理論物理在哪些方面促進了數學的發展？
最早的例子應該是牛頓為了研究運動學而創立了微積分。之後其實都是數學領先於物理。比如Lagrange分析力學之前就有了變分法；Maxwell場論之前就有了矢量分析；廣義相對論之前就有了黎曼幾何。
到了後量子力學時期,即量子場論發展時期Yang-Mills theory促進了Donaldson theory的發展。這是物理學的概念推動數學家解決4維流形分類的一個例子。之後又有了Seiberg-Witten theory, 可以將SW方程理解為超對稱的YM方程。Seiberg-Witten curve對於構造找不到lagragian的理論有著超級的作用。
弦理論誕生後物理學促進數學的例子就比較多了。雖然Calabi-Yau流形是數學家Calabi最先提出一個猜想, 然後丘成桐證明了這個猜想而發現的, 弦理論家發現超弦理論中額外的6維空間是Calabi-Yau 3-fold(復3維,實6維)。之後P. Candelas, B. Greene等物理學家發現Calabi-Yau 3-fold具有一種性質叫mirror symmetry(鏡像對稱性)。mirror symmetry指的不是通常意義下的鏡面對稱性,而是指拓撲意義上不同的CY流形可能對應相同的CFT（共形場論）,這兩個CY 3-fold的Hodge diamond互為鏡像對稱, 所以稱為mirror symmetry。在拓撲弦論中物理學家又發現互為鏡像對稱的兩個CY流形另外一個相關的猜想叫SYZ conjecture, 但是出發點更物理一些。
Perelman在證明Poincaré猜想的文章中用到了熵的概念。Witten發現S-duality和幾何化Langlands綱領的聯繫。Witten發現AdS/CFT correspendence和數論中的monstrous moonshine有聯繫。monstrous moonshine的證明用到了玻色弦在Leech lattice上的緊化, 而Leech lattice的對稱群恰好和monster群有聯繫。之後物理學家在研究K3的elliptic genera時發現了Mathieu moonshine, 之後被推廣為Umbral moonshine。關於數學和物理之間互相促進的例子, 可以參考丘成桐的演講：
鏈接：【精品推薦】丘成桐在上海交大100周年校慶上的演講_14629因為理論需要用數學語言來嚴謹的表達，嶄新的理論就要用嶄新的數學語言來表達，超前的理論就要用超前的數學語言來表達，創新的理論就要用自創的數學語言來表達，
天才的理論就要用天才得數學語言來表達，
無法理解理論卻要用能夠被理解的數學語言來表達。我還沒有學過多少，因此沒有能力給出理論物理，mainly qft，七十年代至今，對代數幾何和復幾何，低維拓撲，表示論等的影響，實際上如果說quantum的話，量子物理誕生之後便開始對各個數學學科產生影響了吧，樓上的回答也只是一些熱點的不完全的問題推薦《數學物理百科全書》，弗朗西斯科等
說到物理對數學的影響，我想除了數學內部需求，eg algebraic geometry in 20th，其餘絕大多數的motivation都來自物理吧
想到一個與最近看的東西有聯繫的：Ricci flow最早來自於string theory的diliton gravity的renormalization group flow，因此Ricci flow裡面的時間其實可以粗略的看成「看」manifold的尺子物理激發的數學
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_光明網
量子幾何
他說有用，我就能算出來！
量子力學使世人注意到早就被發明的矩陣的重要意義
As an example of Witten"s work in pure mathematics, Atiyah cites his application of techniques from quantum field theory to the mathematical subject of low-dimensional topology. In the late 1980s, Witten coined the term topological quantum field theory for a certain type of physical theory in which theexpectation values of observable quantities encode information about the topologyof spacetime. In particular, Witten realized that a physical theory now calledChern–Simons theory could provide a framework for understanding the mathematical theory of knots and 3-manifolds. Although Witten"s work was based on the mathematically ill-defined notion of a Feynman path integral and was therefore notmathematically rigorous, mathematicians were able to systematically develop Witten"s ideas, leading to the theory of Reshetikhin–Turaev invariants.
Another result for which Witten was awarded the Fields Medal was his (nonrigorous[citation needed]) proof in 1981 of thepositive energy theorem in general relativity. This theorem asserts that (under appropriate assumptions) the total energyof a gravitating system is always positive and can be zero only if the geometry of spacetime is that of flat Minkowski space. It establishes Minkowski space as a stable ground state of the gravitational field. While the original proof of this result due to Richard Schoen and Shing-Tung Yau used variational methods, Witten"s proof used ideas from supergravity theory to simplify the argument.
A third area mentioned in Atiyah"s address is Witten"s work relating supersymmetry and Morse theory, a branch of mathematics that studies the topology ofmanifolds using the concept of a differentiable function. Witten"s work gave a physical proof of a classical result, the Morse inequalities, by interpreting the theory in terms of supersymmetric quantum mechanics.
Edward Witten
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