{"id":12238,"date":"2021-03-22T13:14:08","date_gmt":"2021-03-22T17:14:08","guid":{"rendered":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=12238"},"modified":"2021-03-22T13:14:08","modified_gmt":"2021-03-22T17:14:08","slug":"new-spaces-in-mathematics-and-physics","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=12238","title":{"rendered":"New Spaces in Mathematics and Physics"},"content":{"rendered":"<p>Available online today (if your institution is paying&#8230;) from Cambridge University Press are two volumes well-worth spending some time with: <a href=\"https:\/\/www.cambridge.org\/core\/books\/new-spaces-in-mathematics\/2AB1C65DD7F83F5BA2605E8411FDD271\">New Spaces in Mathematics<\/a> and <a href=\"https:\/\/www.cambridge.org\/core\/books\/new-spaces-in-physics\/7CE95474C203AE0A19D66CBE35139D38\">New Spaces in Physics<\/a>. These contain write-ups based on a workshop organized back in 2015 by Mathieu Anel and Gabriel Catren, the videos of which are available <a href=\"https:\/\/www.youtube.com\/playlist?list=PLRxtDuSeiaXYy17D56Era8ns3V4vmWmqE\">here<\/a>.<\/p>\n<p>It would be hard to write in any detail about the wealth of material in these volumes, so I&#8217;ll mainly just link to the essays that seemed especially interesting to me:<\/p>\n<ul>\n<li><a href=\"https:\/\/www.cambridge.org\/core\/books\/new-spaces-in-mathematics\/microlocal-analysis-and-beyond\/A871DFC02C06ACB3B6DA5A72F267DCF5\">Microlocal analysis and beyond<\/a> by Pierre Schapira.<\/li>\n<li><a href=\"https:\/\/www.cambridge.org\/core\/books\/new-spaces-in-mathematics\/spaces-as-infinitygroupoids\/48A62497ACF942C1DF089742C162C776\">Spaces as infinity groupoids<\/a> by Timothy Porter.<\/li>\n<li><a href=\"https:\/\/www.cambridge.org\/core\/books\/new-spaces-in-mathematics\/sheaves-and-functors-of-points\/946DA9D1F78E334A1B23DBAC1206AC4C\">Sheaves and functors of points<\/a> by Michel Vaqui\u00e9.<\/li>\n<li><a href=\"https:\/\/www.cambridge.org\/core\/books\/new-spaces-in-mathematics\/stacks\/21A55519C20E092107074FF3AAD09018\">Stacks<\/a> by Nicole Mestano and Carlos Simpson.<\/li>\n<li><a href=\"https:\/\/www.cambridge.org\/core\/books\/new-spaces-in-mathematics\/geometry-of-ambiguity-an-introduction-to-the-ideas-of-derived-geometry\/0AEA017057D54B951C38021E3C2FAC39\">The geometry of ambiguity: an introduction to the ideas of derived geometry<\/a> by Mathieu Anel.<\/li>\n<li><a href=\"https:\/\/www.cambridge.org\/core\/books\/new-spaces-in-mathematics\/geometry-in-dgcategories\/3761AC14DE8AE1CDB3712CA02DADE468\">Geometry in dg-categories<\/a> by Maxim Kontsevich.<\/li>\n<li><a href=\"https:\/\/www.cambridge.org\/core\/books\/new-spaces-in-physics\/noncommutative-geometry-the-spectral-standpoint\/AE8065DF6E92842107454C3124FF4AE1\">Noncommutative geometry, the spectral standpoint<\/a> by Alain Connes.<\/li>\n<li><a href=\"https:\/\/www.cambridge.org\/core\/books\/new-spaces-in-physics\/supergeometry-in-mathematics-and-physics\/A862F1F476B22BE3757EB73BBD5CE3CD\">Supergeometry in mathematics and physics<\/a> by Mikhail Kapranov.<\/li>\n<li><a href=\"https:\/\/www.cambridge.org\/core\/books\/new-spaces-in-physics\/derived-stacks-in-symplectic-geometry-damien-calaque\/D9BBB810420783F6D6188899E66AC762\">Derived stacks in symplectic geometry<\/a> by Damien Calaque.<\/li>\n<li><a href=\"https:\/\/www.cambridge.org\/core\/books\/new-spaces-in-physics\/twistor-theory-a-geometric-perspective-for-describing-the-physical-world\/0325B5F0E4CD116F31DE1C9CFF0D296B\">Twistor theory: a geometric perspective for describing the physical world<\/a> by Roger Penrose.<\/li>\n<\/ul>\n<p>For several decades now one often hears from prominent theoretical physicists that &#8220;Space-time is doomed&#8221;, to be imminently replaced by something new coming out of the latest ideas about fundamental physics. For a long time the claims of this sort getting the most attention were from string theorists, and in these volumes Marcos Mari\u00f1o explains these in his <a href=\"https:\/\/www.cambridge.org\/core\/books\/new-spaces-in-physics\/stringy-geometry-and-emergent-space\/92A97E8BB63284F5A453A1FE05982646\">Stringy geometry and emergent space<\/a>. More recently, Nima Arkani-Hamed has been making well-publicized claims along these lines, with space-time to be replaced with volumes of objects in Grassmanians such as <a href=\"https:\/\/www.quantamagazine.org\/physicists-discover-geometry-underlying-particle-physics-20130917\/\">the amplitudehedron<\/a>.<\/p>\n<p>A large fraction of the theory community is now working on things like <a href=\"https:\/\/www.simonsfoundation.org\/mathematics-physical-sciences\/it-from-qubit\/\">&#8220;it from qubit&#8221;<\/a>, which propose to somehow get space-time emergent out of things like qubits or quantum information theory. For most of this kind of thing, I&#8217;ve found it hard to figure out exactly what the proposal is for the more fundamental objects from which space-time is supposed to emerge. One <a href=\"https:\/\/arxiv.org\/abs\/2103.09780\">recent extreme proposal, by Sean Carroll<\/a>, has the virtue of specifying what the object is (a self-adjoint matrix acting on a complex vector space), but I don&#8217;t think there&#8217;s a plausible route from that to our observed physics.<\/p>\n<p>As many of the articles linked to above should make clear, mathematicians have over the past centuries developed a range of deep and surprising ideas about new sorts of ways to think about space and geometry. This activity continues: Peter Scholze&#8217;s perfectoid spaces and condensed mathematics are examples of new directions of this kind, too new to make it into these volumes.<\/p>\n<p>Of all of these ideas, the ones that at the moment I find most compelling are the twistor geometry ideas of Roger Penrose, and I&#8217;ll have much more to say about those in another blog post soon.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Available online today (if your institution is paying&#8230;) from Cambridge University Press are two volumes well-worth spending some time with: New Spaces in Mathematics and New Spaces in Physics. These contain write-ups based on a workshop organized back in 2015 &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=12238\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[13],"tags":[],"class_list":["post-12238","post","type-post","status-publish","format-standard","hentry","category-book-reviews"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/12238","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=12238"}],"version-history":[{"count":4,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/12238\/revisions"}],"predecessor-version":[{"id":12242,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/12238\/revisions\/12242"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=12238"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=12238"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=12238"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}