Junior Analysis 2023 - 2024
Welcome to Junior Analysis! Below is a schedule for Spring 2024. If you'd like to give a talk, email the organizer Kenneth DeMason (email: kdemason [at] utexas.edu). Keep in mind that he will be absent from Mar 01 to Mar 31.
This semester Junior Analysis will meet on Mondays at 2pm in PMA 10.176.
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Talk Schedule (Spring 2024)
Title: An Intro to Stochastic Analysis, Part 1 of 2
Abstract: These two seminars will give an overview of the main topics in stochastic analysis, such as martingale theory, Brownian motion, and Ito calculus. The focus of the seminar will be presenting the important definitions, concepts, and the intuition behind them, rather than working through proofs in detail.
Title: An Intro to Stochastic Analysis, Part 2 of 2
Abstract: See previous.
Title: From weak to strong convergence in large fermionic systems.
Abstract: In this talk I will give an overview of the main result of https://arxiv.org/abs/2312.03953. The minimizers of energy functionals in quantum mechanics are states that describe the configuration that the system reaches. In the macroscopic limit, these configurations can be described through an effective theory in the form of a convergence theorem. It had been proven that such convergence holds in a rather weak sense. We will show that it is possible to upgrade the mode of convergence to norm convergence, using physical information of the system (spoiler alert: the Pauli exclusion principle).
Title: Introduction to Tauberian Theorems
Abstract: A classical theorem of real analysis is Abel's theorem, which states that if the sum of the coefficients of a power series converges, then the power series itself converges to the same sum when the variable approaches 1 from the left. Decades later, Alfred Tauber found a condition that allowed one to conclude the converse. As a result, theorems giving conditions under which summability in a weaker sense implies summability in a stronger one are known as Tauberian theorems. In this talk I will introduce some Tauberian theorems and the ideas used to prove them.
Title: Viscous Destabilization of Shock Profiles for Scalar Conservation Laws
Abstract: We will discuss stability properties for viscous perturbations of scalar conservation laws. The recent theory of \(a\)-contraction with shifts gives stability results for the inviscid limit from viscous scalar conservation laws (VSCL) to scalar conservation laws (SCL) for small-amplitude shock profiles. In some sense, this shows that for small shocks, stability properties for the VSCL are very similar to those of the SCL. I will present results showing that for large-amplitude shocks, the situation is markedly different between the two.
Title: The Regularizing Effects of Grazing Collisions
Abstract: The homogeneous Landau equation is one of the fundamental models for collisional plasmas. In this talk, I will review some properties of the equation and show how grazing collisions lead to regularizing effects for solutions. Because the equation is highly nonlinear, these estimates are necessary to construct solutions.
Title: On Uniqueness Properties of Solutions of the Generalized Fourth-Order Schrödinger Equations
Abstract: In this paper, we study uniqueness properties of solutions to the generalized fourth-order Schrödinger equations in any dimension of the following forms, \[ i\partial_t u+\sum_{j=1}^d \partial_{x_j}^4u=V(x,t)u\] and \[ i\partial_t u+\sum_{j=1}^d \partial_{x_j}^4u=F(u,\bar{u}).\] We show that a linear solution u with fast enough decay in certain Sobolev spaces at two different times has to be trivial. Consequently, if the difference between two nonlinear solutions \(u_1\) and \(u_2\) decays sufficiently fast at two different times, it implies that \(u_1 \equiv u_2\).
Title: Existence to solutions to conservation laws via the kinetic formulation
Abstract: I will present some results from the paper Lions, Perthame, and Tadmor "Kinetic Formulation of Conservation laws" (JAMS 1994.) This will lead to a theory different from what I presented last semester which was heavily based on BV estimates and gave existence for initial data with small total variation.
Title: An introduction to Wave Turbulence and the Wave Kinetic Equation
Abstract: As the analog to the Boltzmann equation for a classical system of interacting particles, the wave kinetic equation (WKE) arises as an effective equation from microscopic systems of interacting waves. We will (hopefully) develop some intuition for turbulent systems, define quantities of interest, and sketch a formal derivation of the WKE.
Title: On Isoperimetric Inequalities
Abstract: The isoperimetric inequality is a classical inequality first proven heuristically by Steiner using symmetrization. We'll showcase this technique, as well as two others borrowing ideas from Optimal Transport and the ABP method, to prove the isoperimetric inequality and its rigidity case.
Previous Semesters
- September 01: No talk.
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September 8: Esteban Cardenas
Title: The Hartree Equation
Abstract: The purpose of this talk is to give an introduction to the Hartree equation. This is an example of a well-known semi-linear Schrödinger equation, which describes the mean-field dynamics of many-body quantum-mechanical gases. We will discuss both its mathematical properties as a PDE, and its emergence from physical systems.
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September 15: Justin Toyota
Title: Real Proofs of Complex Theorems and Vice Versa
Abstract: While the "flavor" of real and complex analysis are often quite different, the two are linked in a variety of ways. In this talk, we will examine theorems in complex analysis that can be proved using ideas from real analysis, and conversely theorems in real analysis that can be proved by applying the theory of complex functions.
The paper where I drew these results from (which also contains a lot of other stuff) was
Zalcman, Lawrence. "Real Proofs of Complex Theorems (and Vice Versa)". The American Mathematical Monthly 81 (2), pp 115-137 (1974)
The generalization of Morera's theorem is discussed in sections 1-2 of that paper. That proof and the proof of Morera's theorem for two circles are described in
Zalcman, Lawrence. "Analyticity and the Pompeiu Problem". Archive for Rational Mechanics and Analysis 47, pp 237-254 (1972)
The proof of the Pál--Bohr theorem is given in the Zalcman paper, modulo the trick of adding the function of bounded variation. (It also doesn't include the assumption on the mean of the function, which I think might be a mistake.) The proof of the trick is given in the following paper, which I think is the source of this proof of the theorem:
Salem, R. "On a theorem of Bohr and Pál". Bulletin of the American Mathematical Society 50, pp 579-580 (1944)
My mention that simultaneously uniformizing the Fourier series of two functions is not possible in general was in reference to the following paper, which I admit I didn't read too closely, so my interpretation may be faulty.
Lebedev, Vladimir. "The Bohr--Pál Theorem and the Sobolev Space W_2^{1/2}". arXiv:1508.07167
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September 22: Jeffrey Cheng
Title: (Weighted) Poincaré inequalities
Abstract: We will present multiple proofs of classical Poincaré inequalities and prove a weighted one. Then, we will see an application in fluid mechanics.
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September 29: Zach Richey
Title: The Law(s) of Large Numbers
Abstract: We work through several proofs of the Strong and Weak Laws of Large Numbers, and discuss the different techniques they use. We end with Etemadi's elementary 1981 proof of the Strong Law. No background in probability is required, only some basic real analysis.
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October 06: Cooper Faile
Title: Conservation Laws I
Abstract: I will discuss shock and rarefaction solutions to conservation laws, solve the Riemann problem, and (hopefully) start the existence theory via front tracking for initial value problems with small BV initial data.
The solution to the Riemann problem is covered in Evans Chapter 11 and the Bressan tutorial on conservation laws . The Bressan tutorial outlines front tracking while this short paper by Baiti and Jenssen covers it in more detail.
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October 13: Cooper Faile
Title: Conversation Laws II
Abstract: This is part two of last week's talk.
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October 20: Esteban Cardenas
Title: Title: Quantum Boltzmann dynamics in Fermi gases
Abstract: The derivation of the quantum Boltzmann equation (qBe) from first principles is a longstanding open problem in mathematical physics. On the other hand, in recent years, the field has seen tremendous progress in the development of mathematical tools for the study of mean-field theories. This has led to the rigorous development of key physical ideas describing the dynamics of cold gases of fermions, including the "bosonization" of their statistics. In this talk, we present the main results of https://arxiv.org/abs/2306.03300. We revisit the emergence of the qBe, taking into account the effect of this phenomenon.
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October 27: Kenneth DeMason
Title: Strong Sharp Quantitative Stability for Crystalline Surface Tensions.
Abstract: The classical isoperimetric inequality exemplifies a beautiful relationship between the perimeter and volume of a set. It arises from the goal to find perimeter-minimizing sets with fixed volume, a problem which dates back to antiquity, and characterizes these minimizers as the equality case (balls). Quantitative stability concerns the natural follow-up question: If your set almost minimizes the perimeter, is it close to a ball? This area of study has flourished over the past two decades, leading to many generalizations. One such example is the anisotropic perimeter, where certain directions are preferred. We prove a strong sharp quantitative stability result for so-called crystalline surface tensions, whose anisotropic perimeter minimizers are polytopes.
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November 03: Patricia Ewald
Title: Otto calculus and gradient flows
Abstract: We will discuss how to interpret certain PDEs as gradient flows with respect to the Wasserstein metric, and comment on why that could be useful, following Felix Otto's seminal paper "The geometry of dissipative evolution equations: the porous medium equation".
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November 10: Luisa Velasco
Title: Brownian Motion and the Heat Equation
Abstract: The macroscopic phenomenon of diffusion arises from the random motion of particles. Thus, it is not surprising that there is a deep connection between Brownian motion and heat and other diffusion equations. We will examine the relation between the Brownian motion and the heat equation as well as look at some examples of PDE that can be solved by running the Brownian motion.
- November 17: No talk.
- November 24: Fall break, no talk.
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December 01: Antonio Farah
Title: The Grad-Mercier Free Boundary
Abstract: The Grad Mercier equation is a non-local PDE with applications in plasma physics. In this talk, I will first present a survey of the current existence, uniqueness, and regularity results on the Grad Mercier PDE. Then, I will show that the free boundary associated with the equation is a set of finite n-1 Hausdorff measure. This is based on joint work with Luis Caffarelli, Daniel Restrepo, and Ignacio Tomasetti.
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December 08: Will Porteous [feat. 18th Century all-stars Lagrange, Hamilton, Legendre and more]
Title: Introduction to Optimal Control Theory
Abstract: Yes, it's true - stochastic optimal control problems and Hamilton Jacobi (HJ) equations pervade the applied sciences and play a key role in modern PDE theory. However: most students have never been introduced to optimal control and first-order Hamilton-Jacobi PDE associated to them, let alone their more sophisticated stochastic counterpart (with 2nd order HJ equations). We will introduce the topic by answering these questions: What is an optimal control problem, as opposed to a calculus-of-variations problem? How do such problems yield Hamilton-Jacobi equations? When does calculus of variations reach its limits, and modern PDE theory [i.e. the viscosity solution] begin? Come to find out!