{"id":16364,"date":"2022-09-29T15:47:47","date_gmt":"2022-09-29T15:47:47","guid":{"rendered":"http:\/\/45.119.114.148\/iisernew\/?page_id=16364"},"modified":"2022-11-15T11:07:12","modified_gmt":"2022-11-15T11:07:12","slug":"harshitha-c-report2","status":"publish","type":"page","link":"https:\/\/iiser.ntc-us.com\/iisernew\/pmrf\/harshitha-c-report2\/","title":{"rendered":"Harshitha C Report2"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"16364\" class=\"elementor elementor-16364\">\n\t\t\t\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9993d25 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9993d25\" data-element_type=\"section\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4dcb733\" data-id=\"4dcb733\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e9c5eff elementor-widget elementor-widget-heading\" data-id=\"e9c5eff\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.10.0 - 09-01-2023 *\/\n.elementor-heading-title{padding:0;margin:0;line-height:1}.elementor-widget-heading .elementor-heading-title[class*=elementor-size-]>a{color:inherit;font-size:inherit;line-height:inherit}.elementor-widget-heading .elementor-heading-title.elementor-size-small{font-size:15px}.elementor-widget-heading .elementor-heading-title.elementor-size-medium{font-size:19px}.elementor-widget-heading .elementor-heading-title.elementor-size-large{font-size:29px}.elementor-widget-heading .elementor-heading-title.elementor-size-xl{font-size:39px}.elementor-widget-heading .elementor-heading-title.elementor-size-xxl{font-size:59px}<\/style><h1 class=\"elementor-heading-title elementor-size-default\">WRITE-UP FOR PMR FELLOWSHIP FROM FEBRUARY 2021 - OCTOBER 2021<\/h1>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-73f513d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"73f513d\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-799139d\" data-id=\"799139d\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2ca0ea3 elementor-widget elementor-widget-image\" data-id=\"2ca0ea3\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.10.0 - 09-01-2023 *\/\n.elementor-widget-image{text-align:center}.elementor-widget-image a{display:inline-block}.elementor-widget-image a img[src$=\".svg\"]{width:48px}.elementor-widget-image img{vertical-align:middle;display:inline-block}<\/style>\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"150\" height=\"193\" src=\"https:\/\/iiser.ntc-us.com\/iisernew\/wp-content\/uploads\/2022\/06\/harshita.jpg\" class=\"attachment-large size-large wp-image-4189\" alt=\"Harshitha C Profile pic\" loading=\"lazy\" style=\"width:100%;height:128.67%;max-width:150px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-1057aa3\" data-id=\"1057aa3\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4de5359 elementor-widget elementor-widget-heading\" data-id=\"4de5359\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Harshitha C<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fe1da0c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fe1da0c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1a237fb\" data-id=\"1a237fb\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-8ffd6e9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8ffd6e9\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-wide\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-inner-column elementor-element elementor-element-94901e8\" data-id=\"94901e8\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7e79534 elementor-widget elementor-widget-text-editor\" data-id=\"7e79534\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.10.0 - 09-01-2023 *\/\n.elementor-widget-text-editor.elementor-drop-cap-view-stacked .elementor-drop-cap{background-color:#818a91;color:#fff}.elementor-widget-text-editor.elementor-drop-cap-view-framed .elementor-drop-cap{color:#818a91;border:3px solid;background-color:transparent}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap{margin-top:8px}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap-letter{width:1em;height:1em}.elementor-widget-text-editor .elementor-drop-cap{float:left;text-align:center;line-height:1;font-size:50px}.elementor-widget-text-editor .elementor-drop-cap-letter{display:inline-block}<\/style>\t\t\t\t<p>Our main focus during the past one year was to learn foundational material related to the theory of representations of p-adic groups, the subject area of our interest. As a part of this, we undertook several reading projects.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-352faca elementor-widget elementor-widget-text-editor\" data-id=\"352faca\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><strong><em>1.\u00a0 Representations of finite groups:\u00a0<\/em><\/strong>Basics of the representation theory of finite groups were covered. We also studied the construction of all complex irreducible representations of the group GL2(F) where F is a finite field.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-214d3e6 elementor-widget elementor-widget-text-editor\" data-id=\"214d3e6\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><strong><em>2. Representations of compact groups:\u00a0<\/em><\/strong>We studied the theory of representations of compact groups. Our main aim was to understand the proof of Peter-Weyl theorem which showed that the space of matrix coefficients is dense in the space of continuous functions on a compact group.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f2e4f09 elementor-widget elementor-widget-text-editor\" data-id=\"f2e4f09\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><strong>3.\u00a0<em>Basic algebraic number theory:<\/em><\/strong> We studied the basic results of classical algebraic number theory. After getting a basic understanding of the structure of the ring of integers of an algebraic number field, we studied some famous fundamental theorems in the subject like unique factorization of ideals in the ring of integers, finiteness of class number, and Dirichlet\u2019s unit theorem. We next turned our attention to the local theory. We studied the structure, classification and properties of locally compact non-discrete fields (local fields).<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-886f3fb elementor-widget elementor-widget-text-editor\" data-id=\"886f3fb\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><strong><em>4. Tate\u2019s thesis:\u00a0<\/em><\/strong>As an introduction to the theory of automorphic forms, we read Tate\u2019s thesis. We began with the study of dual of locally compact groups and their properties. This was applied to understand in particular the dual of local fields and the adele group of global fields. The isomorphism obtained on the Schwartz-Bruhat spaces via the Fourier transform was used to prove the analytic continuation and functional equation of the local and global zeta functions as defined by Tate in his thesis. This was used to give an alternate proof of the functional equation and analytic continuation of classical zeta functions considered by Hecke. As a consequence, analytic continuation and functional equations of some well-known L-functions like Riemann zeta function, Dedekind zeta function, etc were obtained.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-443551d elementor-widget elementor-widget-text-editor\" data-id=\"443551d\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><strong><em>5. Representations of p-adic groups: <\/em><\/strong>We started studying the complex representations of <em>l-<\/em>groups, which are locally compact Hausdorff groups that have a fundamental system of neighborhoods of the identity consisting of compact and open subgroups. After looking into the generalities in the subject like the notion of smooth and admissible representations and their basic properties, we studied the properties of two fundamental functors namely smooth induction and Jacquet functor. We also studied the proof that every smooth irreducible representation of a general linear group over a local field is admissible which uses the above-mentioned properties.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-26ea070 elementor-widget elementor-widget-heading\" data-id=\"26ea070\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Future plan:<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-734325a elementor-widget elementor-widget-text-editor\" data-id=\"734325a\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>We plan to continue studying the theory of representations of p-adic groups, in particular, the representation theory of the general linear group over a non-archimedean local field F. In particular, we would like to understand the Zelevinksy and Langlands classification of irreducible admissible representations of GLn(F).<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d2c2455 elementor-widget elementor-widget-heading\" data-id=\"d2c2455\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\">Courses attended:<\/h4>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7cb4be3 elementor-widget elementor-widget-text-editor\" data-id=\"7cb4be3\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul><li>MTH 611- Algebra-1<\/li><li>MTH 612- Analysis-1<\/li><li>MTH 613- Topology-1<\/li><li>MTH 621- Algebra-2<\/li><li>MTH 623- Topology-2<\/li><li>MTH 627- Differential Geometry<\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6bb8573 elementor-widget elementor-widget-heading\" data-id=\"6bb8573\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\">Teaching assistantship:<\/h4>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-44abdb3 elementor-widget elementor-widget-text-editor\" data-id=\"44abdb3\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul><li>MTH 211- Multivariable Calculus<\/li><li>MTH 221- Probability and Statistics<\/li><li>MTH 314- Linear Algebra<\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>he structure of the ring of integers of an algebraic number field, we studied some famous fundamental theorems in the subject like unique factorization of ideals in the ring of integers, finiteness of class number, and Dirichlet\u2019s unit theorem. We next turned our attention to the local theory. We studied the structure, classification and properties&hellip; <a class=\"more-link\" href=\"https:\/\/iiser.ntc-us.com\/iisernew\/pmrf\/harshitha-c-report2\/\">Continue reading <span class=\"screen-reader-text\">Harshitha C Report2<\/span><\/a><\/p>\n","protected":false},"author":4,"featured_media":7974,"parent":4129,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_expiration-date-status":"","_expiration-date":0,"_expiration-date-type":"","_expiration-date-categories":[],"_expiration-date-options":[]},"acf":[],"_links":{"self":[{"href":"https:\/\/iiser.ntc-us.com\/iisernew\/wp-json\/wp\/v2\/pages\/16364"}],"collection":[{"href":"https:\/\/iiser.ntc-us.com\/iisernew\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/iiser.ntc-us.com\/iisernew\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/iiser.ntc-us.com\/iisernew\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/iiser.ntc-us.com\/iisernew\/wp-json\/wp\/v2\/comments?post=16364"}],"version-history":[{"count":10,"href":"https:\/\/iiser.ntc-us.com\/iisernew\/wp-json\/wp\/v2\/pages\/16364\/revisions"}],"predecessor-version":[{"id":17357,"href":"https:\/\/iiser.ntc-us.com\/iisernew\/wp-json\/wp\/v2\/pages\/16364\/revisions\/17357"}],"up":[{"embeddable":true,"href":"https:\/\/iiser.ntc-us.com\/iisernew\/wp-json\/wp\/v2\/pages\/4129"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/iiser.ntc-us.com\/iisernew\/wp-json\/wp\/v2\/media\/7974"}],"wp:attachment":[{"href":"https:\/\/iiser.ntc-us.com\/iisernew\/wp-json\/wp\/v2\/media?parent=16364"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}