2025年3月26日，素有「數學界諾貝爾獎」之稱的阿貝爾獎（Abel Prize）正式公布本年度得主。來自日本京都大學的數學家柏原正樹（Masaki Kashiwara）榮獲此殊榮，成為該獎設立以來首位獲獎的黃種人。現年78歲的柏原教授以其在代數分析（Algebraic Analysis）與表示論（Representation Theory）領域的開創性研究而聞名於世，他的理論成果深刻改變了現代數學的面貌。

柏原正樹的學術生涯幾乎與代數分析的誕生緊密相連。他在23歲撰寫碩士論文時，首次提出了「D-模理論」（D-module theory）的雛形。該理論透過將微分方程與代數結構結合，為分析學與代數幾何之間建立起橋樑，開啟一個嶄新的研究領域。隨後，他運用D-模理論成功證明數學界長期懸而未決的「黎曼–希爾伯特對應」（Riemann–Hilbert correspondence）問題，這一成就被認為是20世紀後半葉最重要的數學突破之一。該對應揭示了解析微分方程的解與拓撲幾何結構之間的深層聯繫，對現代數學的整體架構產生了深遠影響。

除D-模理論外，柏原教授與法國數學家雪維雷（Pierre Schapira）合作，開發出「微觀局部層理論」（Microlocal Sheaf Theory），這是一種結合層論與微觀分析的嶄新方法，用以研究奇點與解析結構的本質。該理論不僅推動了表示論與代數幾何的融合，也對數學物理中的量子場論、弦理論等領域產生了啟發性影響。

在表示論方面，柏原正樹同樣貢獻卓越。他與日本另一位數學家路幸男（Yukio Saito）共同創立了「量子群晶體基理論」（Theory of Crystal Bases of Quantum Groups），證明了量子群的可積最高權表示存在晶體基。這一發現為量子代數的結構研究提供了核心工具，也在理論物理和組合數學中被廣泛應用。

挪威科學與文學院院長在頒獎辭中高度評價柏原的成就，指出他在過去五十多年中「重塑並極大地豐富了代數分析與表示論的整個領域」，其工作不僅處於當代數學的最前沿，更啟發了數代年輕研究者的研究方向。院長強調，柏原正樹的思想使分析學、代數幾何與物理理論之間的界限變得模糊，為現代數學建立了新的統一語言。

阿貝爾獎設立於2003年，由挪威政府設立，以紀念偉大的數學家尼爾斯·亨里克·阿貝爾（Niels Henrik Abel）。該獎每年由挪威科學與文學院頒發，專門表彰在數學領域做出傑出貢獻的學者。獎金為750萬挪威克朗（約合新台幣1650萬元，或人民幣約517萬元）。歷屆得主多來自歐美學術重鎮，如法國、英國與美國。柏原正樹此次獲獎，象徵著亞洲數學研究的實力正受到世界的正式肯定，也被視為日本在數學界長期深耕的成果之一。

值得一提的是，京都大學被譽為日本的「數學搖籃」，多位世界級數學家皆出自該校。柏原教授繼承京都學派嚴謹且重視直覺的研究傳統，他的成就讓日本數學再次站上世界舞台，也進一步提升亞洲在基礎數學領域的國際影響力。

On March 26, 2025, the Abel Prize — often referred to as the “Nobel Prize of Mathematics” — announced its latest laureate. Japanese mathematician Masaki Kashiwara, a professor at Kyoto University, was awarded the prestigious honor, becoming the first Asian and first non-Caucasian recipient in the prize’s history. At 78, Kashiwara is celebrated for his groundbreaking work in algebraic analysis and representation theory, contributions that have profoundly reshaped modern mathematics.

Kashiwara’s mathematical journey is deeply intertwined with the very birth of algebraic analysis. At the young age of 23, in his master’s thesis, he introduced the foundational ideas of D-module theory, a revolutionary framework that bridges differential equations with algebraic geometry. This innovation established a powerful connection between analysis and algebra, leading to entirely new methods in the study of mathematical structures.

Building on this theory, Kashiwara later achieved a major breakthrough by proving the Riemann–Hilbert correspondence, a long-standing problem that had puzzled mathematicians for decades. His proof revealed a profound link between the analytic solutions of differential equations and the topological properties of geometric structures. This result is now regarded as one of the most significant advances in mathematics in the latter half of the 20th century.

In collaboration with French mathematician Pierre Schapira, Kashiwara went on to develop microlocal sheaf theory, an advanced mathematical framework that combines sheaf theory with microlocal analysis. This work has become an essential tool for understanding singularities and analytic structures, influencing not only pure mathematics but also areas of theoretical physics, such as quantum field theory and string theory.

Kashiwara also made major contributions to representation theory. Together with fellow Japanese mathematician Yukio Saito, he co-founded the theory of crystal bases for quantum groups, proving the existence of crystal bases for integrable highest-weight representations. This discovery provided an elegant combinatorial framework that has since become central to quantum algebra and has applications in both physics and modern combinatorics.

In its official statement, the Norwegian Academy of Science and Letters praised Kashiwara’s lifetime achievements, noting that over the past fifty years, he has “reshaped and greatly enriched the fields of algebraic analysis and representation theory.” The Academy emphasized that his ideas have blurred traditional boundaries between analysis, geometry, and physics, offering a new unifying language for modern mathematics.

The Abel Prize, established in 2003 by the Norwegian government to honor the legacy of mathematician Niels Henrik Abel, recognizes outstanding contributions to the field of mathematics. The award carries a cash prize of 7.5 million Norwegian kroner (approximately 517,000 USD or 3.6 million RMB). Until now, most recipients have been from Europe or North America, making Kashiwara’s win not only a personal triumph but also a historic milestone for Asian mathematics.

Kyoto University, often regarded as Japan’s “cradle of mathematics,” has produced many of the country’s leading mathematicians. Kashiwara’s work continues this tradition of intellectual rigor and creative insight, reinforcing Japan’s reputation as a powerhouse of mathematical research and elevating the visibility of Asian scholarship on the global stage.