The proposed project is aimed at discovering a mathematical relation between the "classical" topological invariants and the "quantum" ones of a given space, such as a knot complement, in terms of the analysis on Riemann surfaces. Within the last two years, many new insights have been established around the idea of mirror symmetry and "quantization" of topological invariants for a large class of geometric spaces. The central driving force of the recent development is the discovery, due mainly to physicists Eynard, Orantin, Marino, and others, of a recursive formula for quantum invariants in terms of integral transforms over Riemann surfaces. For example, this recursion formula computes both open and closed Gromov-Witten invariants for all genera of an arbitrary toric Calabi-Yau three-fold. Here what we call the classical invariants of the Calabi-Yau space determine a Riemann surface as its mirror dual. Then the Eynard-Orantin recursion formula computes the quantum invariants, i.e., the higher-genus Gromov-Witten invariants, of the original Calabi-Yau space. The mathematical structure of this miraculous procedure is best understood by the simplest example of the theory discovered by the PI, in collaboration with Dumitrescu and others. This example is based on the quest of mirror symmetric dual of the Catalan numbers, which has led the authors to an unexpectedly rich theory. The application of the mirror symmetry, and then of the quantization process of Eynard-Orantin, to the Catalan numbers, we obtain: (1) the virtual Poincare polynomials of the moduli spaces of smooth pointed curves; and (2) the intersection numbers of the tautological classes of the moduli spaces of stable pointed curves. For this particular example, and later for many other examples, the PI and his collaborators Bouchard, Shadrin, Sulkowski, and others, have discovered that the partition function (a special choice of the generating function of quantum invariants) satisfies the "quantum curve" equation, which is a Schrodinger equation. The idea of quantum curves is due to physicists Aganagic, Dijkgraaf, Klemm, Marino, Vafa, and others. For all examples that admit the quantum curve, it has also been verified that the partition function is a Baker-Akhiezer function of an integrable system of the KdV/KP type. Very surprisingly, a torsion condition of a Steinberg symbol in the second K-group in algebraic K-theory holds for all these examples. Establishing a mathematical understanding of the relation between this higher algebraic K-theory condition and the quantizability of a Riemann surface (in particular a knot A-polynomial), and the existence of a Baker-Akhiezer function behind the scene, in the context of quantum topological invariants, is the goal of the proposed project.
Pure mathematical research is often inspired by radical ideas from theoretical physics. The mirror symmetry is an example of such ideas: there are two very different mathematical ways of describing the physical universe. Since the universe is unique, we must conclude that these two mathematical theories are equivalent. The "mirror symmetry" refers to the relation of these two theories. Further extending the idea of mirror symmetry more mathematically, one arrives at a naive, but also quite radical, question: what is the mirror symmetric partner of Catalan numbers? If we consider Catalan numbers as a "classical" object, then what are the "quantum" generalization of them? The PI and his collaborators have discovered an affirmative answer to these questions. To their surprise, the answer turns out to provide the simplest mathematical example of a powerful speculative theory due to theoretical physicists. The physics theory predicts a concrete and universal formula to calculate an infinite series of characteristic numbers (called invariants) of the possible universe. This is a rather involved theory, and mathematical proofs of the formulas are also complicated. Our simplest example illustrates what is happening in an elementary language, and helps understanding the general theory. Building on the concrete mathematical foundation the PI and his collaborators have established, the PI proposes to study newly proposed conjectures on classical and quantum knot invariants by physicists. The quantum generalization of Catalan numbers count certain topological graphs. It is interesting to note that these numbers also appear in biology as the free energy of complex molecules such as DNA, RNA, and proteins. For example, the quantum generalized Catalan numbers count the "secondary" structures of an RNA. The primary structure is the linear sequence of nucleotides. The secondary structure refers to the complication of its position due to knotted, tangled, and bridged, structures. Being a result of quantization, the generating function of these quantized Catalan numbers satisfies a Schrodinger equation. This also provides a simple example of another set of radical physics predictions related to quantization of surfaces. These predictions include new conjectures in knot theory. The proposed research project is aimed at establishing mathematically rigorous results, verifying speculative predictions from theoretical physics. The work is expected to have impact on several areas of pure mathematics. It is also expected to have an application in the study of secondary structures of complex molecules in biology. Through the construction and analysis of simple examples of the mysterious and miraculous theories with quite involved nature, the PI has been able to attract undergraduate students participating in these research topics. The proposed project contains an REU component to engage undergraduate students in the exciting research frontier.