*To view the video, click
the title of each lecture.
Date:
Tuesday, 9 Sep. 2025,
09:00-10:00 TST
(GMT+8) , in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker:Dr. Sung-Jin Oh
(University of California, Berkeley)
Title:
Integral formulas for under/overdetermined differential
operators via recovery on curves and the finite-dimensional
cokernel condition Abstract: Underdetermined differential operators arise naturally in
diverse areas of physics and geometry, including the
divergence-free condition for incompressible fluids, the
linearized scalar curvature operator in Riemannian geometry,
and the constraint equations in general relativity. The duals
of underdetermined operators, which are overdetermined, also
play a significant role. In this talk, I will present recent
joint work with Philip Isett (Caltech), Yuchen Mao (UC
Berkeley), and Zhongkai Tao (IHÉS) that introduces a novel
approach - called recovery on curves - to constructing integral
solution/representation formulas (i.e., right-/left-inverses)
for a broad class of under/overdetermined operators via solving
ODEs on curves. They are optimally regularizing and have
prescribed support properties (e.g., produce compactly
supported solutions for compactly supported forcing terms). A
key feature of our approach is a simple algebraic condition on
the principal symbol - called the finite-dimensional cokernel
(FC) condition - that implies the applicability of our method.
This condition simplifies and unifies various treatments of
related problems in the literature. If time permits, I will
discuss applications to studying the flexibility of initial
data sets in general relativity.
Date:
Tuesday, 23 Sep. 2025,
15:00-16:00 TST
(GMT+8) , in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker:Dr.
Elia Brué (Università Bocconi)
Title:
Non-Uniqueness and Flexibility in Two-Dimensional
Euler Equations Abstract: In 1962, Yudovich established the well-posedness of the
two-dimensional incompressible Euler equations for solutions
with bounded vorticity. However, uniqueness within the
broader class of solutions with L^p vorticity remains a key
unresolved question. In this talk, I will survey recent
advances on this problem and present new nonuniqueness
results, obtained via the convex integration method. This
work is in collaboration with Colombo and Kumar.
Date: Tuesday,
14 Oct. 2025,
15:00-16:00 TST
(GMT+8) , in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker:Dr.
Jaemin Park (Yonsei University)
Title:
No anomalous dissipation in two dimensional fluids Abstract: In this talk, we will discuss
Leray-Hopf solutions to the incompressible Navier-Stokes equations
with vanishing viscosity. We explore important features of
turbulence, focusing around the anomalous energy dissipation
phenomenon. As a related result, I will present a recent result
proving that for two-dimensional fluids, assuming that the initial
vorticity is merely a Radon measure with nonnegative singular part,
there is no anomalous energy dissipation. Our proof draws on several
key observations from the work of J. Delort (1991) on constructing
global weak solutions to the Euler equation. We will also discuss
possible extensions to the viscous SQG equation in the context of
Hamiltonian conservation and existence of weak solutions for a rough
initial data. This is a joint work with Mikael Latocca (Univ. Evry)
and Luigi De Rosa (GSSI).
Date: Tuesday,
21 Oct. 2025,
15:00-16:00 TST
(GMT+8) , in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker:
Dr. Surjeet Singh Choudhary (National Center for Theoretical
Sciences)
Title: Twisted bilinear spherical maximal functions Abstract: In this talk, we will discuss \(L^{p}\)-estimates for the full and lacunary maximal functions associated with the twisted bilinear spherical averages given by
\[
\mathcal{U}_t(f_1,f_2)(x,y)
= \int_{S^{2d-1}}
f_1(x+t z_1,\, y)\,
f_2(x,\, y+t z_2)\,
d\sigma(z_1,z_2), \ t>0,
\]
for all dimensions \(d \ge 1\).
Date: Tuesday,
28 Oct. 2025,
9:00-10:00 TST
(GMT+8) , in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker:Dr. Bochen
Liu (Southern University of Science and Technology)
Title: Fourier frames on measures with Fourier decay Abstract: In this talk we shall discuss the
(non)existence of Fourier frames on measures with Fourier decay. In
dimension 2 and higher we only focus on surfaces with nonvanishing
Gaussian curvature, while in the unit interval we consider all
existing constructions of Salem measures in the literature.
Date: Tuesday,
4 Nov. 2025,
15:00-16:00 TST
(GMT+8) , in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker:Dr.
Alexander Tyulenev (Steklov Mathematical Institute)
Title: Traces of weighted Sobolev spaces in the limiting
case
Abstract: A complete description of traces on
\( \mathbb{R}^n \)
of functions from the weighted Sobolev space
\( W_1^{l}(\mathbb{R}^{n+1}, \gamma) \),
\( l \in \mathbb{N} \), with weight
\( \gamma \in A_1^{\mathrm{loc}}(\mathbb{R}^{n+1}) \)
will be presented.
Date: Tuesday,
11 Nov. 2025, 09:00-10:00 TST
(GMT+8) , in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker:Dr. Alexander
Nabutovsky (University of Toronto)
Title:
Boxing inequalities, widths, and systolic geometry
Abstract: We will present generalizations of the classical boxing inequality:
For a bounded domain \( \Omega \subset \mathbb{R}^{n+1} \) and a positive
\( m \in (0,n] \)
\( \mathrm{HC}_m(\Omega) \le c(m)\mathrm{HC}_m(\partial \Omega) \),
where \( \mathrm{HC}_m \) denotes the \( m \)-dimensional Hausdorff content.
Recall that \( \mathrm{HC}_m(X) \) is defined as the infimum of
\( \sum_i r_i^m \) over all coverings of \( X \) by metric balls, where
\( r_i \) denote the radii of these balls. The case \( m=n \) here is the classical
boxing inequality that is stronger than the isoperimetric inequality.
Yet this result is only a particular case of our boxing inequality valid also in higher codimensions:
For each Banach space \( B \) and compact \( M \subset B \) there is a “filling” of \( M \) by \( W \)
so that \( W \) is at the distance at most \( c(m)\mathrm{HC}_m^{1/m}(M) \) from \( M \) and
\( \mathrm{HC}_m(W) \le \mathrm{const}(m)\mathrm{HC}_m(M) \).
This result can be further generalized to the case where the ambient space \( B \)
is a metric space with a linear contractibility function.
This result generalizes the high-codimension isoperimetric inequality for Hausdorff contents
proven by B. Lishak, Y. Liokumovich, R. Rotman and the speaker originally motivated by
applications to systolic geometry.
The applications to systolic geometry involve inequalities that provide upper bounds for
the widths of \( M \subset B \) in terms of volume or Hausdorff contents of \( M \).
The widths \( W_m^B(M) \) measure how far \( M \) is from an \( m \)-dimensional simplicial complex in \( B \).
In the second part of the talk we will explain the new inequality\(
W_{m-1}^{l^\infty}(M) \le \mathrm{const}\,\sqrt{m}\,
\operatorname{vol}(M^m)^{1/m} \)
for closed manifolds \( M^m \subset \mathbb{R}^N \) and its implications to systolic geometry.
Here, the width is measured with respect to the \( l^\infty \) distance in the ambient Euclidean space.
Joint work with Sergey Avvakumov.
Date: Tuesday,
18 Nov. 2025, 15:00-16:00 TST
(GMT+8) , in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker:
Dr. Paz Hashash (Ben Gurion University of the Negev)
Title: The refined area formula for Sobolev mappings
Abstract:
We present a refined area formula for Sobolev mappings
\( \varphi : \Omega \to \mathbb{R}^n \). The classical identity
does not hold in general, since Sobolev mappings are not differentiable on large sets.
We show that the formula is valid once we remove an exceptional set of vanishing
Riesz capacity. The argument uses Lipschitz approximation of Sobolev mappings on
subsets where the capacity is large. On these subsets we apply the usual area
formula, and then pass to the limit. This gives an extension of the change of
variables formula beyond the Lipschitz case.