HW4 MA5311. Due February 24

Exercise 1. Let M be a smooth manifold and  V: C^\infty (M,R) \to C^\infty (M,R) be a linear operator such that for every smooth functions f,g: M \to R, V(fg)=fVg+gVf. Show that there exists a vector field U on M such that for every smooth function g, Vg(x)=dg_x (U(x)).

Exercise 2. Let B_n be the open unit ball in R^n. Let y in B_n. Show that there exists a smooth vector field on R^n, such that e^V(0)=y and V(x)=0 if x is not in B_n.

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HW3 MA5311. Due February 15

Exercise. Let X \subset \mathbf{R}^k be a subset homeomorphic to the closed ball B_n \subset \mathbf{R}^n.  Show that if f: X \to X is continuous, then there exists x \in X such that f(x)=x.

Exercise. Let X be a one-dimensional compact manifold with boundary. Show that X is diffeomorphic to a finite union of segments and circles (You may use the appendix in Milnor’s lecture notes).

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HW3 MA5161. Due February 15

Exercise. (First hitting time of a closed set by a continuous stochastic process)
Let (X_t)_{t \ge 0} be a continuous process adapted to a filtration (\mathcal{F}_t)_{t \ge 0}. Let

T=\inf \{ t \ge 0, X_t \in F \},

where F is a closed subset of \mathbb{R}. Show that T is a stopping time of the filtration (\mathcal{F}_t)_{t\ge 0}.

Exercise. (Closed martingale)
Let (\mathcal{F}_t)_{t \ge 0} be a filtration defined on a probability space (\Omega, \mathcal{F},\mathbb{P}) and let X be an integrable and \mathcal{F}-measurable random variable. Show that the process \left( \mathbb{E}(X\mid \mathcal{F}_t) \right)_{t \ge 0} is a martingale with respect to the filtration (\mathcal{F}_t)_{t \ge 0}.

Exercise. Let (\mathcal{F}_t)_{t \ge 0} be a filtration defined on a probability space (\Omega, \mathcal{F},\mathbb{P}) and let (M_t)_{t \ge 0} be a submartingale with respect to the filtration (\mathcal{F}_t)_{t \ge 0}. Show that the function t \rightarrow \mathbb{E} (M_t) is non-decreasing.

Exercise. Let (\mathcal{F}_t)_{t \ge 0} be a filtration defined on a probability space (\Omega, \mathcal{F},\mathbb{P}) and let (M_t)_{t \ge 0} be a martingale with respect to the filtration (\mathcal{F}_t)_{t \ge 0}. Let now \psi : \mathbb{R} \rightarrow \mathbb{R} be a convex function such that for t \ge 0, \mathbb{E} \left( \mid \psi(M_t) \mid \right) < + \infty. Show that the process (\psi(M_t))_{t \ge 0} is a submartingale.

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HW2 MA5161. Due February 3

Exercise 1. Let m:\mathbb{R}_{\ge 0} \rightarrow \mathbb{R} and let R: \mathbb{R}_{\ge 0} \times \mathbb{R}_{\ge 0} \rightarrow \mathbb{R} be a symmetric and positive function. Show that there exists a probability space \left( \Omega , \mathcal{F}, \mathbb{P} \right) and a Gaussian process (X_t)_{t \ge 0} defined on it, whose mean function is m and whose covariance function is R.

Exercise 2. Let (X_t)_{t \ge 0} be a continuous process adapted to a filtration \mathcal{F}_t. Show that (X_t)_{t \ge 0} is progressively measurable.

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HW2 MA5311: Due February 3

Exercise 1. Let g : \mathbf{R}^2 \to \mathbf{R}^4, (u,v) \to (\cos u , \sin u, \cos v, \sin v). Show that g( \mathbf{R}^2) is a 2-dimensional smooth manifold homeomorphic to the torus \mathbf{S}^1 \times \mathbf{S}^1.

Exercise 2. Let h_+: \mathbf{S}^2-N \to \mathbf{C} be the stereographic projection from the north pole N, and h_- be the stereographic projection from the south pole S.

  1. Show that for z \neq 0, h_+ h_-^{-1} =\frac{1}{\bar z}.
  2. Show that if P is a non constant polynomial, the map f=h_+^{-1} P h_+, f(N)=N is smooth.
  3. More generally, if Q: \mathbf{C} \to \mathbf{C} is smooth, find a condition on Q so that f=h_+^{-1} Q h_+, f(N)=N is smooth.
  4. By following Milnor’s argument in the proof of the fundamental theorem of algebra, find sufficient conditions so that a smooth map Q: \mathbf{C} \to \mathbf{C} is onto.

 

Exercise 3.  By using Sard’s theorem, prove that the set of regular values of a smooth map f : M \to N is dense in N.

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MA5161. HW 1 due Wednesday 1/25

Exercise 1. Show that the  \sigma-algebra \mathcal{T}(\mathbb{R}_{\ge 0},\mathbb{R}^d) is also generated by the following families:

  • \{ f \in \mathcal{A}(\mathbb{R}_{\ge 0}, \mathbb{R}^d ), f(t_1) \in B_1,...,f(t_n) \in B_n \} where t_1,...,t_n \in \mathbb{R}_{\ge 0} and where B_1,...,B_n are Borel sets in \mathbb{R}^d.
  • \{ f \in \mathcal{A}(\mathbb{R}_{\ge 0}, \mathbb{R}^d), (f(t_1),...,f(t_n)) \in B \} where t_1,...,t_n \in \mathbb{R}_{\ge 0} and where B is a Borel set in (\mathbb{R}^{d})^{\otimes n}.

 

Exercise 2.  Show that the following sets are in \mathcal{B} ([0,1],\mathbb{R}):

  • \{ f \in \mathcal{C}([0,1], \mathbb{R}), \sup_{t\in [0,1]} f(t) <1 \}
  • \{ f \in \mathcal{C}([0,1], \mathbb{R}), \exists t\in [0,1] f(t) =0 \}
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MA5311. About John Milnor

John Milnor is a renowned mathematician who made fundamental contributions to differential topology and was awarded the Fields medal in 1962. One of his most striking result is the existence of several distinct differentiable structures on the 7 dimensional sphere (!).

The following video is an introductory lecture to differential topology given in 1965. The first 26 minutes will give you a good overview of what differential topology is about.

 

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