# Research

**Seminars and Colloquia**

Seminars are held in room S-104.
*(Click talk titles to see available abstracts.)*

Summer 2019

*(Click to Hide)*- Fri, June 28, 2019 at 15:30--16:30.
**Keisuke Arai**(Tokyo Denki University)

Non-existence of rational points on Shimura curves and its function field analogueThe moduli space of QM-abelian surfaces (i.e., two-dimensional abelian varieties with an action of a maximal order of a quaternion algebra) is called a Shimura curve. I will talk about non-existence of rational points on Shimura curves as well as Hasse principle violations. The moduli space of D-elliptic sheaves (or Drinfeld-Stuhler modules) is a function field analogue of a Shimura curve. I will also talk about analogous results for function fields. The latter is a joint work with Satoshi Kondo and Mihran Papikian.

Spring 2019

*(Click to Show)*Fall 2018

*(Click to Show)*Spring 2018

*(Click to Show)*Fall 2017

*(Click to Show)*Spring 2017

*(Click to Show)*Fall 2016

*(Click to Show)*Spring 2016

*(Click to Show)*Fall 2015

*(Click to Show)*Spring 2015

*(Click to Show)*Fall 2014

*(Click to Show)*- Fri, Dec 26 at 14:40.
**Süleyman Ulusoy**(Zirve University)

Localization, smoothness and convergence to equilibrium for a thin film equationWe investigate the long-time behavior of weak solutions to the thin-film type equation

*v*, which arises in the Hele-Shaw problem. We estimate that the rate of convergence of solutions to the Smyth-Hill equilibrium solution, which has the form_{t}= (x ‧ v – v ‧ v_{xxx})_{x}, in the norm^{1}⁄_{24}(C^{2}– x^{2})^{2}_{+}*‖ f ‖*. We obtain exponential convergence in the^{2}_{m,1}= ∫_{R}(1 + |x|^{2m}) | f(x) |^{2}dx + ∫_{R}| f_{x}(x) |^{2}dx*‖‧‖*norm for all_{m,1}*m*with*1 ≤ m < 2*thus obtaining rates of convergence in norms measuring both smoothness and localization. The localization is the main novelty, and in fact, we show that there is a close connection between the localization bounds and the smoothness bounds: Convergence of second moments implies convergence in the*H*Sobolev norm. We then use methods of optimal mass transportation to obtain the convergence of the required moments. We also use such methods to construct an appropriate class of weak solutions for which all of the estimates on which our convergence analysis depends may be rigorously derived. Though our main results on convergence can be stated without reference to optimal mass transportation, essential use of this theory is made throughout our analysis.^{1}

Spring 2014

*(Click to Show)*
The general seminar is Thursdays 13:40-14:40.
The special seminar is Tuesdays 13:40-14:40.

Fall 2013

*(Click to Show)*
The general seminar is Thursdays 13:40-14:40.
The special seminar is Tuesdays 13:40-14:40.

- Tuesdays at 13:40.
**Munnever Celik**(METU)

(Special Seminar) Drinfeld's Double and R-Matrices - Tue, Sept 24 and Oct 1 at 13:40.
**Ozgur Kisisel**(METU)

(Special Seminar) Overview of Quantum Groups and Knots I/II(Two lectures) Review of lectures on quantum groups and knots from last year, as well as overview of remaining results.(notes) - Thu, Sept 26 and Oct 3 at 13:40.
**Ozgur Kisisel**(METU)

(General Seminar) Tropical Geometry I/II(Two lectures) We will give a general overview of tropical geometry: What things are and how they've been used. Also some new work will be outlined. If interest continues further lectures can be given during the special seminar.(notes) - Thu, Oct 10 and 24 at 13:40.
**Salih Durhan**(METU)

(General Seminar)Valued Fields and Tropical Geometry I/II(two lectures)(notes) - Thu, Nov 7 and 14 at 13:40.
**Ibrahim Unal**(METU)

(General Seminar) φ-free Submanifolds and Convexity in Calibrated ManifoldsF. R. Harvey and H. B. Lawson, Jr. canonically generalized the classical plurisubharmonic functions and convexity in complex geometry to all calibrated manifolds, and called them φ-plurisubharmonic functions and φ-convexity on a calibrated manifold(notes)*(X,φ)*. One of the techniques to construct enormous families of strictly φ-convex domains with different topological types is using φ-free submanifolds, analogues of totally real submanifolds. In this talk, I will speak about the topology of φ-free submanifolds for well-known calibrations in Quaternion-Kähler,*G*and_{2}*Spin(7)*manifolds and explain some recent results about φ-free embeddings, which use*h*-principle. - Thu, Nov 28 and Dec 5 and Dec 12 at 13:40.
**Anar Dosi**(METU)

(General Seminar) Noncommutative Algebraic Geometry - Thu, Dec 19 and Jan 2 at 13:40.
**Benjamin Walter**(METU)

(General Seminar) Chromatic Numbers, Homotopy, and Calculus(Two lectures)

Spring 2012

*(Click to Show)*
The quantum groups seminar is Tuesdays 13:40-14:40.
The general seminar is Thursdays 13:40-14:40.
The Hartshorne reading group is Thursdays 14:40-15:30.

- Thu, May 16 at 13:40.
**Ahmet Beyaz**(METU)

Contact structuresTight and overtwisted contact structures in dimension 5.(notes) - Tue, May 14 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 20Representation data and enhanced R-matrices.(notes) - Thu, May 9 at 13:40.
**Ahmet Beyaz**(METU)

Contact structuresTight and overtwisted contact structures in dimension 3.(notes) - Tue, May 7 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 19Representations of the tangle category.(notes) - Tue, Apr. 30 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 18Tangle category. Generators and relations.(notes) - Tue, Apr. 9 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 17Tensor functors. Equivalence of tensored categories.(notes) - Tue, Mar. 20 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 16Tensored categories. Definition and a little bit of structure.(notes) - Thu, Mar. 14 at 13:40.
**Ahmet Beyaz**(METU)

Symplectic Topology (cont) - Tue, Mar. 12 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 15More basic category theory.

Finish proof that an equivalence of categories is an essentially surjective fully faithful functor. Examples: finite sets and set iso, is equivalent to integers and symmetric groups; finite dimensional vector spaces and v.s. iso, is equivalent to integers and GL_{n}. Define adjoint functors and prove that adjoint functors define natural bijections of Hom. - Thu, Mar 7 at 13:40.
**Murat Baris Paksoy**(Humboldt Universitæt zu Berlin)

Derived Ramanujan Primes*R'*_{n}We study the Ramanujan-prime-counting function along the lines of Ramanujan's original work on Bertrand's Postulate. We show that the number of Ramanujan primes between x and 2x tends to infinity with x. This analysis leads us to define a new sequence of prime numbers, which we call derived Ramanujan primes. For n greater than or equal to 1 we define the nth derived Ramanujan prime as the smallest positive integer with the property that if x is greater than or equal to the nth derived Ramanujan prime, then the number of Ramanujan primes in the interval from x to x/2 is greater than or equal to n. As an application of the existence of derived Ramanujan primes, we prove analogues for Ramanujan primes of Richert's Theorem and Greenfield's Theorem for primes. We give some new inequalities for both the prime-counting function and for the Ramanujan-prime-counting function. Following the recent works of Sondow and Laishram on the bounds of Ramanujan primes, we analyze the bounds of derived Ramanujan primes. Finally, we give another proof of the theorem of Amersi, Beckwith, Miller, Ronan and Sondow. - Tue, Mar. 5 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 14Introduction to basic category theory.

Categories, functors, natural transformations, natural isomorphisms, equivalence of categories, essentially surjective and fully faithful functors. - Thu, Feb. 28 at 13:40.
**Ahmet Beyaz**(METU)

Symplectic Topology (cont) - Tue, Feb. 26 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 13 - Thu, Feb. 22 at 13:40.
**Ahmet Beyaz**(METU)

Symplectic Topology and J-Holomorphic Curves - Wed, Feb. 20 at 15:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 12

Fall 2012

*(Click to Show)*
The geometry/topology seminar is Tuedays 13:40-14:40.
The general seminar is Thursdays 13:40-14:40.
The alternate seminar time is Wednesdays 15:40-17:30.

- Tue, Jan. 8 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 11 - Thurs, Jan. 3 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Hopf Algebras and Quantum Groups - Wed, Jan. 2 at 15:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 10 - Fri, Dec. 21 at 13:40.
**Eduard Emel'yanov**(METU)

Special Seminar - Thu, Dec. 13 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Hopf Algebras and Quantum Groups - Tue, Dec. 11 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 9 - Thu, Dec. 6 at 13:40.
**Kürşat Aker**(METU-NCC)

Schubert Polynomials III - Tue, Dec. 4 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 8 - Thu, Nov. 29 at 13:40.
**Kürşat Aker**(METU-NCC)

Schubert Polynomials II - Tue, Nov. 27 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 7 - Tue, Nov. 20 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 6 - Tue, Nov. 13 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 5 - Thu, Nov. 8 at 13:40.
**Kürşat Aker**(METU-NCC)

Schubert Polynomials - Tue, Nov. 6 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 4 - Tue, Oct. 23 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 3 - Tue, Oct. 16 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum Groups 2 - Thu, Oct. 11 at 13:40.
**Safak Alpay**(METU-NCC)

On a Property of Vector Lattices and Related Classes of Operators - Tue, Oct. 9 at 13:40.
**Ozgur Kisisel**(METU-NCC)

Quantum GroupsQuantum groups are like the group-like objects in noncommutative geometry. This lecture begins a series explaining quantum groups and applications to knot theory. Today is basic definitions and notation: algebra, coalgebra, bialgebra, Hopf algebra. We will show that the involution of a Hopf algebra is unique and acts like an inverse (in particular(notes)S(gh) = S(h)S(g) ). - Thu, Oct. 4 at 13:40.
**Anar Dosi**(METU-NCC)

(Continuation) - Thu, Sept. 27 at 13:40.
**Anar Dosi**(METU-NCC)

Noncommutative Localization for Filtered RingsWe had a series of talks in the last Spring semester on an operator approach to noncommutative algebraic geometry. It was proposed a new approach to noncommutative spectra of Lie-complete associative rings, which is based on operator theory philosophy. The noncommutative rings are represented as linear (unbounded) operators over the quantum domains rather than regular functions (or global sections) over schemes as classically done in the commutative case. Commutative functional rings are replaced by operator rings, whose localizations are reduced to the operators over invariant submodules. As the main result we proved that the formal spectrum Spf(A) of a Lie-operator ring A over the domain X is the Kolmogorov completion of X. Moreover, each Lie-complete ring A turns out to be a Lie-operator ring A over the formal inflation X of Spf(A) up to a topological isomorphism. Today we talk about a certain part of this construction, namely on noncommutative localization for filtered rings. It has an independent interest and shows us how the topological completions can stand for noncommutative algebraic operations.

Spring 2012

*(Click to Show)*
The general seminar is Tuesdays 14:40-15:40. The knot theory seminar is Thursdays 15:40-16:30.

- Thursdays at 15:40.
**Ozgur Kisisel**(METU-NCC)

Knot Theory Seminar Series**Advanced Knot Theory**(ongoing seminar series) - June 7.
**Sergey Kruschev**(Altılım U.)

The Inverse Bernstein Inequality for Polynomials with Roots on the Unit Circle**Theorem 1**Let b be a separable polynomial of degree n with roots on the unit circle T, m_{b}be the smalest local maximum of |b| on T and M_{b}be the greatest local maximum. Then for every z in T, n/2 m_{b}\le |b'| \le n/2 M_{b}.

The proof is based on logarithmic potential theory on the complex plane. - Wed, May 23 at 15:40.
**Kürşat Aker**(Feza Gursey)

TBA - Monday, April 30.
**Muhammed Uludağ**(Galatasaray U.)

Öklid Algoritması, Çarklar ve Sınıf GrupoidiFarkli uzunlukta iki çubuk alalim. Bu çubuklarin uzunluklarini kiyaslamanin dogal yolu, Öklid algoritmasini uygulamaktir, sonuç olarak bir sürekli kesir elde ederiz. Öklid algoritmasini cebirsel islemlere kodlanmasi, modüler grubu verir. (Özel bazi sartlari tasiyan kiyaslamalarin kodlamasi kalandas modüler altgruplari verir). Bu cebirsel islemler, hiperbolik düzlemin bir dönüsüm grubu olarak, veya bu uzayin içinde oturan sonsuz bir agacin otomorfizmleri olarak yorumlanabilir. Hiperbolik düzlemi (agaci), modüler grubun sonsuz mertebeli tek bir elemaninin ürettigi Z-altgruba bölecek olursak, halkalari (taban kenarli çarklari) elde ederiz. (Genel olarak agacin modüler grubun bir altgrubuna bölümü bir kurdela çizge verir). Bu Z-altgruplara ayni zamanda birer belirsiz ikili kuadratik form (bikf) tekabül eder. Z-altgruplarin modüler gruptaki eslenik siniflarinaysa bikf'lerin bildik modüler grubu etkisi altindaki denklik siniflari tekabül eder. Bir çarka "takla (flip)" attirarak, yeni bir çark elde edebiliriz. Soru: Nesneleri çarklar(=bikf'ler) olan ve morfizmleri taklalar tarafindan üretilen "sinif grupoid"inin temel grubu nedir? - April 12.
**Meral Tosun**(Galatasaray U.)

Resolution Graphs of Rational SingularitiesThe first of three related talks. In this talk we will introduce rational singulaities of complex surfaces and some of the combinatorial properties of their resolution. - April 12.
**Ayse Alintas**(Yildiz Technical U.)

Equations Defining Rational SingularitiesThe second of three related talks. In this talk we will show how to use m-coverings to find non-isolated forms of rational singularities. - April 12.
**Gulen Cevik**(Koç U.)

Newton Polygon and Resolution of SingularitiesThe third of three related talks. In this talk we will introduce Newton polygons which are used to construct inimal resolutions of rational singularities. - Februray 28 at 14:40.
**Anar Dosi**(METU-NCC)

Noncommutative algebraic geometry*(continued)**(Postponed to March 6)*In this talk we investigate noncommutative schemes of Lie-complete associative rings based on their operator realizations over quantum domains. The main result asserts that the formal spectrum*Spf(A)*of an Lie-operator ring*A*over the domain*X*is the Kolmogorov completion of*X*. Moreover, each Lie-complete ring*A*turns out to be an Lie-operator ring over a certain domain. - Febrary 23 at 15:40.
**Koray Karabina**(Waterloo)

Efficient Arithmetic and Compact Representations in Finite FieldsThe efficiency of many cryptographic protocols relies on fast arithmetic and compact representation of the underlying group. I will present my recent work on obtaining compressed represenations for finite field elements and speeding up certain finite field operations. I will compare these techniques with the previously-known methods, discuss some applications to pairing-based cryptography, and conclude by stating some open problems and directions for future research. - February 21 at 14:40.
**Anar Dosi**(METU-NCC)

Noncommutative algebraic geometryIn this talk we investigate noncommutative schemes of Lie-complete associative rings based on their operator realizations over quantum domains. The main result asserts that the formal spectrum*Spf(A)*of an Lie-operator ring*A*over the domain*X*is the Kolmogorov completion of*X*. Moreover, each Lie-complete ring*A*turns out to be an Lie-operator ring over a certain domain.

Fall 2011

*(Click to Show)*- Thursdays at 13:40.
**Anar Dosi**(METU-NCC)

**Algebraic and Arithmetic Geometry Seminar Series (lecture 3) (lecture 4)****Noncommutative affine schemes, regularity and the weak operator topology.**(ongoing seminar series)

We propose an operator approach to noncommutative affine schemes based on Kapranov's framework of NC-schemes for NC-complete algebras (Noncommutative geometry based on commutator relations, J. Reine Angwer. Math. (505) (1998) 73-118).

It turns out that NC-schemes unifies many ideas of noncommutative spectral theory, operator theory and the theory of topological radicals. It seems we can easily explain many Grothendieck's extraordinary tricks done in commutative algebra and algebraic geometry in terms of the operator rings. The theory presents an interest even in the commutative case, but appearance of the weak operator topology is something new in the noncommutative case, which is mainly due to Kapranov's ideas. Many assertions have not checked out yet up to the end. Therefore many mistakes could be. All remarks and critics are welcome. - Fridays at 15:40.
**Özgür Kişisel**(METU-NCC)

**Knot Theory Seminar Series****An Introduction to Knot Theory**(ongoing seminar series)

- Wednesday, January 18 at 10:30.
**Cem Tezer**(METU Ankara)

**"Hyperbolicity of Geodesic Flows"**Each Riemannian manifold hosts a natural flow in its unit tangent bundle, the so callled geodesic flow. A general result of D. V. Anosov guarantees that the geodesic flow of a Riemannian manifold of negative sectional curvature is hyperbolic, a property that heralds complicated dynamical behaviour. I will elaborate the well-known instance of the Poincare half-plane whereof the unit tangent bundle can be identified with PSL(R, 2). The exact conditions under which the geodesic flow of a Riemannian manifold is hyperbolic are unknown. I conjecture that these are topological (even homological) rather than geometric conditions. - Wednesday, January 11 at 10:30.
**Guido Scavicco**

**"Introduction to Modal and Temporal Logic"**In this talk (intended for a audience with mathematical, computer science, or philosophy background, with a basic knowledge of propositional and first-order logics) I intend to present modal logics as a subject. We will start from basic definitions and intended semantics, progressively going towards a precise notion of truth in modal logic, and, on the side, of formal deduction. - October 13.
**Salih Durhan**(METU-NCC)

**"Images of Additive Polynomials" (cont)**Continuation of the talk from October 6. - October 6.
**Salih Durhan**(METU-NCC)

**"Images of Additive Polynomials"**Images of additive polynomials in one variable exhibit representative behavior for all polynomials over valued fields of positive characteristic. It was conjectured that this could be carried out to polynomials in several variables. I will present a new class of valued fields which are extremal with respect to additive polynomials(in several variables) but they are not extremal with respect to every polynomial, thus disproving the conjecture.

Spring 2011

*(Click to Show)*- March 4.
**Anar Dosi**(METU-NCC)

**"Quantum Calculus" (notes)**We discuss something about calculus but being equipped with quantum operations. For us the identity $1+1=1$ turns out to be quite natural and it breaks up our vision to classical calculus. The latter is too useless and boring. - March 11.
**Anar Dosi**(METU-NCC)

**"Quantum Calculus II" (notes)**... Continuation of the previous talk. - March 18.
**Ibrahim Unal**(METU-NCC)

**"Calibrated Geometries" (notes)**Calibrated Geometries were introduced by Harvey and Lawson in their foundational paper in 1982. These are the geometries of minimal submanifolds which are determined by a form φ on a Riemannian manifold called calibration. I will talk about the well-known examples of calibrated submanifolds, especially coming from special holonomy. - March 25.
**Ibrahim Unal**(METU-NCC)

**"Calibrated Geometries II"**We continue our discussion of calibrated manifolds. We introduce holonomy groups, discuss Berger's (1950's) classification of holonomy groups, and the natural calibrations induced on the different holonomy types. We are particularly interested in the special holonomy*G*and_{2}*Spin(7)*manifolds. - April 1.
**Ibrahim Unal**(METU-NCC)

**"An Introduction to Potential Theory on Calibrated Manifolds" (notes)**Recently, the notion of plurisubharmonic functions in calibrated geometries was introduced by Harvey and Lawson. These functions generalize the classical plurisubharmonic functions from complex geometry to calibrated manifolds. In this talk, I will give some information about these functions and their properties where the calibration is parallel. - April 8.
**Ozcan Kasal**(METU-NCC)

**"Valued Fields" (notes)**TBA - April 15.
**Salih Azgin**(METU-NCC)

**"On Non-extremal Hanselian Valued Fields" (notes)**TBA - April 22.
**Özgür Kişisel**(METU-NCC)

**"Toric Varieties" (notes)**Introduction to toric varieties. Regular morphisms, rational morphisms, rational varieties, and toric varieties. Uses of toric varieties. - April 29.
**Özgür Kişisel**(METU-NCC)

**"Toric Varieties II" (notes)**Definitions and examples. Pictures of toric varieties and blowups. End with a rough description of some current research. - May 6.
**Özgür Kişisel**(METU-NCC)

**"Toric Varieties III" (notes)**Definitions and examples. Pictures of toric varieties and blowups. End with a rough description of some current research.

Fall 2010

*(Click to Show)*- October 8.
**Benjamin Walter**(METU-NCC)

**"Lie algebras via the Lie algebra configuration pairing"**I will describe a way of understanding free Lie algebras using preLie algebras and graph algebras which generalizes and simplifies the standard method of embedding a Lie algebra in its universal enveloping algebra. I will also outline connections with shuffle algebras, giving new ways of understanding them, as well. The ideas presented have roots in algebraic topology computations of Dev Sinha in 2006, as well as rational homotopy extensions of these by Sinha and myself in subsequent years; however, the precise framing I will describe (purely in algebra) is due to myself and is the product of work this summer. - October 15. (Postponed to Oct 22)
**Benjamin Walter**(METU-NCC)

**"The Lie algebra configuration basis"**I will give an application of the framework presented in the previous seminar, giving a new basis for free Lie algebras and using the configuration pairing to write Lie bracket expressions in terms of this basis. Time-permitting, I will show how to make Grobner basis calculations using this basis in general (non-free) Lie algebras. - October 29.
**HOLIDAY** - November 5.
**Salih Azgın**(METU-NCC)

**"An introduction to Model Theory" (notes)**Basic model theory terminology, definitions, and theorems will be introduced: First order languages, L-structures and L-formulas, elementary equivalences, definable sets, completeness, compactness thoerem, Ax-Kochen and Ershov's Theorem. I will end with Artin's conjecture about solutions of degree*d*forms in*d*variables over rational^{2}+1*p*-adics. - November 12.
**HOLIDAY** - November 19.
**HOLIDAY** - November 26.
**Salih Azgın**(METU-NCC)

**"Model Theory and Valued Fields" (notes)**Valued fields are fields equipped with a valuation compatible with the field structure. I will introduce their value groups and residue fields; equal and mixed characteristic cases; Hahn fields; Henselian fields; and valued fields as first order structures. Time will be spent on classical examples. In this talk I will also give a more detailed description of the Ax-Kochen, Ershov theorem, give an example of its failure in positive characteristic, and introduce the open question of determining the first-order theory of F_p((t)). - December 3.
**Salih Azgın**(METU-NCC)

**"Model Theory and Valued Difference Fields" (notes)**Valued difference fields are valued fields with a compatible automorphism σ. I will introduce also Hahn difference fields, and investigate an Ax-Kochen, Ershov type theorem in this setting. Classical examples are Witt vectors over the closure of F_p with a lifting of the Frobenius automorphism; and R((t))^{LE}, the "log-exp closure of R((t))". Ideas in the previous lecture generalize to Henselian configurations, σ-polynomials, and σ-Henselianity. I will give an Ax-Kochen type theorem in this setting (in the σ-Henselian case for contractive σ), and relate valued fields with positive characteristic to valued difference fields. In this relation, Kaplansky fields correspond to σ-Henselian fields. - December 31.
**HOLIDAY** - December 10. (Postponed to Dec 24)
**Özcan Kasal**(METU-NCC)

**"Ultraproducts and the Compactness Theorem"**