[{"content":"","date":null,"permalink":"https://sayan1729.github.io/posts/","section":"Blog","summary":"","title":"Blog"},{"content":"","date":null,"permalink":"https://sayan1729.github.io/tags/free-groups/","section":"Tags","summary":"","title":"Free-Groups"},{"content":"","date":null,"permalink":"https://sayan1729.github.io/tags/geometric-group-theory/","section":"Tags","summary":"","title":"Geometric-Group-Theory"},{"content":"","date":null,"permalink":"https://sayan1729.github.io/tags/growth-of-groups/","section":"Tags","summary":"","title":"Growth-of-Groups"},{"content":" Automorphism, Geometry and the Growth of Groups #Quasi isometry\nGroup generators induce metric\nquasi isometry invariant properties: growth (focus) dehn fn virtual freeness (focus) hyperbolicity \u0026hellip;\nfree group Nielsen rank formula Cayley graph metric\nEx: \\(\\left(\\mathbb{Z}, \\langle1\\rangle\\right) \\sim_{\\text{QI}} \\left(\\mathbb{Z}, \\langle2,3\\rangle\\right)\\)\nGrowth:\nExponential intermediate (Grigorchuk; open problem: finitely presented but intermediate growth?) polynomial (Gromov \\(\\implies\\) nilpotent) Milnor conjecture (disproved by Grigorchuk) Milnor-Schwarz Thm $$ \\mathbb{R/Z} \\sim S^1 $$ \\(\\mathbb{R}^n\\) is hyperbolic iff \\(n = 1\\) (Gromov thin hyperbolicity)\nNielsen transformation \\(\\text{Aut}(F_n)\\)\nGGT Martin Bridson Bible of GGT Gromov Hyperbolicity\n","date":"28 July 2026","permalink":"https://sayan1729.github.io/posts/how-free-is-free/","section":"Blog","summary":"Growth and virtual freeness of a group.","title":"How free is free ?"},{"content":"I’m a fourth year maths undergrad at Jadavpur University, primarily interested in number theory and arithmetic geometry.\n","date":null,"permalink":"https://sayan1729.github.io/","section":"Sayan Das","summary":"","title":"Sayan Das"},{"content":"","date":null,"permalink":"https://sayan1729.github.io/tags/","section":"Tags","summary":"","title":"Tags"},{"content":"","date":null,"permalink":"https://sayan1729.github.io/tags/virtual-freeness/","section":"Tags","summary":"","title":"Virtual-Freeness"},{"content":" Structure of finite-dimensional semisimple Lie algebras #Let \\(\\mathbb{F}\\) be a field. A vector space \\(\\mathfrak{g}\\) over \\(\\mathbb{F}\\) is called a Lie algebra if there exists an \\(\\mathbb{F}\\)-bilinear map \\(\\mathfrak{g\\times g\\to g}\\) \\((x,y)\\mapsto [x,y]\\) (called the commutator/Lie bracket) such that\n\\([x,x]=0\\) (alternating) \\([x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0\\) (Jacobi identity) for all \\(x,y,z\\in\\mathfrak{g}.\\) Lie algebras are a class of non-associative algebras - the Jacobi identity controls how non-associative the Lie bracket is.\nNote: if \\(\\text{char}(\\mathbb F)\\neq2\\) then \\([x,x]=0\\;\\forall x\\in\\mathfrak g\\) is equivalent to \\([x,y]=-[y,x]\\;\\forall x,y\\in\\mathfrak g.\\) This is because $$[x,y]=[x,y]+0=[x,y]-[y+x,y+x]$$ $$=[x,y]-[y,x]-[y,y]-[x,x]-[x,y]=-[y,x].$$ On the other hand, \\([x,y]=-[y,x]\\implies[x,x]=-[x,x]\\implies 2[x,x]=0\\implies\\text{char}(\\mathbb F)=2\\) or \\([x,x]=0.\\)\nDefinition. A vector subspace \\(s\\) of \\(\\mathfrak{g}\\) is called a\nsubalgebra if \\([x,y]\\in s\\) for all \\(x,y\\in s,\\) and ideal if \\(x\\in\\mathfrak{g},\\) \\(y\\in s\\implies [x,y]\\in s.\\) A Lie algebra \\(\\mathfrak{g}\\) is called\nabelian if \\([x,y]=0\\) for all \\(x,y\\in s,\\) simple if \\(\\{0\\}\\) and \\(\\mathfrak{g}\\) are the only ideals of \\(\\mathfrak{g},\\) and semisimple if \\(\\mathfrak{g}\\) is a direct sum of its simple ideals. Example.\nAny vector space \\(V\\) is an abelian Lie algebra. \\(\\mathbb{R}^3\\) is a Lie algebra with \\([x,y]=\\vec{x}\\times \\vec{y}.\\) Any associative algebra \\(A\\) can be made into a Lie algebra by defining \\([a,b]=ab-ba.\\) The affine Lie algebras # \u003c?xml version=\"1.0\" encoding=\"UTF-8\"?\u003e Theorem. Root generated subalgebras are in bijective correspondence with closed subroot systems. Proof. cf. Biswas-Habib-Venkatesh1.\nD. Biswas, I. Habib, R. Venkatesh, On symmetric closed subsets of real affine root systems, Journal of Algebra, 628 (2023) 212-240.\u0026#160;\u0026#x21a9;\u0026#xfe0e;\n","date":"12 June 2024","permalink":"https://sayan1729.github.io/posts/lie-groups/","section":"Blog","summary":"Lie groups, lie algebras and representation theory.","title":"Lie Groups and Representation Theory"},{"content":"","date":null,"permalink":"https://sayan1729.github.io/tags/lie-algebra/","section":"Tags","summary":"","title":"Lie-Algebra"},{"content":"","date":null,"permalink":"https://sayan1729.github.io/tags/lie-group/","section":"Tags","summary":"","title":"Lie-Group"},{"content":"","date":null,"permalink":"https://sayan1729.github.io/tags/representation-theory/","section":"Tags","summary":"","title":"Representation-Theory"},{"content":"","date":null,"permalink":"https://sayan1729.github.io/tags/semi-simple-lie-algebra/","section":"Tags","summary":"","title":"Semi-Simple-Lie-Algebra"},{"content":" I\u0026rsquo;m Sayan, a 4th year maths undergrad at Jadavpur University, primarily interested in number theory and arithmetic geometry. In particular, I\u0026rsquo;m interested in in automorphic forms, subconvexity of \\(L\\)-functions, equidistribution, sieve theory, elliptic curves and Iwasawa theory.\nIn July 2025 I was a Summer Research Intern at the ISI Kolkata supervised by Prof. Satadal Ganguly.\nLately I\u0026rsquo;ve been reading Rudin\u0026rsquo;s Real \u0026amp; Complex Analysis, Jacobson\u0026rsquo;s Basic Algebra 2, Goldstein\u0026rsquo;s Classical Mechanics, Willard\u0026rsquo;s General Topology, Samuel\u0026rsquo;s Algebraic Theory of Numbers, Fr\\(\\\"\\text{o}\\)hlich \u0026amp; Taylor\u0026rsquo;s Algebraic Number Theory, and Atiyah \u0026amp; MacDonald\u0026rsquo;s Introduction to Commutative Algebra.\n","date":null,"permalink":"https://sayan1729.github.io/about/","section":"Sayan Das","summary":"","title":"About me"},{"content":"","date":null,"permalink":"https://sayan1729.github.io/categories/","section":"Categories","summary":"","title":"Categories"},{"content":" Exposition # Selberg\u0026rsquo;s Elementary Proof of the Prime Number Theory Internship report. Advisor: Prof. Satadal Ganguly.\nFermat\u0026rsquo;s Last Theorem for Regular Primes Article written for JU Math Society magazine.\nTalks and Presentations # Selberg\u0026rsquo;s Elementary Proof of the Prime Number Theory Internship report presentation. 1 April 2026. Jadavpur University. Notes #Here\u0026rsquo;s a collection of lecture notes I\u0026rsquo;ve taken over the years.\nJadavpur University #Semester 7 # Measure Theory Functional Analysis General Mechanics Advanced Topology PDF Module Theory PDF Past Semesters # Field Theory and Canonical Forms of Matrices PDF Group Theory Linear Algebra PDF Ring Theory Algebraic Number Theory (NPTEL) PDF ","date":null,"permalink":"https://sayan1729.github.io/notes/","section":"Sayan Das","summary":"","title":"Research"}]