our goal is to discuss Dedekind-Mertens Lemma for semigroup module M[S] and an open problem about Gaussian elements and zero-divisors of the semigroup module M[S]. We also show that if M is an R-module and S a commutative, cancellative and torsion-free monoid, then the R[S]-module M[S] has few zero-divisors of size n if and only if the R-module M has few zero-divisors of size n and Property (A).

28 December 2011 (7 Dey 1390 ) at 1:30pm in Math203.
Ali Rajaei (Tarbiet Modares University)
Some diophantine equations.

30 November 2011 (9 Azar 1390 ) at 1:30pm in Math203.
Amir Jafari (Sharif University of Technology)
Fundamental theorem of Algebra.

Abstract: قضیه اساسی جبر که بیان می‌کند هر چندجمله ای با ضرایب مختلط در اعداد مختلط جواب دارد از قضایای قدیمی و مهم در ریاضی است. گاوس در طول حیات خود پنج اثبات مختلف برای آن ارائه کرد. در این سخنرانی به یکی از این اثباتها و تعمیمی از آن می پردازیم.

30 November 2011 (9 Azar 1390 ) at 1:30pm in Math203.
Mohammad Gholamzadeh Mahmoudi (Sharif University of Technology)
Clifford algebras and their applications.

Abstract: در اين سخنراني به بيان مثالها و تاريخچه جبرهاي كليفورد و كاربردهايي از آنها ميپردازيم

12 November 2011 (21 Aban 1390 ) at 3pm in Math203.
Behrooz Mirzaii (Institute for Advanced Studies in Basic Sciences, Zanjan, Iran)
Bloch-Wigner exact sequence and K-theory.

Abstract: In this talk I will introduce and study the Bloch-Wigner exact sequence over a field and will present Suslin's version of this theorem. Our version of this theorem is very close to Suslin's version. This result has a close connection to a question of Suslin, that relates the third homology of SL_2 to the indecomposable part of the third K-group.