Applications of formal groups in algebraic topology, number theory, and algebraic geometry

2012 ◽  
pp. 427-477
Keyword(s):  
Number Theory ◽  
Algebraic Geometry ◽  
Algebraic Topology ◽  
Formal Groups

The Applications of Group Theory

2012 ◽  
Vol 430-432 ◽  
pp. 1265-1268
Author(s):  
Xiao Qiang Guo ◽  
Zheng Jun He
Keyword(s):  
Number Theory ◽  
Algebraic Geometry ◽  
Group Theory ◽  
Algebraic Number ◽  
Algebraic Topology ◽  
Polynomial Equation ◽  
Algebraic Number Theory ◽  
Simple Groups ◽  
Finite Simple Groups

Since the classification of finite simple groups completed last century, the applications of group theory are more and more widely. We first introduce the connection of groups and symmetry. And then we respectively introduce the applications of group theory in polynomial equation, algebraic topology, algebraic geometry , cryptography, algebraic number theory, physics and chemistry.

THE MAIN TRENDS OF ALGEBRAIC TOPOLOGY AND ALGEBRAIC GEOMETRY

1964 ◽  
Vol 19 (6) ◽  
pp. 67-73
Author(s):  
S P Novikov ◽  
I I Pyatetskii-Shapiro ◽  
I R Shafarevich
Keyword(s):  
Algebraic Geometry ◽  
Algebraic Topology

scholarly journals Formal groups and Z -entropies

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160143 ◽  
Author(s):  
Piergiulio Tempesta
Keyword(s):  
Algebraic Topology ◽  
Formal Group ◽  
Point Of View ◽  
Trace Form ◽  
Formal Groups ◽  
Form Class ◽  
New Family ◽  
Mathematical Properties ◽  
Composition Process ◽  
Group Law

We shall prove that the celebrated Rényi entropy is the first example of a new family of infinitely many multi-parametric entropies. We shall call them the Z-entropies . Each of them, under suitable hypotheses, generalizes the celebrated entropies of Boltzmann and Rényi. A crucial aspect is that every Z -entropy is composable (Tempesta 2016 Ann. Phys. 365 , 180–197. ( doi:10.1016/j.aop.2015.08.013 )). This property means that the entropy of a system which is composed of two or more independent systems depends, in all the associated probability space, on the choice of the two systems only. Further properties are also required to describe the composition process in terms of a group law. The composability axiom, introduced as a generalization of the fourth Shannon–Khinchin axiom (postulating additivity), is a highly non-trivial requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis entropy are the only known composable cases. However, in the non-trace form class, the Z -entropies arise as new entropic functions possessing the mathematical properties necessary for information-theoretical applications, in both classical and quantum contexts. From a mathematical point of view, composability is intimately related to formal group theory of algebraic topology. The underlying group-theoretical structure determines crucially the statistical properties of the corresponding entropies.

scholarly journals THE UNITY AND IDENTITY OF DECIDABLE OBJECTS AND DOUBLE-NEGATION SHEAVES

2018 ◽  
Vol 83 (04) ◽  
pp. 1667-1679
Author(s):  
MATÍAS MENNI
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Algebraic Topology ◽  
Full Subcategory ◽  
Sufficient Conditions ◽  
Double Negation ◽  
Synthetic Differential Geometry ◽  
Simplicial Sets ◽  
Image Position

AbstractLet ${\cal E}$ be a topos, ${\rm{Dec}}\left( {\cal E} \right) \to {\cal E}$ be the full subcategory of decidable objects, and ${{\cal E}_{\neg \,\,\neg }} \to {\cal E}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity ${\cal E} \to {\cal S}$ for the two subcategories of ${\cal E}$ above, making them Adjointly Opposite. Typical examples of such ${\cal E}$ include many ‘gros’ toposes in Algebraic Geometry, simplicial sets and other toposes of ‘combinatorial’ spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry.

scholarly journals Dirichlet and Poincaré series

1985 ◽  
Vol 27 ◽  
pp. 39-56 ◽  
Author(s):  
A. Good
Keyword(s):  
Functional Equation ◽  
Number Theory ◽  
Algebraic Geometry ◽  
Entire Function ◽  
Cusp Form ◽  
Modular Forms ◽  
Modular Group ◽  
Positive Proportion ◽  
Fundamental Discriminant ◽  
Image Position

The study of modular forms has been deeply influenced by famous conjectures and hypotheses concerningwhere T(n) denotes Ramanujan's function. The fundamental discriminant Δ is a cusp form of weight 12 with respect to the modular group. Its associated Dirichlet seriesdefines an entire function of s and satisfies the functional equationThe most penetrating statements that have been made on T(n) and LΔ(s)are:Of these four problems only A1 has been established so far. This was done by Deligne [1] using methods from algebraic geometry and number theory. While B1 trivially holds with ε > 1/2, it was established in [2] for every ε>1/3. Serre [12] proved A2 for a positive proportion of the integers and Hafner [5] showed that LΔ has a positive proportion of its non-trivial zeros on the line σ=6. The proofs of the last three results are largely analytic in nature.

scholarly journals Tōhoku

2015 ◽  
Vol 1 (3) ◽  
Author(s):  
Rick Jardine
Keyword(s):  
Algebraic Geometry ◽  
Algebraic Topology ◽  
Turning Point ◽  
Homological Algebra ◽  
Modern Mathematics

Rick Jardine evaluates Alexander Grothendieck’s 1957 mathematical masterpiece, Some Aspects of Homological Algebra, a turning point in homological algebra, algebraic topology and algebraic geometry, and for modern mathematics as a whole.

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