Background setting
Carnot groups, especially step-two Carnot groups, are natural generalizations of the Euclidean spaces. The classical Heisenberg group is a prototypical example of such groups.
As a result, many topics on Euclidean spaces, including geometric inequalities, function spaces, harmonic analysis, optimal transport, PDEs, and geometric measure theory, are studied in the setting of Heisenberg groups/Carnot groups intensively. However, the following features make Carnot groups different from the Euclidean spaces and the study on them more challenging and interesting:
1. The geometry on Carnot groups is the sub-Riemannian geometry instead of the Riemannian one. In other words, abnormal geodesics, which may not be written as a solution of some ODE, can appear on Carnot groups;
2. The corresponding Laplacian is not elliptic but hypoelliptic, which means the classical theory on elliptic PDEs of second order cannot apply to this case;
3. The correspnding gradient is noncommutative with the Laplacian;
4. The Hausdorff dimension is strictly larger than the topological one, which makes them an important examples of doubling metric measure spaces.
Researches during PhD
My research during my PhD concentrates on the heat kernel asymptotics in the framework of step-two Carnot groups.
Thanks to the explicit expression of the heat kernel as an oscillatory integral in this background, we can use the method of stationary phase to obtain such asymptotics.
A direct consequence of such asymptotic is the precise bounds of the heat kernel, which allows us to use the heat kernel in this setting as we do in the Euclidean spaces (up to uniform constants). Moreover, these precise bounds implies the Gaussian bound for the heat kernel, which is suffcient for studying problems such as Riesz transforms. Upper bounds for the derivatives of the heat kernel can be deduced in a similar way.
Another consequence is the exact formulae for the sub-Riemannian distance by the well-known Varadhan's formula. This may provide a possible approach to study the sub-Riemannian geometry on step-two Carnot groups.
Researches during Postdoc
During my time in OIST, I developed the interest in the weak notions of curvature-dimension conditions on metric measure space.
Recall that based on the theory of optimal transport, the notion of curvature-dimension condition can be generalized to non-smooth metric measure spaces in a synthetic way. This is a breakthrough since it turns out that these non-smooth spaces still share many important geometric and analytic properties which hold in the smooth setting. However, it can be shown that this condition fails on the Heisenberg group, which plays important roles in many branches of mathematics such as functions of several complex variables, sub-Riemannian geometry, partial differential equations, and quantum mechanics, etc. As a result, finding weak notions of curvature-dimension conditions which also work well on spaces like Heisenberg groups becomes important.
I am now focusing on two weak notions of curvature-dimension conditions: quasi Bakry–Émery curvature condition and Measure Contraction Property, which are already established on Heisenberg group.
Carnot groups, especially step-two Carnot groups, are natural generalizations of the Euclidean spaces. The classical Heisenberg group is a prototypical example of such groups.
As a result, many topics on Euclidean spaces, including geometric inequalities, function spaces, harmonic analysis, optimal transport, PDEs, and geometric measure theory, are studied in the setting of Heisenberg groups/Carnot groups intensively. However, the following features make Carnot groups different from the Euclidean spaces and the study on them more challenging and interesting:
1. The geometry on Carnot groups is the sub-Riemannian geometry instead of the Riemannian one. In other words, abnormal geodesics, which may not be written as a solution of some ODE, can appear on Carnot groups;
2. The corresponding Laplacian is not elliptic but hypoelliptic, which means the classical theory on elliptic PDEs of second order cannot apply to this case;
3. The correspnding gradient is noncommutative with the Laplacian;
4. The Hausdorff dimension is strictly larger than the topological one, which makes them an important examples of doubling metric measure spaces.
Researches during PhD
My research during my PhD concentrates on the heat kernel asymptotics in the framework of step-two Carnot groups.
Thanks to the explicit expression of the heat kernel as an oscillatory integral in this background, we can use the method of stationary phase to obtain such asymptotics.
A direct consequence of such asymptotic is the precise bounds of the heat kernel, which allows us to use the heat kernel in this setting as we do in the Euclidean spaces (up to uniform constants). Moreover, these precise bounds implies the Gaussian bound for the heat kernel, which is suffcient for studying problems such as Riesz transforms. Upper bounds for the derivatives of the heat kernel can be deduced in a similar way.
Another consequence is the exact formulae for the sub-Riemannian distance by the well-known Varadhan's formula. This may provide a possible approach to study the sub-Riemannian geometry on step-two Carnot groups.
Researches during Postdoc
During my time in OIST, I developed the interest in the weak notions of curvature-dimension conditions on metric measure space.
Recall that based on the theory of optimal transport, the notion of curvature-dimension condition can be generalized to non-smooth metric measure spaces in a synthetic way. This is a breakthrough since it turns out that these non-smooth spaces still share many important geometric and analytic properties which hold in the smooth setting. However, it can be shown that this condition fails on the Heisenberg group, which plays important roles in many branches of mathematics such as functions of several complex variables, sub-Riemannian geometry, partial differential equations, and quantum mechanics, etc. As a result, finding weak notions of curvature-dimension conditions which also work well on spaces like Heisenberg groups becomes important.
I am now focusing on two weak notions of curvature-dimension conditions: quasi Bakry–Émery curvature condition and Measure Contraction Property, which are already established on Heisenberg group.