𑐱𑑂𑐬𑐷𑐬𑐵𑐩 𑐱𑑄𑐎𑐬 𑐀𑐨𑐫𑐒𑑂𑐎𑐬

𑐱𑑂𑐬𑐷𑐬𑐵𑐩 𑐱𑑄𑐎𑐬 𑐀𑐨𑐫𑐒𑑂𑐎𑐬 (𑐖𑐸𑐮𑐵𑐃 𑑒𑑒, 𑑑𑑙𑑓𑑐 – 𑐣𑑀𑐨𑐾𑐩𑑂𑐧𑐬 𑑒, 𑑒𑑐𑑑𑑒) 𑐕𑐩𑑂𑐴 𑐴𑐮𑐶𑐩𑐫𑑂 𑐣𑐵𑑄𑐖𑐵𑐩𑑂𑐴 𑐨𑐵𑐬𑐟𑐷𑐫-𑐀𑐩𑐾𑐬𑐶𑐎𑐷 𑐐𑐞𑐶𑐟𑐖𑑂𑐘 𑐏, 𑐐𑐸𑐩𑑂𑐴𑐾𑐳𑑂𑐫𑐵𑑄 𑐧𑐷𑐖𑐐𑐞𑐶𑐟𑐷𑐫 𑐖𑑂𑐫𑐵𑐩𑐶𑐟𑐶 (Algebraic Geometry) 𑐫𑐵 𑐏𑑂𑐫𑐮𑐫𑑂 𑐎𑑂𑐬𑐵𑐣𑑂𑐟𑐶𑐎𑐵𑐬𑐷 𑐖𑑂𑐫𑐵 𑐫𑐵𑐣𑐵𑐡𑐶𑐮। 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐖𑐣𑑂𑐩 𑐨𑐵𑐬𑐟𑐫𑐵 𑐩𑐢𑑂𑐫 𑐥𑑂𑐬𑐡𑐾𑐱 𑐬𑐵𑐖𑑂𑐫𑐫𑐵 𑐄𑐖𑑂𑐖𑐿𑐣𑐫𑑂 𑐕𑐐𑐹 𑐥𑐬𑐩𑑂𑐥𑐬𑐵𑐐𑐟 𑐰𑐶𑐡𑑂𑐰𑐵𑐣 𑐥𑐬𑐶𑐰𑐵𑐬𑐫𑑂 𑐖𑐹𑐐𑐸 𑐏 । 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐀𑐧𑐸 𑐕𑐩𑑂𑐴 𑐟𑑅𑐢𑑄𑐩𑑂𑐴 𑐳𑑄𑐳𑑂𑐎𑐺𑐟 𑐥𑐞𑑂𑐜𑐶𑐟 𑐖𑐸𑐫𑐵𑐔𑑂𑐰𑐣, 𑐐𑐸𑐎𑐶𑐫𑐵 𑐥𑑂𑐬𑐨𑐵𑐰𑑄 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐟𑐬𑑂𑐎𑐱𑐎𑑂𑐟𑐶 𑐰 𑐰𑐶𑐔𑐵𑑅𑐢𑐵𑐬𑐵𑐫𑑂 𑐖𑐷𑐰𑐣𑐨𑐬 𑐟𑑅𑐢𑑄𑐐𑐸 𑐨𑐹𑐩𑐶𑐎𑐵 𑐩𑑂𑐴𑐶𑐟𑐮। 𑐀𑐨𑐫𑐒𑑂𑐎𑐬𑐫𑐵 𑐥𑑂𑐬𑐵𑐬𑐩𑑂𑐨𑐶𑐎 𑐱𑐶𑐎𑑂𑐲𑐵 𑐐𑑂𑐰𑐵𑐮𑐶𑐫𑐬 𑐰 𑐩𑐸𑐩𑑂𑐧𑐃𑐫𑑂 𑐖𑐸𑐮, 𑐐𑐣 𑐰𑐫𑑂‌𑐎𑑅𑐣𑑄 𑐐𑐞𑐶𑐟 𑐰 𑐰𑐶𑐖𑑂𑐘𑐵𑐣𑐫𑑂 𑐠𑑅𑐐𑐸 𑐰𑐶𑐱𑐾𑐲 𑐡𑐎𑑂𑐏𑐮 𑐎𑑂𑐫𑐣𑐵𑐡𑐶𑐮। 𑐳𑐣𑑂 𑑑𑑙𑑕𑑑𑐫𑑂 𑐰𑐫𑑂‌𑐎𑐮𑑄 𑐩𑐸𑐩𑑂𑐧𑐃 𑐰𑐶𑐱𑑂𑐰𑐰𑐶𑐡𑑂𑐫𑐵𑐮𑐫𑑄 𑐐𑐞𑐶𑐟𑐫𑑂 𑐳𑑂𑐣𑐵𑐟𑐎 (B.Sc.) 𑐄𑐥𑐵𑐢𑐶 𑐥𑑂𑐬𑐠𑐩 𑐱𑑂𑐬𑐾𑐞𑐷 𑐴𑐵𑐳𑐶𑐮 𑐫𑐵𑐣𑐵𑐡𑐶𑐮। 𑐠𑑂𑐰 𑐢𑐸𑐎𑐵𑑅 𑐰𑐫𑑂‌𑐎𑑅 𑐄𑐔𑑂𑐔 𑐱𑐶𑐎𑑂𑐲𑐵𑐫𑐵 𑐣𑐶𑐟𑐶𑑄 𑐳𑑄𑐫𑐸𑐎𑑂𑐟 𑐬𑐵𑐖𑑂𑐫 𑐀𑐩𑐾𑐬𑐶𑐎𑐵𑐫𑐵 𑐥𑑂𑐬𑐏𑑂𑐫𑐵𑐟 𑐴𑐵𑐬𑑂𑐰𑐬𑑂𑐜 𑐰𑐶𑐱𑑂𑐰𑐰𑐶𑐡𑑂𑐫𑐵𑐮𑐫𑐫𑑂 𑐰𑐣𑐵𑐡𑐶𑐮। 𑐴𑐵𑐬𑑂𑐰𑐬𑑂𑐜𑐫𑑂 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵𑐟 𑑒𑑐𑐐𑐹 𑐱𑐟𑐵𑐧𑑂𑐡𑐷𑐫𑐵 𑐡𑐎𑐮𑐫𑑂 𑐥𑑂𑐬𑐨𑐵𑐰𑐱𑐵𑐮𑐷 𑐐𑐞𑐶𑐟𑐖𑑂𑐘 𑐌𑐳𑑂𑐎𑐬 𑐖𑐵𑐬𑐶𑐳𑑂𑐎𑐷 (Oscar Zariski) 𑐫𑐵 𑐳𑐵𑐩𑐷𑐥𑑂𑐫𑐫𑑂 𑐱𑑀𑐢 𑐫𑐵𑐫𑐾𑐐𑐸 𑐡𑐸𑐬𑑂𑐮𑐨 𑐴𑑂𑐰𑑅𑐟𑐵𑑅 𑐡𑐟 । 𑐳𑐣𑑂 𑑑𑑙𑑕𑑕𑐫𑑂 𑐰𑐫𑑂‌𑐎𑐮𑑄 𑐴𑐵𑐬𑑂𑐰𑐬𑑂𑐜𑑄 𑐥𑐷.𑐊𑐔.𑐜𑐷. (Ph.D.) 𑐥𑐹𑐬𑐵 𑐫𑐵𑐣𑐵𑐡𑐶𑐮, 𑐐𑐸𑐎𑐶𑐫𑐵 𑐰𑐶𑐲𑐫 '𑐎𑑂𑐫𑐵𑐬𑐾𑐎𑑂𑐚𑐬𑐶𑐳𑑂𑐚𑐶𑐎 𑐥𑐷' (characteristic p) 𑐡𑐹𑐐𑐸 𑐏𑑂𑐫𑐮𑐫𑑂 𑐧𑐷𑐖𑐐𑐞𑐶𑐟𑐷𑐫 𑐳𑐟𑐴𑐫𑐵 𑐰𑐶𑐳𑑄𑐐𑐟𑐶𑐟 𑐳𑐩𑐵𑐢𑐵𑐣 𑐫𑐵𑐫𑐾𑐐𑐸 𑐏। 𑐀𑐨𑐫𑐒𑑂𑐎𑐬𑑄 𑐠𑑅𑐐𑐸 𑐖𑐷𑐰𑐣𑐫𑐵 𑑔𑑐 𑐡𑑃 𑐳𑑂𑐰𑐫𑐵 𑐀𑐥𑑂𑐰 𑐃 𑐥𑐬𑑂𑐜𑑂𑐫𑐹 𑐰𑐶𑐱𑑂𑐰𑐰𑐶𑐡𑑂𑐫𑐵𑐮𑐫𑐫𑑂 (Purdue University) '𑐩𑐵𑐬𑑂𑐱𑐮 𑐜𑐶𑐳𑑂𑐚𑐶𑐒𑑂𑐐𑐸𑐂𑐳𑑂𑐜 𑐥𑑂𑐬𑑀𑐦𑐾𑐳𑐬'𑐫𑐵 𑐬𑐸𑐥𑐫𑑂 𑐳𑐾𑐰𑐵 𑐫𑐵𑐣𑐵𑐡𑐶𑐮। 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐀𑐣𑐸𑐳𑐣𑑂𑐢𑐵𑐣𑑄 '𑐬𑐶𑐖𑑀𑐮𑑂𑐫𑐸𑐳𑐣 𑐀𑐦 𑐳𑐶𑑄𑐐𑐸𑐮𑐵𑐬𑐶𑐚𑐷' (Resolution of Singularities) 𑐢𑐂𑐐𑐸 𑐀𑐟𑑂𑐫𑐣𑑂𑐟 𑐖𑐚𑐶𑐮 𑐐𑐞𑐶𑐟𑐷𑐫 𑐳𑐩𑐳𑑂𑐫𑐵𑐫𑐵𑐟 𑐣𑑂𑐴𑐸𑐐𑐸 𑐡𑐶𑐱𑐵 𑐧𑐶𑐮 । 𑐰𑐫𑑂‌𑐎𑑅 𑐕𑐩𑑂𑐴 𑐳𑐿𑐡𑑂𑐢𑐵𑐣𑑂𑐟𑐶𑐎 𑐐𑐞𑐶𑐟𑐖𑑂𑐘 𑐖𑐎 𑐩𑐖𑐸𑐳𑐾𑑄 𑐕𑐩𑑂𑐴 𑐎𑐸𑐱𑐮 𑐱𑐶𑐎𑑂𑐲𑐎, 𑐳𑑄𑐳𑑂𑐎𑐺𑐟 𑐥𑑂𑐬𑐾𑐩𑐷 𑐰 𑐬𑐵𑐲𑑂𑐚𑑂𑐬𑐰𑐵𑐡𑐷 𑐔𑐶𑐣𑑂𑐟𑐎 𑐣𑑄 𑐏। 𑐳𑐣𑑂 𑑑𑑙𑑙𑑘𑐫𑑂 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐐𑐞𑐶𑐟𑐷𑐫 𑐳𑐾𑐰𑐵𑐫𑐵 𑐎𑐡𑐬 𑐫𑐵𑐳𑐾𑑄 𑐨𑐵𑐬𑐟 𑐳𑐬𑐎𑐵𑐬𑑄 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵𑐟 '𑐥𑐡𑑂𑐩 𑐨𑐹𑐲𑐞' 𑐳𑐩𑑂𑐩𑐵𑐣𑑄 𑐰𑐶𑐨𑐹𑐲𑐶𑐟 𑐫𑐵𑐟। 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐖𑑂𑐫𑐵𑑄 𑐁𑐢𑐸𑐣𑐶𑐎 𑐖𑑂𑐫𑐵𑐩𑐶𑐟𑐷𑐫 𑐳𑐶𑐡𑑂𑐢𑐵𑐣𑑂𑐟𑐟𑐫𑑂𑐟 𑐕𑐐𑐹 𑐣𑑂𑐴𑐸𑐐𑐸 𑐄𑐔𑐵𑐂𑐫𑑂 𑐠𑑂𑐫𑑄𑐎𑐹𑐐𑐸 𑐡𑐸।

## 𑐩𑐸𑐏𑑂𑐫 𑐐𑐞𑐶𑐟𑐷𑐫 𑐫𑑀𑐐𑐡𑐵𑐣
𑐀𑐨𑐫𑐒𑑂𑐎𑐬𑐫𑐵 𑐡𑐎𑐮𑐫𑑂 𑐟𑑅𑐢𑑄𑐐𑐸 𑐫𑑀𑐐𑐡𑐵𑐣 𑐧𑐷𑐖𑐐𑐞𑐶𑐟𑐷𑐫 𑐖𑑂𑐫𑐵𑐩𑐶𑐟𑐶𑐫𑐵 𑐏𑑂𑐫𑐮𑐫𑑂 '𑐬𑐶𑐖𑑀𑐮𑑂𑐫𑐸𑐳𑐣 𑐀𑐦 𑐳𑐶𑑄𑐐𑐸𑐮𑐵𑐬𑐶𑐚𑐷' (Resolution of Singularities) 𑐫𑑂 𑐏𑐣𑐾𑐡𑐸। 𑐠𑑂𑐰 𑐕𑐐𑐹 𑐀𑐖𑐵𑑅𑐐𑐸 𑐳𑐩𑐳𑑂𑐫𑐵 𑐏 𑐐𑐸𑐎𑐶𑐮𑐶𑑄 𑐖𑑂𑐫𑐵𑐩𑐶𑐟𑐷𑐫 𑐳𑑄𑐬𑐔𑐣𑐵𑐫𑑂 𑐰𑐂𑐐𑐸 𑐰𑐶𑐳𑑄𑐐𑐟𑐶𑐟 (Singular points) 𑐟𑐫𑑂𑐟 𑐔𑐶𑐂𑐎𑐾𑐐𑐸 𑐖𑑂𑐫𑐵 𑐫𑐵𑐂। 𑐰𑐫𑑂‌𑐎𑐮𑑄 𑑓-𑐜𑐵𑐫𑐩𑐾𑐣𑑂𑐳𑐣𑐮 𑐳𑐟𑐴𑐫𑐵 𑐣𑐶𑐟𑐶𑑄 𑐠𑑂𑐰 𑐳𑐩𑐳𑑂𑐫𑐵 𑐳𑐩𑐵𑐢𑐵𑐣 𑐫𑐵𑐣𑐵𑐡𑐶𑐮, 𑐐𑐸𑐐𑐸 𑐄𑐐𑐸 𑐃𑐮𑐫𑑂 𑐐𑐞𑐶𑐟𑐖𑑂𑐘𑐟𑐫𑑂𑐐𑐸 𑐣𑐶𑐟𑐶𑑄 𑐟𑑅𑐢𑑄𑐐𑐸 𑐔𑐸𑐣𑑁𑐟𑐷 𑐖𑐸𑐫𑐵𑐔𑑂𑐰𑑄𑐐𑐸 𑐏 । 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐩𑐾𑐐𑐸 𑐩𑐴𑐟𑑂𑐰𑐥𑐹𑐬𑑂𑐞 𑐖𑑂𑐫𑐵 '𑐀𑐨𑐫𑐒𑑂𑐎𑐬𑐫𑐵 𑐀𑐣𑐸𑐩𑐵𑐣' (Abhyankar's Conjecture) 𑐏। 𑐠𑑂𑐰 𑐀𑐣𑐸𑐩𑐵𑐣𑑄 '𑐐𑑂𑐫𑐵𑐮𑑀𑐰𑐵 𑐠𑑂𑐫𑑀𑐬𑐷' (Galois Theory) 𑐰 '𑐎𑐨𑐬𑐶𑐒𑑂𑐐𑑂𑐳 𑐀𑐦 𑐎𑐬𑑂𑐰𑑂𑐳' (Coverings of Curves) 𑐡𑐠𑐸𑐫𑐵 𑐳𑐸𑐎𑑂𑐲𑑂𑐩 𑐳𑑂𑐰𑐵𑐥𑐹𑐫𑐵𑐟 𑐎𑑂𑐫𑐣𑐶। 𑐳𑐣𑑂 𑑑𑑙𑑙𑑔𑐫𑑂 𑐩𑐾𑐩𑐾𑐥𑐶𑑄 𑐐𑐞𑐶𑐟𑐖𑑂𑐘𑐟𑐫𑑂𑐳𑑄 𑐠𑑂𑐰 𑐀𑐣𑐸𑐩𑐵𑐣𑐫𑐵𑐟 𑐥𑑂𑐬𑐩𑐵𑐞𑐶𑐟 𑐫𑐵𑐫𑐾𑐦𑐟, 𑐐𑐸𑐎𑐶𑐫𑐵 𑐁𑐢𑐵𑐬 𑐀𑐨𑐫𑐒𑑂𑐎𑐬𑑄 𑐴𑐾 𑐟𑐫𑐵𑐬 𑐫𑐵𑐣𑐵𑐡𑐷𑐐𑐸 𑐏 । 𑐰𑐫𑑂‌𑐎𑐮𑑄 '𑐜𑐶𑐚𑐬𑐩𑐶𑐣𑐾𑐣𑑂𑐚𑐮 𑐁𑐂𑐜𑐶𑐫𑐮' (Determinantal Ideals) 𑐰 '𑐂𑐣𑑂𑐨𑐾𑐬𑐶𑐫𑐣𑑂𑐚 𑐠𑑂𑐫𑑀𑐬𑐷' (Invariant Theory) 𑐫𑑂 𑐣𑑄 𑐟𑑅𑐢𑑄𑐐𑐸 𑐖𑑂𑐫𑐵 𑐫𑐵𑐣𑐵𑐡𑐶𑐮। 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐖𑑂𑐫𑐵𑑄 𑐳𑐿𑐡𑑂𑐢𑐵𑐣𑑂𑐟𑐶𑐎 𑐥𑐎𑑂𑐲𑐫𑑂 𑐖𑐎 𑐩𑐖𑐸𑐳𑐾𑑄 𑐎𑐩𑑂𑐧𑐶𑐣𑐾𑐚𑑀𑐬𑐶𑐎𑑂𑐳 (Combinatorics) 𑐫𑑂 𑐣𑑄 𑐣𑑂𑐴𑐸𑐐𑐸 𑐰𑐶𑐔𑐵𑑅𑐟 𑐥𑐶𑐠𑐣𑐾𑐐𑐸 𑐖𑑂𑐫𑐵 𑐫𑐵𑐟।

### 𑐐𑐞𑐶𑐟𑐷𑐫 𑐖𑑂𑐫𑐵𑐫𑐵 𑐳𑑄𑐎𑑂𑐲𑐾𑐥 𑐢𑐮𑑅

## 𑐥𑐬𑑂𑐜𑑂𑐫𑐹 𑐰𑐶𑐱𑑂𑐰𑐰𑐶𑐡𑑂𑐫𑐵𑐮𑐫 𑐰 𑐀𑐢𑑂𑐫𑐵𑐥𑐣 𑐱𑐿𑐮𑐷

𑐀𑐨𑐫𑐒𑑂𑐎𑐬𑑄 𑐥𑐬𑑂𑐜𑑂𑐫𑐹 𑐰𑐶𑐱𑑂𑐰𑐰𑐶𑐡𑑂𑐫𑐵𑐮𑐫𑐫𑑂 𑐡𑐱𑐎𑑁𑑄 𑐟𑐎 𑐀𑐢𑑂𑐫𑐵𑐥𑐣 𑐫𑐵𑐣𑐵𑐡𑐶𑐮, 𑐐𑐣 𑐰𑐫𑑂‌𑐎𑑅 𑐐𑐞𑐶𑐟 𑐰𑐶𑐨𑐵𑐐𑐫𑐵 𑐩𑐾𑐬𑐸𑐡𑐞𑑂𑐜 𑐖𑐸𑐫𑐵𑐔𑑂𑐰𑐣। 𑐰𑐫𑑂‌𑐎𑑅 𑐠𑑅𑐐𑐸 𑐰𑐶𑐡𑑂𑐫𑐵𑐬𑑂𑐠𑐷𑐟𑐫𑑂𑐟 𑐟𑐳𑐎𑑄 𑐩𑐶𑐴𑐶𑐣𑐾𑐟 𑐫𑐵𑐫𑐾𑐟 𑐰 𑐩𑑁𑐮𑐶𑐎 𑐰𑐶𑐔𑐵𑑅 𑐥𑐶𑐠𑐣𑐾𑐟 𑐥𑑂𑐬𑐾𑐬𑐶𑐟 𑐫𑐵𑐂𑐩𑑂𑐴 𑐐𑐸𑐬𑐸 𑐏। 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐩𑐵𑐟𑐴𑐟𑐫𑑂 𑑓𑑐 𑐳𑑂𑐰𑐫𑐵 𑐀𑐥𑑂𑐰 𑐰𑐶𑐡𑑂𑐫𑐵𑐬𑑂𑐠𑐷𑐟𑐫𑑂𑐳𑑄 𑐥𑐷.𑐊𑐔.𑐜𑐷. (Ph.D.) 𑐱𑑀𑐢 𑐥𑐹𑐬𑐵 𑐫𑐵𑐣𑐵𑐡𑐶𑐮, 𑐐𑐸𑐥𑐶𑑄 𑐁𑑅 𑐴𑐮𑐶𑐩𑐫𑐵 𑐠𑐷-𑐠𑐷 𑐰𑐶𑐱𑑂𑐰𑐰𑐶𑐡𑑂𑐫𑐵𑐮𑐫𑐫𑑂 𑐥𑑂𑐬𑑀𑐦𑐾𑐳𑐬 𑐖𑐸𑐫𑐵𑐔𑑂𑐰𑑄𑐐𑐸 𑐡𑐸 । 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐮𑐾𑐎𑑂𑐔𑐬𑐟 𑐟𑐳𑐎𑑄 𑐅𑐬𑑂𑐖𑐵𑐰𑐵𑐣 𑐖𑐸𑐂𑐐𑐸 𑐰 𑐰𑐫𑑂‌𑐎𑑅 𑐖𑐚𑐶𑐮 𑐐𑐞𑐶𑐟𑐫𑐵𑐟 𑐣𑑄 𑐳𑐬𑐮 𑐨𑐵𑐳𑐵𑑄 𑐎𑐣𑐾𑐐𑐸 𑐳𑐩𑐬𑑂𑐠 𑐎𑑂𑐫𑐣𑐵𑐡𑐶𑐮। 𑐰𑐫𑑂‌𑐎𑑅 𑐥𑐬𑑂𑐜𑑂𑐫𑐹𑐫𑑂 '𑐩𑐵𑐬𑑂𑐱𑐮 𑐜𑐶𑐳𑑂𑐚𑐶𑐒𑑂𑐐𑐸𑐂𑐳𑑂𑐜 𑐥𑑂𑐬𑑀𑐦𑐾𑐳𑐬'𑐫𑐵 𑐥𑐡𑐫𑑂 𑐔𑑂𑐰𑐣𑐵𑑅 𑐀𑐣𑑂𑐟𑐬𑑂𑐬𑐵𑐲𑑂𑐚𑑂𑐬𑐶𑐫 𑐀𑐣𑐸𑐳𑐣𑑂𑐢𑐵𑐣 𑐎𑐾𑐣𑑂𑐡𑑂𑐬𑐫𑐵 𑐣𑐾𑐟𑐺𑐟𑑂𑐰 𑐣𑑄 𑐫𑐵𑐣𑐵𑐡𑐶𑐮। 𑐰𑐫𑑂‌𑐎𑑅 𑐎𑐾𑐧𑐮 𑐐𑐞𑐶𑐟 𑐰𑐶𑐨𑐵𑐐𑐫𑑂 𑐖𑐎 𑐩𑐖𑐸𑐳𑐾𑑄 𑐎𑐩𑑂𑐥𑑂𑐫𑐸𑐚𑐬 𑐳𑐵𑐂𑐣𑑂𑐳 𑐰 𑐂𑐣𑑂𑐜𑐳𑑂𑐚𑑂𑐬𑐶𑐫𑐮 𑐂𑐣𑑂𑐖𑐶𑐣𑐶𑐫𑐬𑐶𑐒𑑂𑐐 𑐰𑐶𑐨𑐵𑐐𑐣𑐵𑐥 𑐣𑑄 𑐑𑐣𑐶𑐲𑑂𑐛 𑐬𑐸𑐥𑑄 𑐁𑐧𑐡𑑂𑐢 𑐖𑐸𑐫𑐵𑐔𑑂𑐰𑐣। 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐱𑐶𑐎𑑂𑐲𑐞 𑐱𑐿𑐮𑐷𑑄 𑐫𑐎𑑂𑐰 𑐣𑑂𑐴𑐸𑐐𑐸 𑐥𑐸𑐳𑑂𑐟𑐵𑐫𑐵 𑐐𑐞𑐶𑐟𑐖𑑂𑐘𑐟𑐫𑑂𑐟 𑐖𑐣𑑂𑐩 𑐧𑐶𑐮। 

## 𑐎𑐩𑑂𑐥𑑂𑐫𑐸𑐚𑐾𑐱𑐣𑐮 𑐀𑐮𑑂𑐖𑐾𑐧𑑂𑐬𑐵 𑐰 𑐬𑑀𑐧𑑀𑐚𑐶𑐎𑑂𑐳

𑐀𑐨𑐫𑐒𑑂𑐎𑐬 𑐀𑐩𑐹𑐬𑑂𑐟 𑐰𑐵 𑐳𑐶𑐡𑑂𑐢𑐵𑐣𑑂𑐟𑐫𑑂 𑐖𑐎 𑐳𑐷𑐩𑐶𑐟 𑐩𑐖𑐸𑐳𑐾𑑄 𑐐𑐞𑐶𑐟𑐫𑐵 𑐰𑑂𑐫𑐵𑐰𑐴𑐵𑐬𑐶𑐎 𑐥𑑂𑐬𑐫𑑀𑐐𑐫𑑂 𑐣𑑄 𑐟𑐳𑐎𑑄 𑐬𑐸𑐔𑐶 𑐟𑐂𑐩𑑂𑐴 𑐰𑑂𑐫𑐎𑑂𑐟𑐶 𑐏। 𑐰𑐫𑑂‌𑐎𑑅𑐣𑑄 𑐥𑑀𑐮𑐶𑐣𑑀𑐩𑐶𑐫𑐵𑐮 (Polynomial) 𑐳𑐩𑐷𑐎𑐬𑐞𑐟𑐫𑑂𑐟 𑐎𑐩𑑂𑐥𑑂𑐫𑐸𑐚𑐬𑑄 𑐐𑐠𑐾𑑄 𑐫𑐵𑐣𑐵𑑅 𑐀𑑅𑐥𑐸𑐎𑑂𑐎 𑐰 𑐫𑐠𑐵𑐬𑑂𑐠 𑐬𑐸𑐥𑑄 𑐳𑐩𑐵𑐢𑐵𑐣 𑐫𑐵𑐫𑐾𑐕𑐶𑑄 𑐢𑐂𑐐𑐸 𑐏𑑃𑐫𑑂 𑐫𑐎𑑂𑐰 𑐖𑑂𑐫𑐵 𑐫𑐵𑐣𑐵𑐡𑐶𑐮 । 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐠𑑂𑐰 𑐖𑑂𑐫𑐵𑑄 '𑐎𑐩𑑂𑐥𑑂𑐫𑐸𑐚𑐬 𑐊𑐜𑐾𑐜 𑐜𑐶𑐖𑐵𑐂𑐣' (CAD) 𑐎𑑂𑐲𑐾𑐟𑑂𑐬𑐫𑑂 𑐎𑑂𑐬𑐵𑐣𑑂𑐟𑐶𑐎𑐵𑐬𑐷 𑐥𑐬𑐶𑐰𑐬𑑂𑐟𑐣 𑐴𑐮। 𑐬𑑀𑐧𑑀𑐚𑐶𑐎𑑂𑐳𑐫𑑂 𑐬𑑀𑐧𑑀𑐚𑐫𑐵 𑐮𑑂𑐴𑐵𑑅𑐫𑐵 𑐔𑐵𑐮 (Motion Planning) 𑐰 𑐩𑐾𑐎𑐵𑐣𑐶𑐎𑐮 𑐂𑐣𑑂𑐖𑐶𑐣𑐶𑐫𑐬𑐶𑐒𑑂𑐐𑐫𑐵 𑐫𑐎𑑂𑐰 𑐖𑐚𑐶𑐮 𑐖𑑂𑐫𑐵𑐫𑑂 𑐀𑐨𑐫𑐒𑑂𑐎𑐬𑐫𑐵 𑐖𑑂𑐫𑐵𑐩𑐶𑐟𑐷𑐫 𑐳𑐹𑐟𑑂𑐬𑐟 𑐕𑑂𑐫𑐮𑐶𑐐𑐸 𑐫𑐵𑐂। 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 '𑐀𑐮𑑂𑐐𑑀𑐬𑐶𑐡𑐩'𑐟 𑐁𑑅 𑐣𑑄 𑐁𑐢𑐸𑐣𑐶𑐎 𑐂𑐣𑑂𑐖𑐶𑐣𑐶𑐫𑐬𑐶𑐒𑑂𑐐 𑐳𑐦𑑂𑐚𑐰𑐾𑐫𑐬 𑐣𑐶𑐬𑑂𑐩𑐵𑐞𑐫𑑂 𑐁𑐢𑐵𑐬𑐨𑐹𑐟 𑐖𑐸। 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐰𑐶𑐔𑐵𑐬𑐫𑑂 𑐐𑐞𑐶𑐟 𑐕𑐐𑐹 𑐀𑐖𑐵𑑅𑐐𑐸 𑐳𑐵𑐢𑐣 𑐏 𑐐𑐸𑐎𑐶𑐮𑐶𑑄 𑐨𑑁𑐟𑐶𑐎 𑐳𑑄𑐳𑐵𑐬𑐫𑐵 𑐖𑐚𑐶𑐮𑐟𑐵𑐫𑐵𑐟 𑐀𑑅𑐥𑐸𑐎𑐶। 𑐠𑑂𑐰 𑐴𑐾 𑐎𑐵𑐬𑐞𑑄 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵𑐟 𑐂𑐣𑑂𑐖𑐶𑐣𑐶𑐫𑐬𑐶𑐒𑑂𑐐 𑐰 𑐎𑐩𑑂𑐥𑑂𑐫𑐸𑐚𑐬 𑐰𑐶𑐖𑑂𑐘𑐵𑐣𑐫𑐵 𑐏𑑂𑐫𑐮𑐫𑑂 𑐣𑑄 𑐳𑐩𑑂𑐩𑐵𑐣𑐥𑐹𑐬𑑂𑐰𑐎 𑐳𑑂𑐰𑐂।

## 𑐳𑑄𑐳𑑂𑐎𑐺𑐟, 𑐰𑑂𑐫𑐵𑐎𑐬𑐞 𑐰 𑐨𑐵𑐬𑐟𑐷𑐫 𑐡𑐬𑑂𑐱𑐣

𑐀𑐨𑐫𑐒𑑂𑐎𑐬𑐫𑐵 𑐰𑑂𑐫𑐎𑑂𑐟𑐶𑐟𑑂𑐰𑐫𑐵 𑐕𑐐𑐹 𑐁𑐖𑐸𑐔𑐵𑑅𑐐𑐸 𑐰 𑐥𑑂𑐬𑐾𑐬𑐞𑐵𑐡𑐵𑐫𑐷 𑐥𑐎𑑂𑐲 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐳𑑄𑐳𑑂𑐎𑐺𑐟 𑐩𑑀𑐴 𑐏। 𑐰𑐫𑑂‌𑐎𑑅 𑐳𑑄𑐳𑑂𑐎𑐺𑐟 𑐰𑑂𑐫𑐵𑐎𑐬𑐞𑐫𑐵 𑐆𑐲𑐶 '𑐥𑐵𑐞𑐶𑐣𑐶'𑐫𑐵 𑐳𑐹𑐟𑑂𑐬𑐟𑐫𑑂𑐟 𑐴𑐮𑐶𑑄𑐫𑐵 𑐡𑐎𑐮𑐫𑑂 𑐣𑑂𑐴𑐵𑐥𑐵𑑄𑐐𑐸 𑐎𑐩𑑂𑐥𑑂𑐫𑐸𑐚𑐬 𑐥𑑂𑐬𑑀𑐐𑑂𑐬𑐵𑐩𑐶𑐒𑑂𑐐 𑐨𑐵𑐲𑐵 𑐢𑐎𑐵𑑅 𑐎𑐫𑐵𑐡𑐶𑐮 । 𑐰𑐫𑑂‌𑐎𑑅 𑐠𑑅𑐐𑐸 𑐐𑐞𑐶𑐟𑐷𑐫 𑐔𑑂𑐰𑐳𑐸𑐟𑐫𑑂𑐐𑐸 𑐣𑑂𑐴𑑂𑐫𑐠𑐣𑐾𑐖𑑂𑐫𑐵 𑐰𑐵 𑐡𑐠𑐸𑐂 𑐥𑑂𑐬𑐵𑐫𑑅 𑐄𑐥𑐣𑐶𑐲𑐡𑑂, 𑐨𑐐𑐰𑐡𑑂𑐐𑐷𑐟𑐵 𑐰𑐵 𑐳𑑄𑐳𑑂𑐎𑐺𑐟 𑐎𑐵𑐰𑑂𑐫𑐫𑐵 𑐱𑑂𑐮𑑀𑐎𑐟 𑐔𑑂𑐰𑐫𑐾𑐐𑐸 𑐫𑐵𑐣𑐵𑐡𑐷। 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐎𑐠𑑄 𑐐𑐞𑐶𑐟 𑐰 𑐳𑑄𑐳𑑂𑐎𑐺𑐟 𑐣𑐶𑐐𑐸𑐮𑐶𑑄 𑐕𑐐𑐹 𑐴𑐾 𑐟𑐬𑑂𑐎 (Logic) 𑐰 𑐀𑐣𑐸𑐱𑐵𑐳𑐣𑐫𑑂 𑐁𑐢𑐵𑐬𑐶𑐟 𑐡𑐸। 𑐰𑐫𑑂‌𑐎𑑅 𑐕𑐩𑑂𑐴 𑐬𑐵𑐲𑑂𑐚𑑂𑐬𑐰𑐵𑐡𑐷 𑐔𑐶𑐣𑑂𑐟𑐎 𑐣𑑄 𑐖𑐸𑐫𑐵𑐔𑑂𑐰𑐣 𑐐𑐸𑐩𑑂𑐴 𑐨𑐵𑐬𑐟𑐷𑐫 𑐖𑑂𑐘𑐵𑐣 𑐥𑐬𑐩𑑂𑐥𑐬𑐵𑐫𑐵𑐟 𑐴𑐮𑐶𑐩𑐫𑑂 𑐳𑑂𑐠𑐵𑐥𑐶𑐟 𑐫𑐵𑐫𑐾𑐟 𑐥𑑂𑐬𑐫𑐟𑑂𑐣𑐱𑐷𑐮 𑐖𑐸𑐫𑐵𑐔𑑂𑐰𑐣। 𑐰𑐫𑑂‌𑐎𑑅 𑐨𑐵𑐬𑐟𑐷𑐫 𑐐𑐞𑐶𑐟𑐫𑐵 𑐂𑐟𑐶𑐴𑐵𑐳, 𑐰𑐶𑐱𑐾𑐲 𑐫𑐵𑐣𑐵𑑅 𑐧𑑂𑐬𑐴𑑂𑐩𑐐𑐸𑐥𑑂𑐟 𑐰 𑐨𑐵𑐳𑑂𑐎𑐬𑐫𑐵 𑐖𑑂𑐫𑐵𑐫𑑂 𑐟𑐳𑐎𑑄 𑐐𑑁𑐬𑐰 𑐫𑐵𑐣𑐵𑐡𑐷𑐐𑐸 𑐏। 𑐰𑐫𑑂‌𑐎𑑅𑐫𑐵 𑐰𑐶𑐔𑐵𑐬𑐫𑑂 𑐥𑐵𑐱𑑂𑐔𑐵𑐟𑑂𑐫 𑐐𑐞𑐶𑐟𑑄 𑐨𑐵𑐬𑐟𑐷𑐫 𑐐𑐞𑐶𑐟𑐫𑐵 𑐟𑐬𑑂𑐎𑐱𑐷𑐮𑐟𑐵𑐫𑐵𑐟 𑐁𑐟𑑂𑐩𑐳𑐵𑐟 𑐫𑐵𑐫𑐾𑐩𑐵𑑅। 𑐳𑑄𑐳𑑂𑐎𑐺𑐟𑐫𑐵 𑐀𑐲𑑂𑐚𑐵𑐢𑑂𑐫𑐵𑐫𑐷 𑐰 𑐐𑐞𑐶𑐟𑐫𑐵 𑐳𑐹𑐟𑑂𑐬𑐟𑐫𑑂𑐟 𑐳𑑂𑐰𑐵𑐣𑐵𑑅 𑐰𑐫𑑂‌𑐎𑐮𑑄 𑐫𑐎𑑂𑐰 𑐥𑑂𑐬𑐰𑐔𑐣𑐟 𑐣𑑄 𑐧𑐶𑐫𑐵𑐡𑐷𑐐𑐸 𑐡𑐸।

## 𑐩𑐹 𑐥𑐶𑐠𑑄𑐐𑐸 𑐳𑐦𑐹𑐟

𑐰𑐫𑑂‌𑐎𑐮𑑄 𑐐𑐞𑐶𑐟𑐫𑐵 𑐠𑐷-𑐠𑐷 𑐰𑐶𑐲𑐫𑐫𑑂 𑐩𑐴𑐟𑑂𑐰𑐥𑐹𑐬𑑂𑐞 𑐳𑐦𑐹𑐟 𑐔𑑂𑐰𑐫𑐵𑐡𑐷𑐐𑐸 𑐡𑐸:

* Resolution of Singularities of Embedded Algebraic Surfaces (1966)
* Lectures on Expansion Techniques in Algebraic Geometry (1977)
* Weighted Expansions for Canonical Desingularization (1982)
* Enumerative Combinatorics of Young Tableaux (1988)
* Algebraic Geometry for Computer Scientists (1990)
* Lectures on Expansion Techniques in Algebraic Geometry (2001 - 𑐥𑐸𑐣: 𑐥𑑂𑐬𑐎𑐵𑐱𑐣)

## 𑐮𑐶𑐢𑑄𑐳𑐵