#lie group

[geometrymatters]

The Neurogeometry of Perception: A Journey into Geometric Cognition

In the realm of cognitive science and neurology, there exists a fascinating intersection where geometry meets perception, aptly termed “neurogeometry”. This interdisciplinary field seeks to understand how our brains process and interpret the visual world through geometric structures and patterns. Alessandro Sarti and Giovanna Citti, prominent figures in this domain, have extensively explored the fundamental principles of neurogeometry, uncovering the intricate relationship between the architecture of our brains and the geometric forms we perceive.

“Neurogeometry” is not merely a fusion of “neuroscience” and “geometry”. It’s an ambitious endeavor to model the functional architecture of the primary visual cortex and understand how geometric patterns underpin our visual processing. As described by Sarti and Citti,

“We remind some basic principles of the neurogeometrical approach as it has been proposed by various researchers to model the functional architecture of the primary visual cortex.”

This statement underscores the comprehensive nature of the approach and its foundational importance in cognitive science. The very essence of neurogeometry lies in its quest to unravel the architectural blueprint of our perceptual processes. Our brains, complex and intricate, are not just passive receivers of visual stimuli. Instead, they actively construct a coherent understanding of the world through geometric frameworks. Every curve we perceive, every angle we discern, and every spatial relationship we recognize is a testament to the brain’s inherent ability to process the world geometrically. Neurogeometry, therefore, serves as a bridge, connecting the abstract realm of geometric shapes and patterns to the tangible reality of neural processes.

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[futuregrooves]

Subtyl - Lie Group

[curiosamathematica]

Teake Nutma, The Exceptional Symmetry of E8.

This movie attempts to show the beautiful symmetry of the exceptional Lie group E8. Out of all the known Lie groups, E8 stands out as the largest and most complex exceptional group. It has 248 generating elements, which by themselves have an astounding degree of symmetry. This symmetry can only be fully grasped in 8-dimensional space. But luckily it is also possible to project E8 onto a two-dimensional plane, chosen such that the resulting image preserves a small fraction of its total symmetry. There are different choices for these projections, some preserving more symmetry than others. The movie rotates through a selection of projections in succession.

(Source: geometrymatters)

[tablekat]

Lie algebra of strings

In case you haven't noticed, if you've read some of my previous posts, I've been thinking a lot about strings (of characters) lately, and what I can do with them mathematically. Obviously the strings together with concatenation form a monoid. I've been trying to think about what else I can do with them. Firstly, as in my last post, I showed that it's possible to make an infinite dimensional Hilbert space of strings, which I tried to explain in a way clear enough to people who don't exactly know what a Hilbert space is (including myself). It's pretty important to make a vector space out of strings if I want to do any cool stuff with them, because all the cool stuff tends to use vector spaces.

If we take the vector space that I defined previously, then we already have vector addition and inner products defined on the strings. However, neither of these are anything special about strings, and we could've just as easily numbered each dimension with a natural number. However there is one special thing that stand out about strings, and that is string concatenation.

Since string concatenation forms a monoid, it seemed pretty obvious to look around for things that are also monoids. Something that particularly stands out is the set of all real square matrices of size n, Mn(ℝ). This set includes all matrices including ones with zero determinant, which is what makes it different from the general linear group GLn(ℝ). Matrices with zero determinant have no inverse, and therefore Mn(ℝ) does not form a group. However, these two sets of matrices have an interesting relationship: Mn(ℝ) is the Lie algebra of GLn(ℝ)! This led me to the idea of trying to make a Lie algebra for the monoid of strings.

First, I'll explicitly show that the strings form a monoid with concatenation:

Closure: the concatenation of two strings will always form a new string. $$ a\cdot b \in S^*, \forall a,b \in S^* $$

Associativity: string concatenation is associative (but not commutative) $$ a \cdot (b \cdot c) = (a \cdot b) \cdot c, \forall a,b,c \in S^* $$

Identity: the empty string acts as the identity. $$ a\cdot "" = "" \cdot a = a, \forall a \in S^*  $$

Given this definition as multiplication, we can define a Lie bracket, where we use fraktur s as the vector space for the Lie algebra:

$$ [\cdot, \cdot] : \mathfrak{s}^* \times \mathfrak{s}^* \to \mathfrak{s}^* $$

$$ [a, b] = a \cdot b - b \cdot a $$

As an example:

$$ ["what", "meow"] = "whatmeow" - "meowwhat" $$

This lie bracket is alternating and anticommutative:

$$ ["what","what"] = "whatwhat" - "whatwhat" = 0 $$

$$ ["wh","at"] = "what" - "atwh" = -["at","wh"] $$

It is bilinear:

$$ [\alpha x + \beta y, z] = \alpha [x,z] + \beta [y,z] $$

$$ [5 "wh" + 3 "c", "at"] = (5 "what" + 3 "cat") - (5 "atwh" + 3 "atc) $$

$$ 5 ["wh","at"]  + 3 ["c","at"] = 5 ("what" - "atwh") - 3 ("cat" - "atc") $$

The second and third lines are equal, which is exactly what we wanted to show.

And finally, incredibly enough, the Lie bracket satisfies the Jacobi identity:

$$ [x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0 $$

If we let \\( x = "x", y = "b", z = "zz" \\) as an example if you weren't convinced already:

$$ [y,z] = ("bzz" - "zzb") $$

$$ [x,y] = ("xb" - "bx") $$

$$ [z,x] = ("zzx" - "xzz") $$

Then plugging those into the Jacobi identity, we get the following sum:

$$ ("xbzz" - "xzzb" - "bzzx" + "zzbx") + $$

$$ ("zzxb" - "zzbx" - "xbzz" + "bxzz") + $$

$$ ("bzzx" - "bxzz" - "zzxb" + "xzzb") $$

$$ \underline{= 0} $$

As for now, that's all I've got. I don't know if this idea would lead to anything particularly profound. I'll think about it some more. Maybe it would be interesting to define a Lie group given this definition, which shouldn't be too hard. I may also look into other algebras like Jordan algebras over the strings, just cause I find stuff like this fun for some reason.

[powermodules]

In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice. Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum electrodynamics,quantum chromodynamics (QCD) and the Standard Model. Non-perturbative gauge theory calculations in continuous spacetime formally involve evaluating an infinite-dimensional path integral, which is computationally intractable. By working on a discrete spacetime, the path integral becomes finite-dimensional, and can be evaluated by stochastic simulation techniques such as the Monte Carlo method. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum gauge theory is recovered.[1]

In lattice gauge theory, the spacetime is Wick rotated into Euclidean space and discretized into a lattice with sites separated by distance  and connected by links. In the most commonly considered cases, such as lattice QCD, fermion fields are defined at lattice sites (which leads to fermion doubling), while the gauge fields are defined on the links. That is, an element U of the compact Lie group G is assigned to each link. Hence to simulate QCD, with Lie group SU(3), there is a 3×3 special unitary matrix defined on each link. The link is assigned an orientation, with the inverse element corresponding to the same link with the opposite orientation.

[powermodules]

Yang–Mills theory is a gauge theory based on the SU(N) group, or more generally any compact, semi-simple Lie group. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-Abelian Lie groups and is at the core of the unification of the Weak and Electromagnetic force (i.e. U(1) × SU(2)) as well as Quantum Chromodynamics, the theory of the Strong force (based on SU(3)). Thus it forms the basis of our current understanding of particle physics, the Standard Model.

[powermodules]

In mathematics, a Lie group /ˈliː/ is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups.

Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (differential Galois theory), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.

Lie groups are smooth manifolds and, therefore, can be studied using differential calculus, in contrast with the case of more general topological groups. One of the key ideas in the theory of Lie groups is to replace the global object, the group, with its local or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra.

Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold. On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are especially important, and are studied in representation theory.

[arpitgoliya]

The Lie Group and the Universe - I

Groups capture the essence of symmetry. Theory of groups is being used in a wide range of fields - determining possible structure of crystals , conservation laws of physics, string theory and even in sending secure information over internet! So what is the relationship between the lie group, the universe in which we live and the ancient Indian theories?

E8

Well according to one of the 'theory of everything' by Garrett Lisi, our universe can be described by a special Lie Group with root lattice of rank 8 . A 2-D projection of E8 may describes our new universe as per the new theory. And if you look at the figure , it is pretty much similar to the pictures that is presented in ancient scriptures. I specially like description of a lotus-base shaped universe, as described in the Vedic texts. And this is how an E8 looks (2-D projection)

E8- 2D projection Similar to Lotus-Base shape description

See More on E8 @ http://www.aimath.org/E8/ [Images used above have been taken from the internet and are used only for the sake of explanation. No copyright violation is intended. In case of any issues please leave a comment]