\u7bc4\u7587\u5316 (categorification) \u662f\u8fd1\u4e16\u6578\u5b78\u88e1\u9762\u5f88\u91cd\u8981\u7684\u65b9\u6cd5\u8ad6\u3002\u7bc4\u7587\u5316\u5c31\u597d\u50cf\u540d\u5075\u63a2\u67ef\u5357\u7684\u9ed1\u8863\u4eba\u4e00\u6a23\uff0c\u5728\u8a31\u591a\u9818\u57df\u8af8\u5982\u6578\u5b78\u7269\u7406\uff0c\u62d3\u6a38\u5b78\uff0c\u8868\u73fe\u7406\u8ad6\uff0c\u751a\u81f3\u7d44\u5408\u5b78\uff0c\u90fd\u80fd\u898b\u5230\u5b83\u9ed1\u5e55\u4e2d\u7684\u8eab\u5f71\u533f\u4f0f\u5728\u96e3\u984c\u4e4b\u5074\uff0c\u4f3a\u6a5f\u4e0b\u624b\u3002\u6211\u5011\u4eca\u5929\u60f3\u8981\u804a\u804a\u8207\u62d3\u6a38\u5b78\u6709\u95dc\u7684\u7bc4\u7587\u5316\u4f8b\u5b50\u3002<\/p>\n
\u90a3\u4ec0\u9ebc\u662f\u7bc4\u7587\u5316\u5462\uff1f\u5927\u9ad4\u800c\u8a00\uff0c\u7576\u4f60\u624b\u4e0a\u6709\u4e00\u500b\u727d\u6d89\u5230\u6578\u5b78\u7d50\u69cb A \u7684\u554f\u984c\uff0c\u5728 A \u88e1\u9762\u4e0d\u5bb9\u6613\u7406\u89e3\uff0c\u7136\u800c\u6709\u5929\u4f60\u767c\u73fe\u5230\u9019\u500b A \u7684\u80cc\u5f8c\u5b58\u5728\u4e00\u500b\u66f4\u6df1\u523b\u7684\u6578\u5b78\u7d50\u69cb A<\/u> \uff0c\u7528\u52a0\u5e95\u7dda\u4f86\u8868\u793a\uff0c\u5177\u5099\u5982\u4e0b\u7279\u6027\uff1a \u9996\u5148\u7d50\u69cb A \u53ef\u4ee5\u5f88\u76f4\u89c0\u5730\u770b\u6210\u7d50\u69cb A<\/u> \u7684\u5316\u7c21\uff0c\u4e5f\u5c31\u662f\u5b58\u5728 A \u5230 A \u7684\u5c0d\u61c9\u95dc\u4fc2\u4fdd\u6301\u5169\u8005\u7d50\u69cb\uff0c\u4e14\u7d50\u69cb A \u88e1\u9762\u7684\u554f\u984c\u80fd\u88ab\u63d0\u5347\u6210\u7d50\u69cb A<\/u> \u88e1\u9762\u7684\u554f\u984c\uff08\u8acb\u5225\u554f\u6211\u70ba\u4ec0\u9ebc\u52a0\u4e86\u5e95\u7dda\u53eb\u505a\u63d0\u5347\uff0c\u6211\u53ea\u80fd\u8aaa\u4e2d\u6587\u5f88\u96e3\uff09\u3002 \u518d\u8005\uff0c\u96d6\u7136\u770b\u4e0a\u53bb\u7d50\u69cb A<\/u> \u8981\u8003\u616e\u7684\u4e8b\u60c5\u6bd4\u8f03\u591a\uff0c\u7b49\u65bc\u8aaa\u9700\u8981\u6aa2\u9a57\u7684\u908f\u8f2f\u9673\u8ff0\u6bd4\u8f03\u591a\uff0c\u4f46\u662f\u5f9e\u6211\u5011\u4eba\u985e\u7684\u773c\u795e\u89c0\u4f86\uff0c\u7d50\u69cb A<\/u> \u7684\u6a5f\u5236\u7adf\u7136\u662f\u6bd4\u8f03\u81ea\u7136\uff0c\u4e5f\u8b93\u6211\u5011\u6709\u6a5f\u6703\u5f15\u5165\u66f4\u591a\u7684\u5de5\u5177\u4f86\u91d0\u6e05\u6240\u5c6c\u554f\u984c\u3002<\/p>\n
\u7528\u79d1\u5b78\u53f2\u4e0a\u7684\u7d93\u5178\u6848\u4f8b\u4f86\u985e\u6bd4\u7684\u8a71\uff0c\u7bc4\u7587\u5316\u5c31\u50cf\u662f\u6d77\u68ee\u5821\u8207\u73bb\u6069\u7814\u7a76\u6c2b\u539f\u5b50\u5149\u8b5c\u80fd\u968e\u6a21\u578b\u7684\u6642\u5019\uff0c\u767c\u73fe\u5230\u5c07\u4e0d\u540c\u968e\u7684\u5149\u8b5c\u767c\u5c04\u5438\u6536\u983b\u7387\u9069\u7576\u6392\u5217\u6210\u65b9\u9663\u5f8c\uff0c\u5176\u6578\u91cf\u95dc\u4fc2\u53ef\u4ee5\u7528\u7dda\u6027\u4ee3\u6578\u77e9\u9663\u7684\u4e58\u6cd5\u4f86\u8868\u793a\uff0c\u5efa\u7acb\u4e86\u5f8c\u4e16\u7a31\u4f5c\u77e9\u9663\u529b\u5b78\u7684\u91cf\u5b50\u529b\u5b78\u57fa\u790e\u3002\u9019\u88e1\u7684 A \u5c31\u597d\u6bd4\u5be6\u969b\u53ef\u4ee5\u89c0\u6e2c\u5230\u7684\u53e4\u5178\u7269\u7406\u91cf\uff0c\u800c A<\/u> \u662f\u7b97\u5b50\u7b26\u865f\u69cb\u6210\u7684\u91cf\u5b50\u7269\u7406\u91cf\uff0cA<\/u> \u5230 A \u7684\u5c0d\u61c9\u95dc\u4fc2\u5247\u662f\u77e9\u9663\u5728\u91cf\u5b50\u614b\u4e0a\u7684\u6295\u5f71\u9577\u5ea6\uff0c\u800c\u9019\u500b\u5c0d\u61c9\u95dc\u4fc2\u5c07\u77e9\u9663\u9593\u7684\u904b\u7b97\u5316\u7c21\u70ba\u53e4\u5178\u7269\u7406\u91cf\u9075\u5faa\u7684\u5b9a\u5f8b\u6240\u9700\u7684\u904b\u7b97\u3002\u96d6\u7136\u5f9e\u53e4\u5178\u529b\u5b78\u5230\u91cf\u5b50\u529b\u5b78\u9019\u500b\u53cd\u63a8\u904e\u7a0b\uff0c\u5176\u6578\u5b78\u6a21\u578b\u4e26\u975e\u6578\u5b78\u56b4\u683c\u610f\u7fa9\u4e0a\u7684\u7bc4\u7587\u5316\uff0c\u4f46\u7cbe\u795e\u5df2\u7d93\u975e\u5e38\u8fd1\u4f3c\u4e86\u3002<\/p>\n
\u56de\u5230\u6578\u5b78\u672c\u8eab\u3002\u6578\u5b78\u6700\u57fa\u672c\u7684\u7d50\u69cb\u5c0d\u8c61\u5c31\u662f\u6b63\u6574\u6578\u69cb\u6210\u7684\u6578\u5b57\uff0c\u6578\u5b57\u7684\u52a0\u6cd5\u8207\u4e58\u6cd5\u5c31\u662f\u4ed6\u7684\u7d50\u69cb\u3002\u56de\u60f3\u4e00\u4e0b\u6700\u521d\u6211\u5011\u662f\u600e\u9ebc\u4f7f\u7528\u6578\u5b57\u7684\uff1a\u7528\u4ed6\u4f86\u6578\u6578 (count)\u3002 \u90a3\u9ebc\u300c\u88ab\u6578\u6578\u300d\u7684\u5c0d\u8c61\u5c31\u53ef\u4f5c\u70ba\u6578\u5b57\u7684\u7bc4\u7587\u5316\uff0c\u9019\u4e9b\u5c0d\u8c61\u7684\u8853\u8a9e\u5c31\u53eb\u505a\u96c6\u5408 (set)\u3002 \u90a3\u9ebc\u6211\u5011\u5c31\u7522\u751f\u4e86\u5c0d\u61c9\u95dc\u4fc2\uff1a\u7d66\u5b9a\u4e00\u500b\u96c6\u5408\uff0c\u4ed6\u7684\u5143\u7d20\u500b\u6578\u5c31\u662f\u6578\u5b57\uff0c\u9019\u88e1\u5c0d\u61c9\u65bc\u52a0\u6cd5\u8207\u4e58\u6cd5\u7684\u7d50\u69cb\u5c31\u662f\u96c6\u5408\u7684\u806f\u96c6\u8207\u4e58\u7a4d\u3002\u4f46\u770b\u5230\u9019\u88e1\u4f60\u4e00\u5b9a\u6703\u60f3\uff0c\u96c6\u5408\u672c\u8eab\u662f\u5e36\u6709\u5197\u9918\u7684\u7d50\u69cb\uff0c\u4f8b\u5982\u96c6\u5408 {X,Y,Z} \u8207\u96c6\u5408 {x,y,z} \u770b\u4e0d\u51fa\u4f86\u6709\u4ec0\u9ebc\u5206\u5225\uff0c\u4f46\u4ed6\u5011\u90fd\u5c0d\u61c9\u5230 3 \u9019\u500b\u6578\u5b57\uff0c\u53c8\u4f8b\u5982\u96c6\u5408 {X,Y,Z} \u8ddf {Y,Z,X} \u90fd\u4ee3\u8868\u540c\u4e00\u500b\u96c6\u5408\uff0c\u53ea\u662f\u5143\u7d20\u5beb\u51fa\u9806\u5e8f\u4e0d\u540c\u800c\u5df2\u3002\u70ba\u4e86\u89e3\u6c7a\uff0c\u6211\u5011\u53ef\u4ee5\u66f4\u9032\u4e00\u6b65\u628a\u7bc4\u7587\u5316 {X,Y,Z} \u5347\u7d1a\u6210\u4ee5 X,Y,Z \u70ba\u57fa\u5e95\u7684\u5411\u91cf\u7a7a\u9593 V=span[X,Y,Z]\uff0c\u5982\u6b64\u4e00\u4f86\u4e0d\u5149\u662f X,Y,Z \u7684\u9806\u5e8f\u7121\u95dc\u7dca\u8981\uff0c\u4f60\u751a\u81f3\u53ef\u4ee5\u628a X,Y,Z \u96a8\u610f(\u4f46\u4e0d\u4efb\u610f)\u5730\u63db\u6210\u53e6\u4e00\u7d44\u57fa\u5e95 X-Y,Y-Z,Z+X \uff0c\u6240\u5f35\u51fa\u7684\u5411\u91cf\u7a7a\u9593 V \u90fd\u4e0d\u8b8a\u3002 \u9019\u88e1\u7684\u5c0d\u61c9\u95dc\u4fc2\u5c31\u662f\u628a\u5411\u91cf\u7a7a\u9593\u5c0d\u61c9\u5230\u5b83\u7684\u7dad\u5ea6 (dimension)\uff0c\u4f8b\u5982\u525b\u8b1b\u7684 V \u5c0d\u61c9\u5230 3\uff0c\u800c\u7dda\u6027\u4ee3\u6578\u544a\u8a34\u6211\u5011\uff0c\u9019\u500b\u5c0d\u61c9\u95dc\u4fc2\u4e26\u4e0d\u53d7\u57fa\u5e95\u9078\u53d6\u5f71\u97ff\uff0c\u53ea\u53d6\u6c7a\u65bc\u5411\u91cf\u7a7a\u9593\u672c\u8eab\u3002<\/p>\n
\u4ee5\u4e0b\u6211\u5011\u5c31\u4ee5\u4e0d\u52d5\u9ede\u5b9a\u7406\u70ba\u4f8b\u8aaa\u660e\u5c07\u6578\u6578\u5b57\u7bc4\u7587\u5316\u70ba\u5411\u91cf\u7a7a\u9593\u7684\u7dad\u5ea6\u7684\u5999\u7528\u3002\u8003\u616e f \u70ba\u5f9e\u7dca\u7dfb\uff08\u5373\u5c01\u9589\u6709\u754c\uff09\u7a7a\u9593 M \u5230\u81ea\u5df1\u7684\u9023\u7e8c\u6620\u5c04\uff0c\u4e0d\u52d5\u9ede\u6307\u7684\u662f\u6eff\u8db3 f(x)=x \u7684\u9019\u4e9b\u9ede x \u3002\u4e0d\u52d5\u9ede\u5b9a\u7406\u65b7\u8a00\u5728\u67d0\u4e9b\u60c5\u6cc1\u4e0b\uff0cM \u81ea\u8eab\u7684\u62d3\u6a38\u6027\u8cea\u8db3\u4ee5\u5f70\u986f\u4e0d\u52d5\u9ede\u7684\u5b58\u5728\u6027\u3002\u4f8b\u5982\u7576 M \u70ba\u55ae\u4f4d\u5713\u76e4\u6642\uff0c f \u70ba\u65cb\u8f49\uff0c\u90a3\u9ebc\u539f\u9ede\u5373\u70ba f \u7684\u4e0d\u52d5\u9ede\u3002\u4f46\u5982\u679c\u53d6 f \u70ba\u7b49\u6bd4\u4f8b\u7e2e\u6c34\u5230\u534a\u5f91 \u00bd \u5f8c\u518d\u5e73\u79fb\u2153 \uff0c\u5c31\u4e0d\u5bb9\u6613\u7528\u773c\u775b\u770b\u51fa\u9019\u6a23\u7684 f \u662f\u5426\u5177\u5099\u4e0d\u52d5\u9ede\u3002\u7d93\u5178\u7684\u840a\u592b\u8b1d\u8328(Lefschetz)\u4e0d\u52d5\u9ede\u5b9a\u7406\u505a\u4e86\u5982\u4e0b\u7684\u8003\u616e\uff0c\u65e2\u7136 f \u662f\u5f9e M \u5230 M \u7684\u6620\u5c04\uff0c\u90a3\u9ebc f \u4e5f\u8a98\u5c0e\u51fa H_j(M) \u5230H_j(M) \u7684\u7dda\u6027\u6620\u5c04\uff0c\u9019\u88e1 H_j(M) \u662f M \u7684 j \u7dad\u540c\u8abf\u7fa4\uff0cj \u5f9e 0 \u5230 m (M\u7684\u7dad\u5ea6\uff09\uff0c\u662f\u53ea\u548c M \u7684\u62d3\u6a38\u6709\u95dc\u7684\u5411\u91cf\u7a7a\u9593\u3002f \u7684\u4e0d\u52d5\u9ede\u53ef\u4ee5\u770b\u6210\u662f MxM \u9019\u500b\u7a7a\u9593\u4e2d\u5c0d\u89d2\u7dda\u8207 f \u7684\u5716\u5f62\u7684\u4ea4\u6703\u8655\uff0c\u82e5\u4ee5\u5411\u91cf\u7a7a\u9593\u4f86\u8aaa\uff0c\u4e0d\u52d5\u9ede\u7684\u300c\u7bc4\u7587\u5316\u300d\u53ef\u4ee5\u7528 f \u5728 H_j(M) \u7684\u8de1 (trace) T_j \u4f86\u8868\u793a\u3002\u5982\u679c\u53d6 H_j(M) \u7684\u4e00\u7d44\u57fa\u5e95\uff0c\u5c07 f \u8868\u793a\u6210\u77e9\u9663\u7684\u8a71\uff0c\u90a3\u9ebc T_j \u5c31\u662f\u5c0d\u89d2\u7dda\u4e0a\u7684\u5143\u7d20\u4e4b\u548c\u4e14\u8207\u57fa\u5e95\u7684\u9078\u53d6\u7121\u95dc\u3002\u4ee4\u840a\u592b\u8b1d\u8328\u6578 L = T_0 – T_1 + T_2 – T_3 \u2026. +- T_m \u70ba\u8af8\u8de1\u4ea4\u932f\u548c\uff0c\u840a\u592b\u8b1d\u8328\u4e0d\u52d5\u9ede\u5b9a\u7406\u65b7\u8a00\uff0c\u82e5 L \u4e0d\u7b49\u65bc\u96f6\uff0c\u5247 f \u5fc5\u5b58\u5728\u4e0d\u52d5\u9ede\u3002\u9019\u88e1 L \u626e\u6f14\u7684\u89d2\u8272\u76f8\u7576\u65bc\u6240\u6709\u4e0d\u52d5\u9ede\u7684\u52a0\u6b0a\u7e3d\u548c\u3002\u7576 M \u70ba\u5713\u76e4\u6642\uff0c\u6709 H_1(M)=H_2(M)=0 \uff0cH_0(M) \u70ba\u4e00\u7dad\u5411\u91cf\u7a7a\u9593\u4e14 f \u8a98\u5c0e\u51fa\u975e\u96f6\u7684\u6578\u91cf\u4e58\u7a4d\uff0c\u65bc\u662f L = T_0 -0+0 = T_0 \u70ba\u975e\u96f6\u6578\uff0c\u9019\u6642\u5019\u7684\u840a\u592b\u8b1d\u8328\u4e0d\u52d5\u9ede\u5b9a\u7406\u53eb\u505a\u5e03\u52de\u5a01\u723e(Brouwer)\u4e0d\u52d5\u9ede\u5b9a\u7406\uff0c\u4ed6\u65b7\u8a00\u5713\u76e4\u4e0a\u7684\u4efb\u610f\u9023\u7e8c\u81ea\u6620\u5c04 f \u5fc5\u9808\u5b58\u5728\u4e0d\u52d5\u9ede\u3002<\/p>\n
\u5982\u679c\u8aaa\u62d3\u6a38\u5b78\u662f\u7814\u7a76\u5f62\u72c0\u5728\u5e7e\u4f55\u9023\u7e8c\u8b8a\u63db\u4e0b\u7684\u6027\u8cea\uff0c\u90a3\u9ebc\u8f9b\u62d3\u6a38(symplectic topology)\u5c31\u662f\u7814\u7a76\u7269\u7406\u76f8\u7a7a\u9593\u4e2d\u7684\u5f62\u72c0\u5728\u6b63\u5247\u8b8a\u63db(canonical transform)\u4e0b\u7684\u6027\u8cea\uff0c\u9019\u88e1\u7684\u6b63\u5247\u8b8a\u63db\u6307\u7684\u662f\u4fdd\u6301\u7d66\u5b9a\u529b\u5b78\u5b9a\u5f8b(\u4f8b\u5982\u7d66\u5b9a\u7684\u54c8\u5bc6\u9813\u529b\u5b78\u7cfb\u7d71)\u7684\u5e7e\u4f55\u8b8a\u63db\u3002\u6b63\u5247\u8b8a\u63db\u4e00\u822c\u4f86\u8aaa\u662f\u6703\u6539\u8b8a\u91cf\u6e2c\u8ddd\u96e2\u7684\u65b9\u5f0f\uff0c\u4f7f\u5f97\u7121\u6cd5\u6709\u6548\u4f7f\u7528\u66f2\u7387\u7b49\u5e7e\u4f55\u5de5\u5177\u4f86\u8861\u91cf\u4ed6\u7684\u9b06\u6563\u7a0b\u5ea6\uff0c\u4f46\u6b63\u5247\u8b8a\u63db\u53c8\u5177\u5099\u81f3\u5c11\u4fdd\u9ad4\u7a4d\u7684\u7279\u8cea\uff0c\u6bd4\u4e00\u822c\u62d3\u6a38\u5b78\u7814\u7a76\u7684\u540c\u80da\u8b8a\u63db (homeomorphism) \u9084\u8981\u5805\u97cc\uff0c\u53ef\u4ee5\u8aaa\u662f\u4ecb\u65bc\u5e7e\u4f55\u5b78\u8ddf\u62d3\u6a38\u5b78\u4e4b\u9593\u65e2\u525b\u4e14\u67d4\u7684\u7814\u7a76\u5c0d\u8c61\u3002\u5728\u9019\u9818\u57df\u4e2d\uff0c\u7b46\u8005\u7684\u8ca2\u737b\u662f\u63d0\u4f9b\u4e86\u4e00\u500b\u7bc4\u7587\u5316\u7684\u65b9\u6848\uff0c\u8b93\u90a3\u4e9b\u300c\u6709\u9ad4\u7a4d\u300d\u7684\u5f62\u72c0\u4e5f\u80fd\u88ab\u7bc4\u7587\u5316 (\u5728\u6b64\u4e4b\u524d\u7684\u7814\u7a76\u4e3b\u6d41\u662f\u5c07\u4e00\u985e\u7279\u5b9a\u7684\u62c9\u683c\u6717\u65e5\u5b50\u6d41\u5f62\u7bc4\u7587\u5316\uff0c\u4f46\u62c9\u683c\u6717\u65e5\u5b50\u6d41\u5f62\u662f\u6c92\u6709\u9ad4\u7a4d\u7684) \u3002 \u7c21\u8a00\u4e4b\uff0c\u6211\u5011\u80fd\u5c07\u5f62\u72c0\u5c0d\u61c9\u5230\u67d0\u500b\u80cc\u666f\u5c64\u5c0e\u51fa\u7bc4\u7587\uff08\u53ef\u4ee5\u60f3\u6210\u5411\u91cf\u7a7a\u9593\u7684\u62bd\u8c61\u5347\u7d1a\uff09\u7684\u5b50\u7bc4\u7587\uff08\u60f3\u6210\u5b50\u7a7a\u9593\uff09\uff0c\u4e26\u4e14\u7576\u9019\u500b\u5f62\u72c0\u7d93\u904e\u6b63\u5247\u8b8a\u63db\u5f8c\uff0c\u6211\u5011\u80fd\u5920\u63cf\u8ff0\u65b0\u7684\u5f62\u72c0\u5c0d\u61c9\u5230\u7684\u5b50\u7bc4\u7587\u5982\u4f55\u96a8\u4e4b\u8b8a\u5316\uff08\u60f3\u6210\u5411\u91cf\u7a7a\u9593\u7684\u57fa\u5e95\u8b8a\u63db\uff09\u3002\u300c\u5c0e\u51fa\u300d\u7684\u610f\u601d\u662f\uff0c\u6211\u5011\u8003\u616e\u7684\u7269\u4ef6\u662f\u5982\u540c\u4e00\u689d\u934a\u5b50\u822c\u6210\u4e32\u7684\u5411\u91cf\u7a7a\u9593\u5c64 M_0 -> M_1 -> M_2 -> \uff0c\u9019\u7a2e\u934a\u5b50\u7684\u7e3d\u6de8\u7dad\u5ea6\u662f\u4ea4\u932f\u548c H(M_0)-H(M_1)+H(M_2)-…. \uff0c\u4f46\u6211\u5011\u4e0d\u53bb\u62c6\u6578\u5b57\u53d6\u7e3d\u7dad\u5ea6\uff0c\u800c\u662f\u76e1\u53ef\u80fd\u4ee5\u6574\u689d\u934a\u5b50\u70ba\u55ae\u4f4d\u9032\u884c\u904b\u7b97\u8207\u51fd\u6578\u6620\u5c04\u3002\u66f4\u91cd\u8981\u7684\u662f\uff0c\u6211\u5011\u80fd\u5920\u5efa\u7acb\u4ee5\u5b50\u7bc4\u7587\u70ba\u76ee\u6a19\u7684\u6295\u5f71\u51fd\u5b50\uff08\u60f3\u6210\u6295\u5f71\u77e9\u9663\uff09\uff0c\u9019\u500b\u6295\u5f71\u542b\u5b50\u7684\u4ee3\u6578\u6027\u8cea\u860a\u542b\u4e86\u80fd\u88ab\u7a31\u4e4b\u70ba\u5f62\u72c0\u4e4b\u7279\u5fb5\u7684\u4e0d\u8b8a\u91cf(symplectic invariant)\u3002\u9019\u4e9b\u4e0d\u8b8a\u91cf\u53ef\u6bd4\u689d\u78bc\uff0c\u5f9e\u689d\u78bc\u80fd\u8b80\u51fa\u8a72\u5f62\u72c0\u7684\u76f8\u95dc\u8cc7\u8a0a\u3002\u4e8b\u5be6\u4e0a\uff0c\u9019\u4e9b\u4e0d\u8b8a\u91cf\u4e5f\u78ba\u5be6\u53ef\u4ee5\u7528\u62d3\u6a38\u8cc7\u6599\u5206\u6790\u5b78(topological data analysis)\u7684\u689d\u78bc(barcode)\u7684\u5f62\u5f0f\u5beb\u4e0b\u3002\u4f8b\u5982\u7576 R>r \u6642\uff0c\u53d6\u5f62\u72c0\u70ba\u76f8\u7a7a\u9593\u4e2d\u7684\u7403\u9ad4(\u534a\u5f91 R )\u8207\u6a62\u7403\u9ad4(\u534a\u5f91\u70ba r \u8207 \u7121\u7aae\u5927)\u6642\uff0c\u7d93\u7531\u8a08\u7b97\u9019\u4e9b\u689d\u78bc\uff0c\u6211\u5011\u53ef\u4ee5\u77e5\u9053\u8a72\u7403\u9ad4\u4e0d\u53ef\u80fd\u88ab\u6b63\u5247\u904b\u7b97\u585e\u9032\u67d0\u4e9b\u5c3a\u5bf8\u7684\u6a62\u7403\u9ad4\uff0c\u9019\u5c31\u63d0\u4f9b\u4e86\u8f9b\u62d3\u6a38\u9818\u57df\u91cc\u7a0b\u7891\u7684\u683c\u7f85\u83ab\u592b\u6297\u64e0\u58d3\u5b9a\u7406 (Gromov\u2019s non-squeezing theorem) \u7684\u4e00\u500b\u7bc4\u7587\u5316\u8b49\u660e\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u7bc4\u7587\u5316 (categorification) \u662f\u8fd1\u4e16\u6578\u5b78\u88e1\u9762\u5f88\u91cd\u8981\u7684\u65b9\u6cd5\u8ad6\u3002\u7bc4\u7587\u5316\u5c31\u597d\u50cf\u540d\u5075\u63a2\u67ef\u5357\u7684\u9ed1\u8863\u4eba\u4e00\u6a23 […]<\/p>\n","protected":false},"author":1,"featured_media":19521,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[990,983],"tags":[],"acf":[],"_links":{"self":[{"href":"https:\/newsletter.sinica.edu.tw\/wp-json\/wp\/v2\/posts\/19496"}],"collection":[{"href":"https:\/newsletter.sinica.edu.tw\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/newsletter.sinica.edu.tw\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/newsletter.sinica.edu.tw\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/newsletter.sinica.edu.tw\/wp-json\/wp\/v2\/comments?post=19496"}],"version-history":[{"count":0,"href":"https:\/newsletter.sinica.edu.tw\/wp-json\/wp\/v2\/posts\/19496\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/newsletter.sinica.edu.tw\/wp-json\/wp\/v2\/media\/19521"}],"wp:attachment":[{"href":"https:\/newsletter.sinica.edu.tw\/wp-json\/wp\/v2\/media?parent=19496"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/newsletter.sinica.edu.tw\/wp-json\/wp\/v2\/categories?post=19496"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/newsletter.sinica.edu.tw\/wp-json\/wp\/v2\/tags?post=19496"}],"curies":[{"name":"wp","href":"https:\/api.w.org\/{rel}","templated":true}]}}