Geometry of the Abel Equation of the first kind

We study the first kind Abel differential equation $$\dfrac{dy}{dx}=c_0(x)+3c_1(x)y+3c_2y^2+c_3(x)y^3,$$ where the functions $c_i$ are real analytic. The first step of our analysis is through the Cartan equivalence method, then we use techniques from representation theory; this latter mean allows us to exhibit an affine connection and hence a covariant derivative on the space of differential invariants of the Abel equation. In the context of Abel equation, this affine connection is similar to the connection (also called Frobenius-Stickelberger connexion) given by the quasi-modular Eisenstein series $E_2$, which permits to define a covariant derivative in the space of modular forms and is a solution of a Chazy type equation.Comment: 12 page

Lectures on Finslerian Geometry

Lecture notes on Finsler GeometryComment: 74 page

Quantitative genetics and metabolomics of aerobic metabolism

The quantitative genetic, biochemical, and physiological bases of variation in maximal aerobic metabolic rate (MMR) are important for understanding exercise physiology and the evolution of aerobic performance, but they still are not well understood. To this end, I studied three aspects of MMR. First, I estimated the genetic variances and covariances of MMR and basal metabolic rate (BMR). Second, I identified how membrane fatty acid (FA) composition changed in response to selection for increased MMR. Third, I measured metabolite expressions in organs primarily responsible for MMR and BMR. Such an approach allowed me to better understand the mechanistic connection (e.g., shared organs) between MMR and BMR and the evolution of aerobic energy metabolism.In my first chapter, I determined the genetic variances and covariances of MMR and BMR. The genetic variances and covariances of metabolic traits must be known to predict how they respond to selection and how covariances among them might affect their evolutionary trajectories. To this end, I used the animal model to estimate the genetic variances and covariances of MMR and BMR in a genetically heterogeneous stock of laboratory mice. Narrow-sense heritability (h2) was approximately 0.38+0.08 for body mass, 0.24+0.07 for whole-animal MMR, 0.26+0.08 for whole-animal BMR, 0.16+0.06 for mass-independent MMR, and 0.19+0.07 for mass-independent BMR. All h2 estimates were significantly different from zero. The phenotypic correlation of whole animal MMR and BMR was 0.56+0.02, and the corresponding genetic correlation was 0.79+0.12. The phenotypic correlation of mass-independent MMR and BMR was 0.13+0.03, and the corresponding genetic correlation was 0.72+0.03. The genetic correlations of metabolic rates were significantly different from zero, but not significantly different from one. The genetic correlation is not so high as to preclude independent evolution of MMR and BMR.For second chapter of my dissertation, I tested how selection for increased MMR changes membrane fatty acid (FA) composition in a genetically heterogeneous stock of laboratory mice. The membrane pacemaker hypothesis predicts that the unsaturation index (UI) of membrane FAs is positively linked to the high BMR in endotherms. To test this hypothesis, I examined the membrane FA composition of liver and gastrocnemius muscle in mice after 7 generations of selection for increased MMR (high-MMR). Although mass-independent BMR was 3.5% higher in high-MMR mice, the liver UI was not higher than in control mice. Concentration of 16:0 and 18:0 FAs were lower in the liver of high-MMR mice, whereas a greater concentration of 18:1 n-7 FA was found in the gastrocnmeius muscle of high-MMR mice. Moreover, individual variation in UI had no influence on either BMR or MMR. However, concentration of 16:1 n-7, 18:1 n-9, and 22:5 n-3 FAs in the gastrocnemius were significant predictors of BMR, but none of the liver FAs were significant predictors of BMR. In both muscle and liver 20:4 n-6 FA was a significant predictor of MMR and in liver 20:3 n-6 FA was another significant predictor of MMR. The findings did not support the prediction that UI is positively correlated with BMR, but more broadly MMR and BMR were linked to membrane FA composition changes in the skeletal muscle and liver.For third chapter of my dissertation, I examined how 7 generations of selection for high MMR changes metabolite expression of the organs primarily responsible for resting metabolic rate (i.e., the liver) and of organs primarily responsible for MMR (i.e., skeletal muscle as represented by the gastrocnemius and plantaris muscles). One of the pivotal challenges in evolutionary physiology is elucidating the functional connection between MMR and BMR because the main contributors to MMR are skeletal muscles wheras the main contributors to BMR are visceral organs. To this end, I used an untargeted global metabolomic analysis of the gastrocnemius and plantaris muscles and of the liver during resting metabolism to reveal adaptive metabolic responses to selection for increased MMR in a genetically heterogeneous stock of laboratory mice. In the plantaris muscle, metabolic profiles of high-MMR and control mice did not differ. In the liver, amino acid and tricarboxylic acid cycle (TCA cycle) metabolite amounts were lower in high-MMR mice than in controls. For the gastrocnemius muscle, amino acid and TCA cycle metabolite amounts were higher in high-MMR mice than in controls, indicating elevated amino acid and energy metabolism. Moreover, amounts of free fatty acids and triacylglycerol fatty acids in gastrocnemius muscle were lower in high-MMR mice than in controls, indicating elevated energy metabolism. Selection for increased MMR resulted in elevated amino acid and energy metabolism in the gastrocnemius muscle of high-MMR mice. These mice also exhibited a 3.5% correlated increase in mass-independent BMR. Because the untargeted metabolomic profiles were at resting metabolic rate and not at MMR, the elevated amino acid and energy metabolism in the gastrocnemius muscle of high-MMR mice may account for their correlated increase in mass-independent BMR. This dissertation provided quantitative genetic parameter estimates on MMR and BMR, tested the membrane pacemaker hypothesis of metabolism with a manipulative experiment using whole animals, and examined the biochemical variation between resting metabolism and increased MMR. Overall, the estimated genetic correlation between MMR and BMR is consistent with the assumption of the aerobic capacity model. In addition, the metabolic and fatty acid profiles suggest that increased MMR and BMR in high-MMR mice might be mechanistically linked via elevated amino acid and energy metabolism in the musculature. Lastly, my results add a genetic component to the already demonstrated roles of diet and exercise in determining membrane and intra-muscle fatty acid compositio

Geometric Methods in Partial Differential Equations

We study the interplay between geometry and partial differential equations. We show how the fundamental ideas we use require the ability to correctly calculate the dimensions of spaces associated to the varieties of zeros of the symbols of those differential equations. This brings to the center of the analysis several classical results from algebraic geometry, including the Cayley-Bacharach theorem and some of its variants as Serret’s theorem, and the Brill-Noether Restsatz theorem

Geometric Methods in Partial Differential Equations

We study the interplay between geometry and partial differential equations. We show how the fundamental ideas we use require the ability to correctly calculate the dimensions of spaces associated to the varieties of zeros of the symbols of those differential equations. This brings to the center of the analysis several classical results from algebraic geometry, including the Cayley-Bacharach theorem and some of its variants as Serret’s theorem, and the Brill-Noether Restsatz theorem