UCSD Junior Analysis

• May 21: Rahul Sahjwani

Title: Euler equations in 2D with bounded initial vorticity 

Abstract: We will discuss the Yudovich theory for the well posedness of 2D Euler Equations for bounded initial vorticity. Time permitting we will discuss some explicit special solutions of these Equations exhibiting the creation of small scales.

• May 14: Zuo Lin

Title: Projection theorems and homogeneous dynamics

Abstract: Marstrand's projection theorem says that for a set in R^3 with Hausdorff dimension 0 < α < 1, its orthogonal projection to almost every direction is still α. A natural question is whether similar result holds for a more restrictive set of direction, for example, directions along some non-degenerate curve. Recent results by Käenmäki-Orponen-Venieri, Gan-Guo-Guth-Harris-Maldague-Wang, and Gan-Guo-Wang provide complete answers to this question. These results play a crucial role in recent progress on effective results in homogeneous dynamics by Lindenstrauss-Mohammadi-Wang-Yang. In this talk I'll give a brief overview for the projection theorems and how they come into place of homogeneous dynamics. If time permits, I'll talk about the proof idea of the restricted projection theorems and also my joint work with Ben Johnsrude for analogue in p-adic world.

• May 7: Yuchao Yi

Title: The Lorentzian Calderón Problem 

Abstract: Given a Lorentzian manifold (M, g) with timelike, null-convex boundary, the boundary value problem:

(\Box_g +q)u = 0                        u|_{\partial M} = f                    \operatorname{supp}(u) \subset J^+(\operatorname{supp}(f))

has a unique solution. The Dirichlet-to-Neumann (DN) map is defined by (\Lambda: f \mapsto \partial{\nu} u|{\partial M}). The Lorentzian Calderón Problem asks whether the metric is uniquely determined, up to gauge equivalence, by the DN map. The Riemannian Calderón Problem—posed in the same setting with a Riemannian metric—is one of the most important open problems in inverse problems, and the Lorentzian version also remains widely open. In this talk, I will give an overview of its history and recent progress, and present my own recent works on the problem.

• April 30: Minyoung Jeong

Title: Besicovitch 1/2 problem

Abstract: Rectifiable sets are a natural generalization of differentiable manifolds, just as Sobolev functions generalize differentiable functions. We will begin by discussing characterizations of rectifiable sets, which naturally lead to the Besicovitch 1/2 problem. Then, we will outline a rough idea of recent progress on this problem by Camillo De Lellis, Federico Glaudo, Annalisa Massaccesi, and Davide Vittone.

• April 23: Robert Koirala

Title: Quantitative differentiation for Lipschitz functions

Abstract: By Rademacher's theorem, a Lipschitz function is almost everywhere differentiable. In this talk, we present a quantitative version of this statement introducing the idea of quantitative differentiation due to Peter Jones.

• April 16: Vitor Borges

Title: Symmetries, vector fields and the long-time behavior of waves

Abstract: Decay estimates are central to the modern analysis of nonlinear dispersive equations. In this talk, we will discuss Klainerman's vector field method: a robust way to obtain pointwise decay estimates by using L2 based energy estimates and the symmetries of the Minkowski spacetime.  

• April 9: Tik Chan

Title: The 3D Kakeya set conjecture

Abstract: In this talk, we will discuss the paper "Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions" by Hong Wang and Joshua Zahl which proves the Kakeya set conjecture in R^3. There is no reasonable way to discuss a paper of this magnitude in any significant detail in a 50 minute talk so we will instead focus on the background of the Kakeya conjecture, past results and previous obstacles to proving the conjecture, and describing the very big picture idea of Wang and Zahl's proof. In particular, this will be a fairly informal talk.

• March 12: Sharv Laad

Title: Fredholm operators on pseudoconvex domains 

Abstract: Let D be the unit disk in C and let L2(D, n) be the set of L2 C^n valued square integrable functions on D. Let P be the orthogonal projection on H2(D, n), the closed subspace of holomorphic C^n valued square integrable functions. Let s be a nxn matrix of smooth complex valued functions continuous up to the boundary. A theorem of Gohberg and Krein states that the operator S:H2(D, n)->H2(D, n) given by multiplication by s followed by the orthogonal projection P is Fredholm if det s is non zero on the boundary. We will look at a generalisation of this result to strictly pseudoconvex domains due to Venugopalakrishna and examine the proof in detail. 

• March 5: Jasmine Browne

Title: The multiplicity of solutions to a nonlinear problem on Riemannian orbifolds

Abstract: It has been shown that the multiplicity of solutions of a certain nonlinear problem on compact, connected, closed Riemannian manifolds is greater than the manifold's Lusternik–Schnirelmann category. When trying to extend these results to an orbifold, however, there may be technical complications due to the structure of the space, such as the singular points. This talk will sketch the ideas involved in navigating these complications and how one may find an analogous result regarding the multiplicity of solutions to a similar problem on Riemannian orbifolds.

• February 26: Rahul Sahjwani

Title: Geometric Properties of Rectangles with sides parallel to the axes

Abstract: The Lebesgue differentiation theorem states that for locally integrable functions on Euclidean space we have 

$$ lim_{r\to 0} \frac{1}{|B_r(x)|} \int_{B_r(x)} f(y)dy = f(x) \ \ a.e \ x$$

i.e. averages of the function converge to the value of the function at almost every point. Naturally, we ask what happens if we change the class of averaging sets. It is known that if the class of averaging sets is of "bounded eccentricity", then this differentiation theorem still holds. However, this type of theorem fails if we consider, for instance, the class of sets to be arbitrary rectangles. In 1935, Jessen, Marcinkiewicz and Zygmund showed that if we restrict the rectangles to have sides parallel to the axes and impose slightly higher regularity conditions on $f$ then we can recover a similar type of differentiation theorem. Their proof relies essentially on a maximal function estimate now known as the strong maximal theorem. In 1975 Cordoba and Fefferman provided a geometric proof of this theorem. In this talk, we aim to illustrate the main idea behind this geometric proof (albeit for a slightly more restricted class than just rectangles with sides parallel to the axes).

• February 19: JJ Garzella

Title: A taste of estimates in arithmetic geometry

Abstract: We will define two (p-adic) Banach spaces, define a map between them, and prove that this map is surjective with a hairy, down to earth argument. The surjectivity of this map is a catastrophe. We will explain why.

• February 12: Liang Xu

Title: ABP method and applications to geometric inequalities

Abstract: In this talk I will survey some recent progress of geometric inequalities, mainly made by S. Brendle.  I will start by reviewing the Alesandrov-Bakelman-Pucci estimate which serves as the cornerstone in the theory of fully nonlinear elliptic equations. I will then move on to explain how the same technique can be applied to derive isoperimetric inequality, Sobolev inequality, Gagliardo-Nirenberg interpolation, and eigenvalue estimate. 

• January 29: Robert Koirala

Title: Geometric and Analytic Estimates and the Monge–Ampère Equation

Abstract: The PDE-based approach developed by Guo, Phong, and Tong—distinct from Kołodziej's pluripotential theory—has proven remarkably effective in establishing L∞ estimates for complex Monge–Ampère equations. This method has played a central role in deriving Green's function estimates, Sobolev inequalities, and diameter bounds in Kähler geometry. In this talk, we will outline the fundamental ideas behind this approach and discuss its applications to diameter estimates for Kähler manifolds, with potential extensions to the Kähler–Ricci flow if time permits.

• January 22: Tik Chan

Title: Multilinear restriction estimates

Abstract: In this talk we will discuss multilinear restriction estimates in general with a particular focus on the so-called n-linear estimate first studied by Bennett-Carbery-Tao in 2006, with further results by Guth (2008), Guth (2015), and Tao (2020) as well as ongoing work by Bejenaru and the speaker. Along the way we will see the significance of multilinearization and transversality, the techniques and ideas used to prove multilinear restriction estimates, and some of the interesting open problems which people are still thinking about. If there is time, we may also briefly discuss applications of these estimates to problems in harmonic analysis and PDE.

• January 15: Vitor Borges

Title: Global well-posedness and scattering for the massive Dirac-Klein-Gordon system in two dimensions

Abstract: The Dirac-Klein-Gordon (DKG) system provides a fundamental model for the interaction between fermions and bosons, often used to describe particle dynamics such as proton-proton interactions. In the context of dispersive equations, proving global well-posedness presents significant challenges, particularly as the spatial dimension decreases. In this talk, we will explore the difficulties in establishing global existence for solutions to the DKG system in low-dimensional settings. This is based on recent joint work with Ioan Bejenaru.

• December 6: Zhaolong Han

Title: Variational Analysis of a Parametrized Family of Transmission Problems Coupling Nonlocal and Fractional Models 

Abstract: I will present a recent work on the analysis of a parameterized family of energies associated with transmission problems that effectively couple two distinct models across an interface. Specifically, we examine the coupling between a model based on the regional fractional Laplacian and another model employing a nonlocal operator with a position-dependent interaction kernel. Both operators are inherently nonlocal and act on functions defined within their respective domains. The coupling occurs via a transmission condition across a hypersurface interface. The heterogeneous interaction kernel of the nonlocal operator leads to an energy space endowed with a well-defined trace operator. This, combined with well-established trace results of fractional Sobolev spaces, facilitates the imposition of a transmission condition across an interface. The family of problems will be parametrized by two key parameters that measure nonlocality and differentiability. For each pair of parameters, we demonstrate the existence of a solution to the resulting variational problems. Furthermore, we investigate the limiting behavior of these solutions as a function of the parameters.

• November 22: Yuchao Yi

Title: Holomorphic function vanishing on an open set must vanish everywhere 

Abstract: In this talk I will prove a well-known result, as stated in the title, from a purely PDE point of view. The main technique used is called Carleman Estimate, which is an extremely important tool in the study of Unique Continuation Problem, Inverse Problem, and Control Theory. Most of the talk will focus on proving this toy model, and connecting it to the strongly pseudoconvexity condition and Hörmander’s solvability condition, which hopefully offers a different perspective on the rigidity of holomorphic functions. If time permits, I will discuss its application in the Inventiones paper Lorentzian Calderón Problem Under Curvature Bounds by Spyros Alexakis, Ali Feizmohammadi and Lauri Oksanen. 

• November 15: Minyoung Jeong

Title: Variational approach to the regularity of the One-phase free boundary in dimension 2 

Abstract: The Alt-Caffarelli functional is defined as the sum of the Dirichlet energy and the measure of the positivity set of a non-negative function. Given a minimizer u of this functional, the free boundary is defined as the boundary of the positivity set of u, intersected with the domain of u. In 1981, Alt and Caffarelli showed that, in dimension 2, the free boundary is C^{1,alpha}-regular with no singularities. Their proof primarily relied on the improvement of flatness, a technique from minimal surface theory. In this talk, we will discuss a variational approach to the C^{1,alpha}-regularity, following the paper An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2-Dimensional Case by Luca Spolaor and Bozhidar Velichkov. 

• November 8: Tik Chan

Title: The pointwise convergence problem for the Schrödinger equation

Abstract: When do solutions to the Schrödinger equation pointwise converge almost everywhere back to the initial data? In this talk, we will discuss the paper Sharp L^2 estimates of the Schrödinger maximal function in higher dimensions by Xiumin Du and Ruixiang Zhang. In this paper, the authors prove a weighted restriction estimate which, when combined with a previous result by Jean Bourgain, fully answers this question up to the endpoint. Their proof uses several fundamental tools from modern harmonic analysis such as multilinear restriction estimates, decoupling, and induction on scales, which are combined together in a very interesting way in order to prove the result. Our goal in this talk is to describe these tools and where they come from and then give a high-level overview of how they are used to solve a problem which, a priori, does not seem to be related.

• November 1: Sharv Laad

Title: Geometry of the boundary and regularity of the Neumann operator

Abstract: In this talk we will discuss various notions of convexity of domains in C^n. We will define the complex Laplacian and its inverse, the Neumann operator. In the case of the complex Laplacian, ellipticity fails at the boundary and thus a natural question is how regular the Neumann operator is. We will see how the geometry at the boundary influences these questions. 

• October 25: Junekey Jeon

Title: Local well-posedness of touching g-SQG patches

Abstract: Singularity formation has been an important topic in the field of mathematical fluid dynamics, and the g-SQG patch equation is one of a few incompressible fluid equations whose global ill-posedness has been explicitly shown, by the work of Kiselev, Ryzhik, Yao and Zlatos. In their counterexample, the presence of the boundary in the domain plays a key role in stabilizing the dominant motion that eventually drives the singularity formation. For this reason, the authors of the aforementioned work also had to prove local well-posedness of the g-SQG patch equation in a domain with boundary, specifically the half-plane which is the domain they used for their global ill-posedness result, and it turned out, the presence of boundary does impose nontrivial complications compared to the free space case (whose local well-posedness already had been established by earlier works by Gancedo et al) due to possible touches of patches with their reflections along the boundary. In the presented paper, we prove an extension of this local well-posedness result on the half-plane, concerning patches on the free space which are not only allowed to touch their reflections along a line but also allowed to touch any other patches in any way, as long as the signs of the touching patches are aligned with the direction of touch, from inside or from outside. A notable feature of this work is that the solution we are concerning does not live in a vector space. Depending on the time pressure, we will take a brief look at some general setup, the a priori estimate, construction of a solution, and the uniqueness argument in that order.

• October 18: Robert Koirala

Title: Shapes and their singularities

Abstract: The main question we will address in this talk is the following. ''What is the size of a geometric shape and its singularity?” The shapes we will focus on will be defined by level sets of functions that satisfy some PDEs. You can think of zero sets of homogeneous polynomials if you like algebraic varieties. Else, you can think of zero sets of harmonic functions. In any case, we will talk about dimension and volume estimates of such shapes and their singularities. 

• October 4: Vitor Borges

Title: Frequency and modulation analysis of nonlinear dispersive PDEs

Abstract: The Dirac-Klein-Gordon (DKG) system is a basic model for the strong interaction of two atomic nucleons. From a PDE perspective, it is a coupled system of quadratic semilinear dispersive equations possessing a null structure similar to some other well-studied models, like the Dirac-Maxwell and Wave Maps equations. In this talk, I will use the DKG system as the backdrop to discuss important dispersive PDE techniques, such as paraproduct decompositions and the modulation spaces X^{s,b}.

• June 7: Vaki Nikitopoulos

Title: Higher derivatives of perturbed operator functions

Abstract: I'll talk about my paper, "Higher derivatives of operator functions in ideals of von Neumann algebras."  It concerns the problem of "Taylor expanding" the map that takes a(n unbounded) self-adjoint operator A on a Hilbert space and returns f(A), where f is a scalar function and f(A) is defined via the functional calculus.

• May 31: Yuchao Yi

Title: Application of Microlocal analysis in Nonlinear Inverse Problem

Abstract: Inverse problem is the study of recovering parameters of a PDE given partial information on the relationship between initial data and solution of PDE. We will first spend some time on talking about what inverse problems are, and hopefully some motivation of why microlocal analysis can be used to help solve inverse problems. I'll then introduce some of the core ideas in microlocal analysis, including elliptic regularity and propagation of singularity. Finally we shall go over how to use them for nonlinear hyperbolic type PDE, where the method was first developed by Yaroslav Kurylev, Matti Lassas and Gunther Uhlmann in 2018: https://link.springer.com/article/10.1007/s00222-017-0780-y.

• May 24: Minyoung Jeong

Title: Regularity of the Free Boundary in Obstacle Problem

Abstract: The obstacle problem is a classical topic within the field of free boundary problems. This talk will begin by examining classical results and the proof strategy developed by Caffarelli, inspired by De Giorgi's blow-up argument for proving the regularity of minimal surfaces. We will then see recent advancements in the field, including improvements in the stratification of singular sets and the generalization of the obstacle problem to non-linear operators.

• May 3: Soumya Ganguly

Title: A Bergman kernel localization technique, and its applications in several complex variables.

Abstract: Bergman kernel and metrics are one of the most useful tools for proving many analytic/geometric results in several complex variables. We will go over a technique of localizing the Bergman kernel near the boundary point of some special domains. We will also see quite a few deep applications of this localization result.   

• April 26: Robert Koirala

Title: Quantitative Stratification of Singularities in Geometric Analysis

Abstract: Ricci flow, as a geometric partial differential equation (PDE), forms singularities. If we were to define a notion of weak solutions to the PDE, we would have to understand singularity formation. Furthermore, understanding singularities allows us to have topological applications, such as Perelman's resolution of the Poincaré conjecture. In this talk, I will sketch out, rather informally and mostly with drawings and intuition, the idea of quantitative stratification of singularities in Ricci flow, following a series of three papers by Richard Bamler: arxiv:2008.07093, arxiv:2008.09298, and arxiv:2009.03243. Bamler’s theory serves as a parabolic analog of the quantitative stratification results of Cheeger, Jiang, Naber in the setting of Riemmannian manifolds with bounded Ricci curvature. I will clarify the analogy at least heuristically. Finally, I will talk about some topological questions that could possibly be resolved by (a) Ricci flow(er). 

• April 19: Vitor Borges

Title: Testing by wave packets and modified scattering in nonlinear dispersive PDEs

Abstract: In this talk, I will discuss the eponymous 2022 paper "Testing by wave packets and modified scattering in nonlinear dispersive PDEs" by Mihaela Ifrim and Daniel Tataru. The method of testing by wave packets consists in constructing approximate solutions to dispersive PDEs that interpolate between physical and Fourier space asymptotic techniques, leading to results which are near optimal in terms of regularity and decay of the initial data. While the method was originally devised to study the long-time behavior of the 1d cubic nonlinear Schrödinger equation, it also has been successfully used to study water waves and other dispersive PDEs (such as the SQG and KdV equations). To make the talk more accessible, I will provide the basic framework of dispersive PDEs and scattering theory as it applies to our goals.

• April 12: Tik Chan

Title: Polynomial Partitioning in Fourier Restriction

Abstract: In this talk, I will discuss Hong Wang's 2020 paper "A restriction estimate in R3 using brooms" which proved the best-known 3D Fourier restriction estimates at the time. The proof is based on polynomial partitioning, which was originally introduced by Guth and Katz to make significant progress on the Erdős distinct distance problem and then later adapted by Guth to the harmonic analysis setting to prove new restriction estimates. This paper extends Guth's polynomial partitioning ideas by iterating the partitioning process and using additional incidence geometric inputs, namely Wolff's "two-ends" argument and analyzing "broom" structures. Due to the technical nature of the proof, I will mostly focus on describing polynomial partitioning and its place in the greater context of Fourier restriction theory without too many technical details.