{"id":392,"date":"2025-11-27T13:02:30","date_gmt":"2025-11-27T06:02:30","guid":{"rendered":"https:\/\/algebra.math-ugm.id\/?p=392"},"modified":"2025-11-27T13:02:30","modified_gmt":"2025-11-27T06:02:30","slug":"online-lecture-series-on-algebraic-geometry-2","status":"publish","type":"post","link":"https:\/\/algebra.math-ugm.id\/en\/online-lecture-series-on-algebraic-geometry-2\/","title":{"rendered":"Online Lecture Series on Algebraic Geometry"},"content":{"rendered":"

<\/p>\n

Lecturer:<\/strong> Prof. Bharath Sethuraman<\/em>
Emeritus Professor of Mathematics, California State University, Northridge (USA)<\/p>\n

Organized by:<\/strong> Algebra Research Group,
Department of Mathematics, Universitas Gadjah Mada<\/p>\n

Dates:<\/strong> 3rd, 10th, 17th, and 24th November 2025
Platform:<\/strong> Zoom<\/p>\n

The Algebra Research Group of Universitas Gadjah Mada hosted a four-week online lecture series delivered by Prof. Bharath Sethuraman.<\/strong>\u00a0The lectures attracted participants from various universities, offering a clear and accessible introduction to fundamental concepts in algebraic geometry and their deep connections to modern applications.<\/p>\n

\"\"\"\"<\/p>\n

Prof. Sethuraman began the series by introducing the zero set of a polynomial<\/strong> and explaining its relation to the affine space<\/strong>. He discussed the structure of polynomials over the complex numbers \u2102 and developed the notion of algebraic sets<\/strong> along with the ideals<\/strong> defining them. He then presented Hilbert\u2019s Nullstellensatz<\/strong>, emphasizing its central role in establishing the correspondence between ideals and varieties. Before concluding the first session, he highlighted foundational topics in commutative algebra<\/strong>\u2014such as integral extensions, Noether normalization, projective spaces, and projective varieties\u2014that would be developed in subsequent lectures. He also mentioned a broad range of applications of algebraic geometry<\/strong>, including physics, cryptography, error-correcting codes, robotics, computer graphics, and biology.<\/p>\n

\n

\"\"The second lecture opened with a brief review of the key ideas from the previous week. Prof. Sethuraman then transitioned to the concept of projective space<\/strong>, defining it as the set of lines passing through the origin. He illustrated this using visual examples in 3-dimensional space<\/strong>, showing how classical conic sections \u2014 circles, ellipses, hyperbolas, and parabolas \u2014 can be viewed as the same geometric object in the projective plane<\/strong> P^2(R)<\/span><\/span>. Near the end of the lecture, he introduced the notion of integral ring extensions<\/strong>, recalling definitions and fundamental results to prepare for deeper discussions.<\/p>\n

\"\"\"\" \"\"\u00a0<\/span><\/p>\n

Responding to participant requests, Prof. Sethuraman opened the third lecture with a more detailed explanation of projective spaces<\/strong>, especially P2(C)<\/span><\/span>.
He shared a philosophical perspective often taught in geometry courses:<\/p>\n

\n

\u201cMoral: In P^2(C)<\/span><\/span>, the fundamental objects are lines through the origin in C^3<\/span><\/span>. If we want to represent these fundamental objects as points \u2014 as our human intuition prefers \u2014we examine where these lines intersect the planes x_0=1, <\/span><\/span>x_1 = 1, <\/span><\/span>x_2<\/span>=<\/span><\/span>1<\/span><\/span><\/span><\/span>. Each plane captures most of the space, and together, they capture all of P^2(C)<\/span><\/span>.\u201d<\/strong><\/p>\n<\/blockquote>\n

The lecture then moved to a clearer explanation of homogeneous polynomials<\/strong>, prompted by a question from Prof. Indah Emilia Wijayanti. Prof. Sethuraman discussed how homogenization ties affine and projective geometry together. He concluded by returning to integral extensions<\/strong> and introduced the celebrated Noether Normalization Theorem<\/strong>, laying the groundwork for the final lecture.<\/p>\n

\"\" \"\" \"\"<\/p>\n

\n

In the final lecture, Prof. Sethuraman began with the statement and proof of the Weak Nullstellensatz<\/strong>, ensuring participants gained a conceptual and computational understanding. He then progressed to the Strong Nullstellensatz<\/strong>, demonstrating the proof using Rabinowitsch\u2019s trick<\/strong>, a clever algebraic method widely used in textbooks. With remaining time, he revisited the Noether Normalization Theorem<\/strong>, providing a complete proof along with concrete examples to clarify its importance in algebraic geometry and commutative algebra.<\/p>\n

\"\"<\/p>\n

<\/p>","protected":false},"excerpt":{"rendered":"

Lecturer: Prof. Bharath SethuramanEmeritus Professor of Mathematics, California State University, […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-392","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/posts\/392","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/comments?post=392"}],"version-history":[{"count":1,"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/posts\/392\/revisions"}],"predecessor-version":[{"id":416,"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/posts\/392\/revisions\/416"}],"wp:attachment":[{"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/media?parent=392"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/categories?post=392"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/tags?post=392"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}