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BEGIN:VEVENT
SUMMARY:Ryan Matzke (University of Minnesota)
DTSTART:20200504T150000Z
DTEND:20200504T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/1
DESCRIPTION:Title:
Discreteness of energy-minimizing measures.\nby Ryan Matzke (Universit
y of Minnesota) as part of HA-GMT-PDE Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/anpdews/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guillermo Rey (Wing)
DTSTART:20200511T150000Z
DTEND:20200511T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/2
DESCRIPTION:Title:
Another counterexample to Zygmund's conjecture\nby Guillermo Rey (Wing
) as part of HA-GMT-PDE Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/anpdews/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Burkart (Stony Brook University)
DTSTART:20200618T150000Z
DTEND:20200618T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/3
DESCRIPTION:Title:
Transcendental Julia sets with fractional packing dimension\nby Jack B
urkart (Stony Brook University) as part of HA-GMT-PDE Seminar\n\n\nAbstrac
t\nIn this talk\, we will define and compare different definitions of dime
nsion (Hausdorff\, Minkowski\, and packing) used to analyze fractal sets.
We will then define the basic objects in complex dynamics\, and discuss so
me history of results about the dimension of the fractal Julia sets that f
amously show up in this area. No prior knowledge of complex dyanmics will
be assumed. We will conclude by discussing my recent construction of a Jul
ia set of a non-polynomial entire function with packing dimension strictly
between one and two. We will see that Whitney decompositions\, a foundati
onal tool in harmonic analysis\, play a vital role in the dimension calcul
ation.\n
LOCATION:https://researchseminars.org/talk/anpdews/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wenjie Lu (University of Minnesota)
DTSTART:20200521T150000Z
DTEND:20200521T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/4
DESCRIPTION:Title:
On the De Gregorio modification of the Constantin-Lax-Majda model\nby
Wenjie Lu (University of Minnesota) as part of HA-GMT-PDE Seminar\n\n\nAbs
tract\nI will introduce the Constantin-Lax-Majda model with De Gregorio mo
difications. More specifically\, I will focus on the problems related to s
tationary solutions and well-posedness.\n
LOCATION:https://researchseminars.org/talk/anpdews/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lisa Naples (University of Connecticut)
DTSTART:20200526T150000Z
DTEND:20200526T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/5
DESCRIPTION:Title:
Rectifiability of pointwise doubling measures in Hilbert space\nby L
isa Naples (University of Connecticut) as part of HA-GMT-PDE Seminar\n\n\n
Abstract\nJones’ beta numbers measure the flatness of a set at various
scales and windows. Since their introduction\, beta numbers have served a
s an important tool to relate the geometric structure of sets and measures
and to measure-theoretic quantities. We will extend results of Badger an
d Schul to show that an $L^2$ variant of the beta numbers can be used to c
haracterize rectifiable pointwise doubling measures in Hilbert space. We
will also discuss results for the related notions of graph rectifiability
and fractional rectifiability.\n
LOCATION:https://researchseminars.org/talk/anpdews/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gianmarco Brocchi (University of Birmingham)
DTSTART:20200528T150000Z
DTEND:20200528T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/6
DESCRIPTION:Title:
Random dyadic grids and what they can do for you\nby Gianmarco Brocchi
(University of Birmingham) as part of HA-GMT-PDE Seminar\n\n\nAbstract\nI
n the last decade dyadic analysis and probabilistic methods have been succ
essfully used to obtain optimal weighted estimates. In this talk we intro
duce dyadic grids and their random shifted analogue. We discuss in an info
rmal way some of the advantages of this tool and how it can be used to dec
ompose your operator.\n
LOCATION:https://researchseminars.org/talk/anpdews/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Turner (University of Birmingham)
DTSTART:20200604T150000Z
DTEND:20200604T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/8
DESCRIPTION:Title:
Solvability of boundary value problems for the Schrodinger equation with n
onnegative potentials\nby Andrew Turner (University of Birmingham) as
part of HA-GMT-PDE Seminar\n\n\nAbstract\nAbstract pdf: https://drive.goog
le.com/file/d/17Kzg-Zp2a9fhr8_2E9IWM_w4v7O3b383/view\n
LOCATION:https://researchseminars.org/talk/anpdews/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dimitrije Cicmilovic (University of Bonn)
DTSTART:20200608T150000Z
DTEND:20200608T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/9
DESCRIPTION:Title:
Symplectic non-squeezing and Hamiltonian PDE\nby Dimitrije Cicmilovic
(University of Bonn) as part of HA-GMT-PDE Seminar\n\n\nAbstract\nIn this
talk we shall discuss infinite dimensional generalization of\nGromov's sym
pelctic nonsqueezing result. As an application we will present\nmass subcr
itical and critical nonlinear Schrodinger equation. Nonsqueezing\nproperty
of the said flows was already known\, however the techniques used\nare ba
sed on finite dimensional Gromov's result\, while ours presents a\nmore na
tural way of looking at the Hamiltonian structure of the equations.\nAddit
ionally\, we shall remark on future projects in terms of application\nof t
he non-squeezing property.\nJoint work with Herbert Koch.\n
LOCATION:https://researchseminars.org/talk/anpdews/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damian Dabrowski (Universitat Autonoma de Barcelona)
DTSTART:20200615T150000Z
DTEND:20200615T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/10
DESCRIPTION:Title: Cones\, rectifiability and singular integral operators.\nby Damian Da
browski (Universitat Autonoma de Barcelona) as part of HA-GMT-PDE Seminar\
n\n\nAbstract\nLet K(x\, V\, s) be the open cone centred at x\, with direc
tion V\, and aperture s. It is easy to see that if a set E satisfies for s
ome V and s the condition:\n"if x belongs to E\, then E has an empty inter
section with K(x\, V\, s)"\,\nthen E is a subset of a Lipschitz graph. To
what extent can we weaken the condition above and still get meaningful inf
ormation about the geometry of E? It depends on what we mean by "meaningfu
l information''\, of course. For example\, one could ask for rectifiabilit
y of E\, or if E contains big pieces of Lipschitz graphs\, or if nice sing
ular integral operators are bounded in L^2(E). In the talk I will discuss
these three closely related questions.\n
LOCATION:https://researchseminars.org/talk/anpdews/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yijing Wu (University of Maryland)
DTSTART:20200629T150000Z
DTEND:20200629T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/12
DESCRIPTION:Title: Existence\, uniqueness and regularity of the minimizer of energy related
to perimeter minus fractional perimeter\nby Yijing Wu (University of M
aryland) as part of HA-GMT-PDE Seminar\n\n\nAbstract\nWe are interested in
the asymptotic behaviors of the following energy functional $E(\\Omega)=\
\sigma Per(\\Omega)+\\beta V_K(\\Omega)$ defined for $|\\Omega|=m$. Here t
he perimeter tries to keep the mass together in a ball\, and $V_K$ is a no
n-local repulsive interaction energy trying to spread the mass around. We
will then discuss the existence\, uniqueness snd regularity properties of
the minimizers of the energy especially in the regime where the energy $E(
\\Omega)$ converges to Perimeter minus fractional perimeter.\n
LOCATION:https://researchseminars.org/talk/anpdews/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gael Diebou (University of Bonn)
DTSTART:20200716T150000Z
DTEND:20200716T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/14
DESCRIPTION:Title: The Dirichlet problem for weakly harmonic maps with rough data\nby Ga
el Diebou (University of Bonn) as part of HA-GMT-PDE Seminar\n\n\nAbstract
\nIn this talk\, we will discuss the well-posedness issues for weakly harm
onic maps subject to Dirichlet boundary data assuming a minimal regularity
. After a brief description of the problem we will present our techniques
which partly rely on certain fundamental notions in harmonic analysis such
as Carleson measures\, its intrinsic connection to the John-Nirenberg spa
ce BMO and the Laplace operator... With an appropriate reformulation of ou
r problem\, various solvability results (existence\, uniqueness and regula
rity) will then be reviewed. Our approach (nonvariational)\, as we will se
e\, is suitable for the analysis of critical or endpoint elliptic boundary
value problems and hence can unambiguously be applicable to similar type
of equations or systems driven by classical operators. For this talk\, we
only mention a generalization of our results to second-order constant elli
ptic systems.\n\nThis is a joint work with Herbert Koch.\n
LOCATION:https://researchseminars.org/talk/anpdews/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lois Okereke (African University of Science and Technology)
DTSTART:20200720T150000Z
DTEND:20200720T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/15
DESCRIPTION:Title: Iterative methods for nonlinear optimisation problems - Prospects and app
lications\nby Lois Okereke (African University of Science and Technolo
gy) as part of HA-GMT-PDE Seminar\n\n\nAbstract\nAn important class of ext
remal problems in nonlinear functional analysis is the nonlinear optimisat
ion problem where some of the objective functions are nonlinear. In many
cases where existence of solutions is guaranteed\, these solutions are not
usually affordable in a direct way. Iterative methods (or algorithms)\, t
herefore provide a convenient way of approximating these solutions. To a l
arge extent\, most of these iterative methods can be traced to the popular
gradient descent algorithm. This talk presents the prospects that may res
ult in using a different approach\, and its usefulness even to equivalent
reformulations of the nonlinear optimisation problem. Its applicability in
some areas of science and technology is highlighted and a spectacular ap
plication in radiotherapy treatment planning where algorithmic efficiency
is especially required is demonstrated.\n\nThis work is Joint with Charles
Ejike Chidume.\n
LOCATION:https://researchseminars.org/talk/anpdews/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:José Luis Luna García (University of Missouri)
DTSTART:20200727T150000Z
DTEND:20200727T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/16
DESCRIPTION:Title: Critical Perturbations for Linear Elliptic Equations\nby José Luis L
una García (University of Missouri) as part of HA-GMT-PDE Seminar\n\n\nAb
stract\nIn this talk we develop a perturbation theory for the L^2 solvabil
ity of certain Boundary Value Problems for linear elliptic equations with
complex coefficients in the upper half space. While we expect the methods
to apply to general systems and higher order equations\, we will focus her
e on the general scalar second order equation\, for which most of the main
difficulties are already present: For instance a lack of boundedness and
continuity of solutions\, precluding the use of a pointwise-defined fundam
ental solution.\n\nOur theory is based on solvability via the method of la
yer potentials. As such the main points to consider are boundedness and in
vertibility\, in the appropriate functional spaces\, of the corresponding
operators and their boundary traces. For the boundedness issue we employ t
he theory of local Tb theorems\, to obtain control on certain square funct
ions that allow us to conclude the desired bounds on the layer potentials.
The invertibility will be treated through the analyticity of the boundar
y traces as a function of the coefficients of the equation.\n\nOf technica
l interest is that our methods allow us to obtain nontangential maximal fu
nction estimates for the layer potential solutions so constructed.\n\nThis
is joint work with Simon Bortz\, Steve Hofmann\, Svitlana Mayboroda\, and
Bruno Poggi.\n
LOCATION:https://researchseminars.org/talk/anpdews/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zihui Zhao (University of Chicago)
DTSTART:20200810T150000Z
DTEND:20200810T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/17
DESCRIPTION:Title: Boundary regularity of area-minimizing currents: a linear model with anal
ytic interface\nby Zihui Zhao (University of Chicago) as part of HA-GM
T-PDE Seminar\n\n\nAbstract\nGiven a curve \\Gamma \, what is the surface
T that has least area among all surfaces spanning \\Gamma? This classical
problem and its generalizations are called Plateau's problem. In this tal
k we consider area minimizers among the class of integral currents\, or ro
ughly speaking\, orientable manifolds. Since the 1960s a lot of work has b
een done by De Giorgi\, Almgren\, et al to study the interior regularity o
f these minimizers. Much less is known about the boundary regularity\, in
the case of codimension greater than 1. I will speak about some recent pro
gress in this direction and my joint work with C. De Lellis.\n
LOCATION:https://researchseminars.org/talk/anpdews/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jungang Li (Brown University)
DTSTART:20200813T150000Z
DTEND:20200813T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/21
DESCRIPTION:Title: The L^p-ellipticity and L^p-Dirichlet problems of second order elliptic s
ystems\nby Jungang Li (Brown University) as part of HA-GMT-PDE Seminar
\n\n\nAbstract\nIn this talk we will discuss a structural condition of sec
ond order elliptic systems with complex coefficients\, namely the L^p-elli
pticity condition\, which can be viewed as an L^p version of the classical
ellipticity condition. Such condition naturally implies both interior and
boundary estimates\, which act as a proper substitution of the De Giorgi-
Nash-Moser regularity theory. The new regularity result will help us to pr
ove an extrapolation theorem of the L^p-Dirichlet problem and we will appl
y it to two well-studied cases: Lam\\'e equations and homogenization probl
ems. This is a joint work with M. Dindos and J. Pipher.\n
LOCATION:https://researchseminars.org/talk/anpdews/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rajula Srivastava (University of Wisconsin-Madison)
DTSTART:20200820T150000Z
DTEND:20200820T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/24
DESCRIPTION:Title: Orthogonal Systems of Spline Wavelets as Unconditional Bases in Sobolev S
paces\nby Rajula Srivastava (University of Wisconsin-Madison) as part
of HA-GMT-PDE Seminar\n\n\nAbstract\nWe exhibit the necessary range for wh
ich functions in the Sobolev spaces $L^s_p$ can be represented as an uncon
ditional sum of orthonormal spline wavelet systems\, such as the Battle-Le
marié wavelets. We also consider the natural extensions to Triebel-Lizork
in spaces. This builds upon\, and is a generalization of\, previous work o
f Seeger and Ullrich\, where analogous results were established for the Ha
ar wavelet system.\n
LOCATION:https://researchseminars.org/talk/anpdews/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jennifer Duncan (University of Birmingham)
DTSTART:20200723T150000Z
DTEND:20200723T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/25
DESCRIPTION:Title: An Algebraic Brascamp-Lieb Inequality\nby Jennifer Duncan (University
of Birmingham) as part of HA-GMT-PDE Seminar\n\n\nAbstract\nThe Brascamp-
Lieb inequalities are a natural generalisation of many familiar multilinea
r inequalities that arise in mathematical analysis\, classical examples of
which include Holder’s inequality\, Young’s convolution inequality\,
and the Loomis-Whitney inequality. Each Brascamp-Lieb inequality is unique
ly defined by a 'Brascamp-Lieb datum'\, which is a pair consisting of a se
t of linear surjections between euclidean spaces and a set of exponents co
rresponding to these maps. It is common in applications to encounter nonli
near variants\, where the linear maps are replaced with nonlinear maps bet
ween manifolds. By incorporating a dampening factor that compensates for l
ocal degeneracies\, we establish a global nonlinear Brascamp-Lieb inequali
ty for a broad class of maps that exhibit a certain algebraic structure\,
with a constant that explicitly depends only on the associated 'degrees' o
f these maps.\n
LOCATION:https://researchseminars.org/talk/anpdews/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiao He (University of Evry - Paris Saclay)
DTSTART:20200709T150000Z
DTEND:20200709T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/27
DESCRIPTION:Title: Regularity criteria for weak solutions to the three-dimensional MHD syste
m\nby Jiao He (University of Evry - Paris Saclay) as part of HA-GMT-PD
E Seminar\n\n\nAbstract\nIn this talk we will first review various known r
egularity criteria and partial regularity theory for 3D incompressible Nav
ier-Stokes equations.\n\nI will then present two generalizations of partia
l regularity theory of Caffarelli\, Kohn and Nirenberg to the weak solutio
ns of MHD equations. The first one is based on the framework of parabolic
Morrey spaces. We will show parabolic Hölder regularity for the "suitable
weak solutions" to the MHD system in small neighborhoods. This type of pa
rabolic generalization using Morrey spaces appears to be crucial when stud
ying the role of the pressure in the regularity theory and makes it possib
le to weaken the hypotheses on the pressure.\n\nThe second one is a regula
rity result relying on the notion of "dissipative solutions". By making us
e of the first result\, we will show the regularity of the dissipative sol
utions to the MHD system with a weaker hypothesis on the pressure ($P \\in
\\mathcal{D}'$).\n\nThis is a joint work with Diego Chamorro.\n
LOCATION:https://researchseminars.org/talk/anpdews/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Montie Avery (University of Minnesota)
DTSTART:20200625T150000Z
DTEND:20200625T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/28
DESCRIPTION:Title: Nonlinear stability of critical pulled fronts via resolvent expansions\nby Montie Avery (University of Minnesota) as part of HA-GMT-PDE Seminar
\n\n\nAbstract\nWe consider invasion processes mediated by propagating fro
nts in spatially extended systems\, in which a stable rest state invades a
n unstable rest state. We focus on the case of pulled fronts\, for which t
he speed of propagation is the linear spreading speed\, which marks the tr
ansition between pointwise decay and pointwise growth for the linearizatio
n about the unstable rest state in a co-moving frame. In a general setting
of scalar parabolic equations on the real line of arbitrary order\, we es
tablish sharp decay rates and temporal asymptotics for perturbations to th
e front\, under conceptual assumptions on the existence and spectral stabi
lity of fronts. Some of these results are known for the specific example o
f the Fisher-KPP equation\, and so our work can be viewed as establishing
universality of certain aspects of this classical model. Technically\, our
approach is based on a detailed study of the resolvent operator for the l
inearization about the critical front\, near its essential spectrum.\n
LOCATION:https://researchseminars.org/talk/anpdews/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgios Sakellaris (Autonomous University of Barcelona)
DTSTART:20200611T150000Z
DTEND:20200611T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/29
DESCRIPTION:Title: Green's function for second order elliptic equations with singular lower
order coefficients and applications\nby Georgios Sakellaris (Autonomo
us University of Barcelona) as part of HA-GMT-PDE Seminar\n\n\nAbstract\nW
e will discuss Green's function for second order elliptic \noperators of t
he form $\\mathcal{L}u=-\\text{div}(A\\nabla u+bu)+c\\nabla \nu+du$ in dom
ains $\\Omega\\subseteq\\mathbb R^n$\, for $n\\geq 3$. We will \nassume th
at $A$ is elliptic and bounded\, and also that \n$d\\geq\\text{div}b$ or $
d\\geq\\text{div}c$ in the sense of distributions.\n\nIn the setting of Lo
rentz spaces\, we will explain why the assumption \n$b-c\\in L^{n\,1}(\\Om
ega)$ is optimal in order to obtain a pointwise \nbound of the form $G(x\,
y)\\leq C|x-y|^{2-n}$. Under the assumption \n$d\\geq\\text{div}b$\, we wi
ll also discuss why this assumption is \nnecessary to even have weak type
bounds on Green's function. Finally\, \nfor the case $d\\geq\\text{div}c$\
, we will deduce a maximum principle \nand a Moser type estimate\, showing
again that the assumption $b-c\\in \nL^{n\,1}(\\Omega)$ is optimal.\n\nOu
r estimates will be scale invariant and no regularity on \n$\\partial\\Ome
ga$ will be imposed. In addition\, $\\mathcal{L}$ will not \nbe assumed to
be coercive\, and there will be no smallness assumption \non the lower or
der coefficients.\n
LOCATION:https://researchseminars.org/talk/anpdews/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:María Ángeles García-Ferrero (University of Heidelberg)
DTSTART:20200730T150000Z
DTEND:20200730T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/30
DESCRIPTION:Title: Unique continuation properties for nonlocal operators\nby María Áng
eles García-Ferrero (University of Heidelberg) as part of HA-GMT-PDE Semi
nar\n\n\nAbstract\nRoughly speaking\, a unique continuation property state
s that a solution of certain partial differential equation is determined b
y its behaviour in a subset. In this talk we will see this kind of propert
ies\, including their strong and quantitative versions\, for some classes
of nonlocal operators like the Hilbert transform\, which arise in medical
imaging\, or the (higher order) fractional Laplacian. The results I will p
resent rely on commonly used tools as Carleman estimates and the Caffareli
-Silvestre extension\, but also on two alternative mechanisms. As an appli
cation we will see Runge approximation results.\n\nThis is joint work with
Angkana Rüland.\n
LOCATION:https://researchseminars.org/talk/anpdews/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joao Pedro Ramos (Instituto Nacional de Matematica Pura e Aplicada
- ETH Zurich)
DTSTART:20200713T150000Z
DTEND:20200713T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/31
DESCRIPTION:Title: Recent progress on Fourier uncertainty\nby Joao Pedro Ramos (Institut
o Nacional de Matematica Pura e Aplicada - ETH Zurich) as part of HA-GMT-P
DE Seminar\n\n\nAbstract\nThe classical Heisenberg Uncertainty Principle s
hows that a function and its Fourier transform cannot be too concentrated
around a point simultaneously. In other words\, if we force a function and
its Fourier transform to vanish outside a small neighborhood of a point\,
then the function is zero. This classical principle has been generalized
to many levels in the past\, including results of Hardy\, Beurling and man
y others. In this talk\, we will recall old and new results about Fourier
ncertainty\, focusing more on the most recent developments on the field an
d its relationship to various topics\, such as the sphere packing problem\
, interpolation formulae and many others.\n
LOCATION:https://researchseminars.org/talk/anpdews/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oscar Jarrin (Universidad Tecnica de Ambato)
DTSTART:20200806T150000Z
DTEND:20200806T155000Z
DTSTAMP:20260422T212936Z
UID:anpdews/32
DESCRIPTION:Title: On the Liouville problem for the stationary Navier-Stokes equations\n
by Oscar Jarrin (Universidad Tecnica de Ambato) as part of HA-GMT-PDE Semi
nar\n\n\nAbstract\nUniqueness of weak solutions of the 3D Navier-Stokes eq
uations is a challenging open problem. In this talk\, we will discuss some
recent results of this problem for the 3D stationary Navier-Stokes equati
ons. More precisely\, within the framework of the Lebesgue\, Lorentz and M
orrey spaces\, we will observe that the null solution of these equations
is the unique one. This kind of results are also known as Liouville-type r
esults.\n
LOCATION:https://researchseminars.org/talk/anpdews/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruno Poggi Cevallos (University of Minnesota)
DTSTART:20200914T160000Z
DTEND:20200914T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/33
DESCRIPTION:Title: Additive and scalar-multiplicative Carleson perturbations of elliptic ope
rators on domains with low dimensional boundaries.\nby Bruno Poggi Cev
allos (University of Minnesota) as part of HA-GMT-PDE Seminar\n\n\nAbstrac
t\nAt the beginning of the 90s\, Fefferman\, Kenig and Pipher (FKP) obtain
ed a rather sharp (additive) perturbation result for the Dirichlet problem
of divergence form elliptic operators. Without delving into details\, the
point is that if the (additive) disagreement of two operators satisfies w
hat is known as a Carleson measure condition\, then quantitative absolute
continuity of the elliptic measure is transferred from one operator to the
other\, if one of the operators already possesses this property. Their (a
dditive) perturbation result has since then been generalized to increasing
ly weaker geometric and topological assumptions on boundaries of co-dimens
ion 1\, by multiple authors. \n\nThis talk will consist of two main parts
. In the first part\, we will see an extension of the FKP result to the d
egenerate elliptic operators of David\, Feneuil and Mayboroda\, which were
developed to study geometric and analytic properties of sets with boundar
ies whose co-dimension is higher than 1. These operators are of the form -
div A∇ \, where A is a degenerate elliptic matrix crafted to weigh the d
istance to the low-dimensional boundary in a way that allows for the nouri
shment of an elliptic theory. When this boundary is a d-Alhfors-David regu
lar set in R^n with d in [1\, n-1)\, and n≥ 3\, we prove that the member
ship of the elliptic measure in A_∞ is preserved under (additive) Carle
son measure perturbations of the matrix of coefficients\, yielding in turn
that the L^p-solvability of the Dirichlet problem is also stable under th
ese perturbations (with possibly different p). If the Carleson measure pe
rturbations are suitably small\, we establish solvability of the Dirichlet
problem in the same L^p space. One of the corollaries of our results toge
ther with a previous result of David\, Engelstein and Mayboroda\, is that\
, given any d-ADR boundary Γ with d in [1\, n-2)\, n≥ 3\, there is a f
amily of degenerate operators of the form described above whose elliptic m
easure is absolutely continuous with respect to the d-dimensional Hausdor
ff measure on Γ. Our method of proof uses the method of Carleson measure
extrapolation\, as developed by Lewis and Murray\, and adapted to a dyadi
c setting by Hofmann and Martell in the past decade. This is joint work wi
th Svitlana Mayboroda.\n\nIn the second part of the talk\, we will adopt a
slightly different perspective than has been customary in the literature
of these perturbation results\, by considering scalar-multiplicative Carle
son perturbations\, as communicated to us by Joseph Feneuil and inspired b
y the work on equations with drift terms of Hofmann and Lewis\, and Kenig
and Pipher\, at the start of the 21st century. Essentially\, if we may wr
ite A=bA_0 with b a scalar function bounded above and below by a positive
number\, and ∇b·dist(· \,Γ) satisfying a Carleson measure condition\,
then we still retain the transference of the quantitative absolute contin
uity of the elliptic measure for -div A∇\, if -div A_0∇ already has th
is property. By way of examples in the setting of low dimensional boundari
es\, we will see that one ought to consider these two types of perturbatio
ns (namely\, additive and scalar-multiplicative) to reckon a more complete
picture of the absolute continuity of elliptic measure. This is joint wor
k with Joseph Feneuil.\n
LOCATION:https://researchseminars.org/talk/anpdews/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arunima Bhattacharya (University of Washington)
DTSTART:20200909T160000Z
DTEND:20200909T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/34
DESCRIPTION:Title: Hessian Estimates for the Lagrangian mean curvature equation\nby Arun
ima Bhattacharya (University of Washington) as part of HA-GMT-PDE Seminar\
n\n\nAbstract\nIn this talk\, we will derive a priori interior Hessian est
imates for the Lagrangian mean curvature equation under certain natural re
strictions on the Lagrangian phase. As an application\, we will use these
estimates to solve the Dirichlet problem for the Lagrangian mean curvature
equation with continuous boundary data\, on a uniformly convex\, bounded
domain in R^n.\n
LOCATION:https://researchseminars.org/talk/anpdews/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bingyang Hu (Purdue University)
DTSTART:20200924T160000Z
DTEND:20200924T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/35
DESCRIPTION:Title: On the adjacency of general dyadic grids in Euclidean spaces .\nby Bi
ngyang Hu (Purdue University) as part of HA-GMT-PDE Seminar\n\n\nAbstract\
nIn this talk\, we will briefly introduce the recent development on study
the adjacent grids in Euclidean spaces. This talk contains three 3 parts\,
the real line case\, the higher dimension case and the different bases ca
se. The first part is a joint work with Tess Anderson\, Liwei Jiang\, Conn
or Olson and Zeyu Wei\, which is due to a summer REU project at UW-Madison
in 2018\; while the second and third parts are taken from joint works wit
h Tess Anderson.\n
LOCATION:https://researchseminars.org/talk/anpdews/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hyunwoo Kwon (Republic of Korea Air Force Academy)
DTSTART:20200928T160000Z
DTEND:20200928T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/36
DESCRIPTION:Title: Elliptic equations with singular drifts term on Lipschitz domains.\nb
y Hyunwoo Kwon (Republic of Korea Air Force Academy) as part of HA-GMT-PDE
Seminar\n\n\nAbstract\nIn this talk\, we consider linear elliptic equatio
n of second-order with the first term given by a singular vector field $\\
mathbf{b}$ on bounded Lipschitz domains $\\Omega$ in $\\mathbb{R}^n$\, $(n
\\geq 3)$. Under the assumption $\\mathbf{b}\\in L^n(\\Omega)^n$\, we esta
blish unique solvability in $L_{\\alpha}^p(\\Omega)$ for Dirichlet and Neu
mann problems. Here $L_{\\alpha}^p(\\Omega)$ denotes the standard Sobolev
spaces(or Bessel potential space) with the pair $(\\alpha\,p)$ satisfying
certain condition. These results extend the classical works of Jerison-Ken
ig (1995) and Fabes-Mendez-Mitrea (1999) for the Poisson equation. In addi
tion\, we study the Dirichlet problem for such linear elliptic equation wh
en the boundary data is in $L^2(\\partial\\Omega)$. Necessary review on th
is topics is also presented in this talk. This is a joint work with Hyunse
ok Kim(Sogang University).\n
LOCATION:https://researchseminars.org/talk/anpdews/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hong Wang (Institute for Advanced Study\, New Jersey)
DTSTART:20201005T160000Z
DTEND:20201005T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/37
DESCRIPTION:Title: Distinct distances on the plane\nby Hong Wang (Institute for Advanced
Study\, New Jersey) as part of HA-GMT-PDE Seminar\n\n\nAbstract\nGiven N
distinct points on the plane\, what is the minimal number of distinct dist
ances between them? This problem was posed by Paul Erdos in 1946 and essen
tially solved by Guth and Katz in 2010. \n\nWe are going to consider a co
ntinuous analogue of this problem\, the Falconer distance problem. Given
a set $E$ of dimension $s>1$\, what can we say about its distance set $\\D
elta(E)=\\{ |x-y|: x\,y\\in E\\}$? Falconer conjectured in 1985 that $\\De
lta(E)$ should have positive Lebesgue measure. In the recent years\, pe
ople have attacked this problem in different ways (including geometric mea
sure theory\, Fourier analysis\, and combinatorics) and made some progress
for various examples and for some range of $s$.\n
LOCATION:https://researchseminars.org/talk/anpdews/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dallas Albritton (Courant Institute\, New York)
DTSTART:20201019T160000Z
DTEND:20201019T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/38
DESCRIPTION:Title: Self-similar solutions of active scalars with critical dissipation\nb
y Dallas Albritton (Courant Institute\, New York) as part of HA-GMT-PDE Se
minar\n\n\nAbstract\nIn PDE analyses of fluid models\, often we may identi
fy a so-called critical space that lives precisely at the borderline betwe
en well-posedness and ill-posedness. What happens at this borderline? We e
xplore this question in two active scalar equations with critical dissipat
ion. In the surface quasi-geostrophic equations\, we investigate the conne
ction between non-uniqueness and large self-similar solutions that was est
ablished by Jia\, Sverak\, and Guillod in the Navier-Stokes equations. Thi
s is joint work with Zachary Bradshaw. In the critical Burgers equation\,
and more generally in scalar conservation laws\, the analogous self-simila
r solutions are unique\, and we show that all front-like solutions converg
e to a self-similar solution at the diffusive rates. This is joint work wi
th Raj Beekie.\n
LOCATION:https://researchseminars.org/talk/anpdews/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Barron (University of Illinois-Urbana Champaign)
DTSTART:20201026T160000Z
DTEND:20201026T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/39
DESCRIPTION:Title: A sharp global Strichartz estimate for the Schrodinger equation on the cy
linder\nby Alex Barron (University of Illinois-Urbana Champaign) as pa
rt of HA-GMT-PDE Seminar\n\n\nAbstract\nThe classical Strichartz estimates
show that a solution to the linear Schrodinger equation on Euclidean spac
e is in certain Lebesgue spaces globally in time provided the initial data
is in $L^2$. On compact manifolds one can no longer have global control\,
and some loss of derivatives is necessary (meaning the initial data needs
to be in a Sobolev space rather than $L^2$). In 'intermediate' cases it i
s a challenging question to understand when one can have good space-time e
stimates with no loss of derivatives. \n\nIn this talk we discuss a global
-in-time Strichartz-type estimate for the linear Schrodinger equation on t
he cylinder. Our estimate is sharp\, scale-invariant\, and requires only $
L^2$ data. Joint work with M. Christ and B. Pausader.\n
LOCATION:https://researchseminars.org/talk/anpdews/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Feneuil (Australian National University)
DTSTART:20201109T170000Z
DTEND:20201109T175000Z
DTSTAMP:20260422T212936Z
UID:anpdews/41
DESCRIPTION:Title: Uniform rectifiability implies $A_{\\infty}$-absolute continuity of the h
armonic measure with respect to the Hausdorff measure in low dimension.\nby Joseph Feneuil (Australian National University) as part of HA-GMT-PD
E Seminar\n\n\nAbstract\nUnder mild conditions of topology on the domain $
\\Omega\\subset\\mathbb R^n$\, the harmonic measure is $A_{\\infty}$-absol
utely continuous with respect to the surface measure if and only if the bo
undary ∂Ω is uniformly rectifiable of dimension n − 1.\n\nWe shall pr
esent the state of the art around the above statement\, and then discuss t
he strategy employed by Guy David\, Svitlana Mayboroda\, and the speaker t
o extend this characterization of uniform rectifiability to sets of dimens
ion d < n − 1.\n
LOCATION:https://researchseminars.org/talk/anpdews/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zanbing Dai (University of Minnesota)
DTSTART:20201116T170000Z
DTEND:20201116T175000Z
DTSTAMP:20260422T212936Z
UID:anpdews/42
DESCRIPTION:Title: The Regularity boundary value problem in domains with lower dimensional b
oundaries.\nby Zanbing Dai (University of Minnesota) as part of HA-GMT
-PDE Seminar\n\n\nAbstract\nRecently\, Guy David\, Joseph Feneuil and Svit
lana Mayboroda developed an elliptic theory in domains with lower dimensio
nal boundaries. They studied a class of degenerate second order elliptic o
perators $-\\textup{div} A\\nabla$ \, where A is a weighted matrix. The Di
richlet boundary value problem associated with these operators in\nhigher
codimension has already been solved by Joseph Feneuil\, Svitlana Mayboroda
and Zihui Zhao. We currently focus on the regularity boundary problem. Ro
ughly speaking\, we are interested in the relation between the gradient of
weak solutions and the gradient of boundary data whenever the boundary ha
s higher regularity and coefficients satisfy a certain smoothness conditio
n . In this talk\, I will introduce our main results about the solvability
of the regularity boundary value problem in the higher codimension. This
is joint work with Svitlana Mayboroda and Joseph Feneuil.\n
LOCATION:https://researchseminars.org/talk/anpdews/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liding Yao (University of Wisconsin-Madison)
DTSTART:20201123T170000Z
DTEND:20201123T175000Z
DTSTAMP:20260422T212936Z
UID:anpdews/43
DESCRIPTION:Title: Frobenius Theorem for Log-Lipschitz Subbundles\nby Liding Yao (Univer
sity of Wisconsin-Madison) as part of HA-GMT-PDE Seminar\n\n\nAbstract\nIn
differential geometry\, Frobenius theorem says that if a (smooth) real ta
ngential subbundle is involutive\, i.e. that X\,Y are sections implies tha
t [X\,Y] is also a section\, then this subbundle is spanned by some coordi
nate vector fields. Recently we prove the Frobenius theorem in the log-Lip
schitz setting. In the talk I will go over the formulation of the theorem
and show how harmonic analysis involves in the proof.\n
LOCATION:https://researchseminars.org/talk/anpdews/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Hoffman (University of Missouri-Columbia)
DTSTART:20201130T170000Z
DTEND:20201130T175000Z
DTSTAMP:20260422T212936Z
UID:anpdews/44
DESCRIPTION:Title: Regular Lip(1\,1/2) Approximation of Parabolic Hypersurfaces\nby John
Hoffman (University of Missouri-Columbia) as part of HA-GMT-PDE Seminar\n
\n\nAbstract\nA classical result of David and Jerison states that a regula
r\, n-dimensional set in R^{n+1} satisfying a two sided corkscrew conditio
n is quantitatively approximated by Lipschitz graphs. After reviewing thi
s result\, we will discuss some recent advances in extending this result t
o the parabolic setting. The proofs of these results are quite difficult\
, but many of the underlying principles are easy to understand and quite g
eometric and presenting these geometric ideas will be the focus of this ta
lk. As such\, this talk will feature lots of pictures! Crucially\, we hi
ghlight how fundamental differences of the parabolic setting require us to
consider additional nuances which are not present in the elliptic setting
. We will sketch the ideas of how to circumvent these difficulties.\n
LOCATION:https://researchseminars.org/talk/anpdews/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Li Chen (Massachusetts Institute of Technology)
DTSTART:20201207T170000Z
DTEND:20201207T175000Z
DTSTAMP:20260422T212936Z
UID:anpdews/45
DESCRIPTION:Title: Effective equations of quantum mechanics and phase transitions\nby Li
Chen (Massachusetts Institute of Technology) as part of HA-GMT-PDE Semina
r\n\n\nAbstract\nEffective equations of many-body quantum mechanics form t
he backbone of many fields of modern physics. Notable examples of effectiv
e equations include the Hartree-Fock\, Kohn-Sham\, and Bogoliubov-de Genne
s (BdG) equations. Although their physical derivations vary\, we will revi
ew an unified formal mathematical frame work for their derivations (if tim
e permits). In this frame work\, the BdG equations are the most general fo
rm of effective equations. Physically\, they form a microscopic descriptio
n of superconductivity. When the temperature T is lower than a certain cri
tical Tc\, superconducting solutions emerge. In this talk\, we will demons
trate the the existence of solutions to the BdG equations via variational
arguments and show energy instability (hence the formation of a supercondu
cting order parameter) when T < Tc. \n\nThis is a joint work with I. M. Si
gal\n
LOCATION:https://researchseminars.org/talk/anpdews/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max Engelstein (University of Minnesota)
DTSTART:20210119T170000Z
DTEND:20210119T175000Z
DTSTAMP:20260422T212936Z
UID:anpdews/46
DESCRIPTION:Title: Lojasiewicz Inequalities and the Zero Sets of Harmonic Functions\nby
Max Engelstein (University of Minnesota) as part of HA-GMT-PDE Seminar\n\n
\nAbstract\nWhereas $C^\\infty$ functions can vanish (almost) arbitrarily
often to arbitrarily high order (e\,g\, $f(x) = e^{-1/x}$ vanishes to infi
nite order at zero)\, the zero sets of analytic functions have a lot more
structure. For example\, you learn in intro to complex analysis that the z
eroes of a Holomorphic function are isolated.\n\nThe Lojasiewicz inequalit
ies (partially) quantify this extra structure possessed by analytic functi
ons. Developed originally by algebraic geometers\, Lojasiewicz inequalitie
s have been used with great success to study geometric flows. In this talk
\, I will give a brief introduction to these inequalities and then discuss
some joint work (and maybe some work in progress) with Matthew Badger (UC
onn) and Tatiana Toro (U Washington)\, in which we use Lojasiewicz inequal
ities to study the zero sets of harmonic functions and\, more interestingl
y\, sets which are infinitesimally approximated by the zero sets of harmon
ic functions.\n
LOCATION:https://researchseminars.org/talk/anpdews/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Darío Mena Arias (Universidad de Costa Rica)
DTSTART:20210125T170000Z
DTEND:20210125T175000Z
DTSTAMP:20260422T212936Z
UID:anpdews/47
DESCRIPTION:Title: Sparse bounds for the Discrete Spherical Maximal Function\nby Darío
Mena Arias (Universidad de Costa Rica) as part of HA-GMT-PDE Seminar\n\n\n
Abstract\nWe prove sparse bounds for the spherical maximal operator of Mag
yar\, Stein and Wainger. The bounds are conjecturally sharp\, and contain
an endpoint estimate. The new method of proof is inspired by ones by Bourg
ain and Ionescu\, is very efficient\, and has not been used in the proof o
f sparse bounds before. The Hardy-Littlewood Circle method is used to deco
mpose the multiplier into major and minor arc components. The efficiency a
rises as one only needs a single estimate on each element of the decomposi
tion.\n
LOCATION:https://researchseminars.org/talk/anpdews/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Bortz (University of Alabama)
DTSTART:20210201T170000Z
DTEND:20210201T175000Z
DTSTAMP:20260422T212936Z
UID:anpdews/48
DESCRIPTION:Title: New Developments in Parabolic Uniform Rectifiability\nby Simon Bortz
(University of Alabama) as part of HA-GMT-PDE Seminar\n\n\nAbstract\nIn th
e 1980’s the $L^2$ boundedness of the Cauchy integral was established (b
y Coifman\, McIntosh and Meyer) and this $L^2$ boundedness was quickly gen
eralized to `nice’ singular integral operators (Coifman\, David and Meye
r / David) on Lipschitz graphs. David and Semmes then asked the natural qu
estion: For what sets are all `nice’ singular integral operators $L^2$ b
ounded? They were remarkably successful in this endeavor\, providing more
than 15 equivalent notions and called these sets “uniformly rectifiable
” or UR. These sets are still studied extensively today\, most recently
in their connection to elliptic partial differential equations.\n\n \nAmon
g the characterizations of UR sets provided by David and Semmes is a quadr
atic estimate on the so-called $\\beta$-numbers\, which measure the flatne
ss of the set at a particular location and scale. In a paper of Hofmann\,
Lewis and Nyström a notion of “parabolic uniform rectictifiable sets”
was introduced by taking the definition as a “quadratic estimate on the
parabolic $\\beta$-numbers”. There are no correct proofs of ANY of the
analogues of the David Semmes theory\; for instance\, it is not known if $
L^2$ boundedness of parabolic singular integral operators characterizes pa
rabolic uniformly rectifiable sets.\n\n \nIn this talk I will discuss some
recent progress in the direction of establishing the parabolic David-Semm
es theory and some open problems that remain. On one hand\, we have made s
ignificant progress and provided some useful characterizations of paraboli
c uniform rectifiability. On the other hand\, we have also discovered that
many of the `elliptic’ characterizations do not hold in this parabolic
setting. This is joint work with J. Hoffman\, S. Hofmann\, J.L. Luna and K
. Nyström.\n
LOCATION:https://researchseminars.org/talk/anpdews/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:José Manuel Conde Alonso (Universidad Autónoma de Madrid)
DTSTART:20210208T170000Z
DTEND:20210208T175000Z
DTSTAMP:20260422T212936Z
UID:anpdews/49
DESCRIPTION:Title: Weak endpoint estimates for Calderón-Zygmund operators in von Neumann al
gebras\nby José Manuel Conde Alonso (Universidad Autónoma de Madrid)
as part of HA-GMT-PDE Seminar\n\n\nAbstract\nThe classical Calderón-Zygm
und decomposition is a fundamental tool that helps one study endpoint esti
mates for numerous operators near L1. In this talk\, we will discuss an ex
tension of the decomposition to a particular operator valued setting where
noncommutativity makes its appearance. Noncommutativity will allow us to
get rid of the -usually necessary- UMD property of the Banach space where
functions take values. Based on joint work with L. Cadilhac and J. Parcet.
\n
LOCATION:https://researchseminars.org/talk/anpdews/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:George Dosidis (Charles University)
DTSTART:20210215T170000Z
DTEND:20210215T175000Z
DTSTAMP:20260422T212936Z
UID:anpdews/50
DESCRIPTION:Title: The uncentered spherical maximal function and Nikodym sets\nby George
Dosidis (Charles University) as part of HA-GMT-PDE Seminar\n\n\nAbstract\
nStein's spherical maximal function is an analogue of the Hardy-Littlewood
maximal function\, where the averages are taken over spheres instead of b
alls. While the uncentered Hardy-Littlewood maximal function is bounded on
Lp for all p>1 and pointwise equivalent to its centered counterpart\, the
corresponding uncentered spherical maximal function is not as well-behave
d.\n\nWe provide multidimensional versions of the Kakeya\, Nikodym\,and Be
sicovitch constructions associated with spheres. These yield counterexampl
es indicating that maximal operators given by translations of spherical av
erages are unbounded on Lp for all finite p.\n\nHowever\, for lower-dimens
ional sets of translations\, we obtain Lp boundedness for the associated m
aximally translated spherical averages for a certain range of p that\ndepe
nds on the Minkowski dimension of the set of translations. This is joint w
ork with A. Chang and J. Kim.\n
LOCATION:https://researchseminars.org/talk/anpdews/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Blair Davey (Montana State University)
DTSTART:20210222T170000Z
DTEND:20210222T175000Z
DTSTAMP:20260422T212936Z
UID:anpdews/51
DESCRIPTION:Title: A quantification of the Besicovitch projection theorem and its generaliza
tions\nby Blair Davey (Montana State University) as part of HA-GMT-PDE
Seminar\n\n\nAbstract\nThe Besicovitch projection theorem asserts that if
a subset E of the plane has finite length in the sense of Hausdorff and i
s purely unrectifiable (so its intersection with any Lipschitz graph has z
ero length)\, then almost every linear projection of E to a line will have
zero measure. As a consequence\, the probability that a randomly dropped
line intersects such a set E is equal to zero. This shows us that the Bes
icovitch projection theorem is connected to the classical Buffon needle pr
oblem. Motivated by the so-called Buffon circle problem\, we explore what
happens when lines are replaced by more general curves. This leads us to
discuss generalized Besicovitch theorems and the ways in which we can qua
ntify such results by building upon the work of Tao\, Volberg\, and others
. This talk covers joint work with Laura Cladek and Krystal Taylor.\n
LOCATION:https://researchseminars.org/talk/anpdews/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Polona Durcik (Chapman University)
DTSTART:20210301T170000Z
DTEND:20210301T175000Z
DTSTAMP:20260422T212936Z
UID:anpdews/52
DESCRIPTION:Title: Multilinear singular and oscillatory integrals and applications\nby P
olona Durcik (Chapman University) as part of HA-GMT-PDE Seminar\n\n\nAbstr
act\nWe give an overview of some recent results in the area of multilinear
singular and oscillatory integrals. We discuss their connection with cert
ain questions about point configurations in subsets of the Euclidean space
and convergence of some ergodic averages. Based on joint works with Micha
el Christ\, Vjekoslav Kovac\, and Joris Roos.\n
LOCATION:https://researchseminars.org/talk/anpdews/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Silvia Ghinassi (University of Washington)
DTSTART:20210308T170000Z
DTEND:20210308T175000Z
DTSTAMP:20260422T212936Z
UID:anpdews/53
DESCRIPTION:Title: On the regularity of singular sets of minimizers for the Mumford-Shah ene
rgy\nby Silvia Ghinassi (University of Washington) as part of HA-GMT-P
DE Seminar\n\n\nAbstract\nThe Mumford-Shah functional was introduced by Mu
mford and Shah in 1989 as a variational model for image reconstruction. Th
e most important regularity problem is the famous Mumford-Shah conjecture\
, which states that (in 2 dimensions) the closure of the jump set can be d
escribed as the union of a locally finite collection of injective $C^1$ ar
cs that can meet only at the endpoints\, in which case they have to form t
riple junctions. If a point is an endpoint of one (and only one) of such a
rcs\, it is called cracktip. In this talk\, I plan to survey some older re
sults concerning the regularity of Mumford-Shah minimizers and their singu
lar sets\, and discuss more recent developments (the talk is based on join
t work with Camillo De Lellis and Matteo Focardi).\n
LOCATION:https://researchseminars.org/talk/anpdews/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sean McCurdy (Carnegie Mellon University)
DTSTART:20210315T160000Z
DTEND:20210315T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/54
DESCRIPTION:Title: The Analysts' Traveling Salesman Problem in Banach spaces\nby Sean Mc
Curdy (Carnegie Mellon University) as part of HA-GMT-PDE Seminar\n\n\nAbst
ract\nThis talk discusses recent work (joint with Matthew Badger\, UCONN)
on generalizations of the Analysts' Traveling Salesman Theorem to uniforml
y smooth and uniformly convex Banach spaces (e.g.\, l_p spaces). In 1990\
, motivated by problems in Singular Integral Operators\, Peter Jones posed
and solved his celebrated Analysts' Traveling Salesman Problem: namely\,
to characterize all subsets of rectifiable curves in the plane. Since the
n\, many authors have contributed\, proving similar results in Euclidean s
paces\, Hilbert Spaces\, Carnot groups\, for 1-rectifiable measures\, etc.
This talk will give a broad overview of some of these results and their
core ideas. In the end\, we will discuss the challenges in Banach spaces
and what generalizations hold there. This talk will include lots of pictu
res and examples.\n
LOCATION:https://researchseminars.org/talk/anpdews/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laurel Ohm (Courant Institute)
DTSTART:20210322T160000Z
DTEND:20210322T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/55
DESCRIPTION:Title: Mathematical foundations of slender body theory\nby Laurel Ohm (Coura
nt Institute) as part of HA-GMT-PDE Seminar\n\n\nAbstract\nMarch 22\, Mond
ay\, 11:00am-11:50am (CDT) - Laurel Ohm (Courant Institute\, New York) Zo
om link. https://umn.zoom.us/j/97214138395\nTitle. Mathematical foundation
s of slender body theory \nAbstract. Slender body theory (SBT) facilitate
s computational simulations of thin filaments in a 3D viscous fluid by app
roximating the hydrodynamic effect of each fiber as the flow due to a line
force density along a 1D curve. Despite the popularity of SBT in computat
ional models\, there had been no rigorous analysis of the error in using S
BT to approximate the interaction of a thin fiber with fluid. In this talk
\, we develop a PDE framework for analyzing the error introduced by this a
pproximation. In particular\, given a 1D force along the fiber centerline\
, we define a notion of `true' solution to the full 3D slender body proble
m and obtain an error estimate for SBT in terms of the fiber radius. This
places slender body theory on firm theoretical footing. In addition\, we p
erform a complete spectral analysis of the slender body PDE in a simple ge
ometric setting\, which sheds light on the use of SBT in approximating the
`slender body inverse problem\,' where we instead specify the fiber veloc
ity and solve for the 1D force density. Finally\, we make some comparisons
to the method of regularized Stokeslets and offer thoughts on improvement
s to SBT.\n
LOCATION:https://researchseminars.org/talk/anpdews/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robin Neumayer (Northwestern University)
DTSTART:20210329T160000Z
DTEND:20210329T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/56
DESCRIPTION:Title: Quantitative stability for minimizing Yamabe metrics\nby Robin Neumay
er (Northwestern University) as part of HA-GMT-PDE Seminar\n\n\nAbstract\n
The Yamabe problem asks whether\, given a closed Riemannian manifold\, one
can find a conformal metric of constant scalar curvature (CSC). An affirm
ative answer was given by Schoen in 1984\, following contributions from Ya
mabe\, Trudinger\, and Aubin\, by establishing the existence of a function
that minimizes the so-called Yamabe energy functional\; the minimizing fu
nction corresponds to the conformal factor of the CSC metric.\n\n\nWe addr
ess the quantitative stability of minimizing Yamabe metrics. On any closed
Riemannian manifold we show—in a quantitative sense—that if a functio
n nearly minimizes the Yamabe energy\, then the corresponding conformal me
tric is close to a CSC metric. Generically\, this closeness is controlled
quadratically by the Yamabe energy deficit. However\, we construct an exam
ple demonstrating that this quadratic estimate is false in the general. Th
is is joint work with Max Engelstein and Luca Spolaor.\n
LOCATION:https://researchseminars.org/talk/anpdews/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mariana Smit Vega Garcia (Western Washington University)
DTSTART:20210405T160000Z
DTEND:20210405T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/57
DESCRIPTION:Title: Almost minimizers for obstacle problems\nby Mariana Smit Vega Garcia
(Western Washington University) as part of HA-GMT-PDE Seminar\n\n\nAbstrac
t\nIn the applied sciences one is often confronted with free boundaries\,
which arise when the solution to a problem consists of a pair: a function
u (often satisfying a partial differential equation)\, and a set where thi
s function has a specific behavior. Two central issues in the study of fre
e boundary problems are: \n\n(1) What is the optimal regularity of the sol
ution u? \n\n(2) How smooth is the free boundary? \n\nThe study of the cla
ssical obstacle problem - one of the most renowned free boundary problems
- began in the ’60s with the pioneering works of G. Stampacchia\, H. Lew
y\, and J. L. Lions. During the past decades\, it has led to beautiful dev
elopments\, and its study still presents very interesting and challenging
questions. In contrast to the classical obstacle problem\, which arises fr
om a minimization problem (as many other PDEs do)\, minimizing problems wi
th noise lead to the notion of almost minimizers. In this talk\, I will in
troduce obstacle type problems and overview recent developments in almost
minimizers for the thin obstacle problem\, illustrating techniques that ca
n be used to tackle questions (1) and (2) in various settings. This is joi
nt work with Seongmin Jeon and Arshak Petrosyan.\n
LOCATION:https://researchseminars.org/talk/anpdews/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cruz Prisuelos-Arribas (Universidad de Alcalá)
DTSTART:20210412T160000Z
DTEND:20210412T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/58
DESCRIPTION:Title: Vertical square functions and other operators associated with an elliptic
operator\nby Cruz Prisuelos-Arribas (Universidad de Alcalá) as part
of HA-GMT-PDE Seminar\n\n\nAbstract\nAlthough\, in general\, vertical and
conical square functions are equivalent operators just in $L^2$\, in this
talk we show that\, when this square functions are defined through the hea
t or Poisson semigroup that arise from an elliptic operator\, there exist
open intervals of p's containing 2 where the equivalence holds in $L^p$. A
s a consequence we obtain new boundedness results for some square function
s. We also show how similar ideas lead us to improve the known range where
a non-tangential maximal function associated with the Poisson semigroup i
s bounded.\n
LOCATION:https://researchseminars.org/talk/anpdews/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Moritz Egert (Université Paris-Sud (Orsay))
DTSTART:20210419T160000Z
DTEND:20210419T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/59
DESCRIPTION:Title: Boundary value problems for elliptic systems with block structure\nby
Moritz Egert (Université Paris-Sud (Orsay)) as part of HA-GMT-PDE Semina
r\n\n\nAbstract\nI’ll consider a very simple elliptic PDE in the upper h
alf-space: divergence form\, transversally independent coefficients and no
mixed transversal-tangential derivatives. In this case\, the Dirichlet pr
oblem can formally be solved via a Poisson semigroup\, but there might not
be a heat semigroup. The construction is rigorous for L2 data. For other
data classes X (Lebesgue\, Hardy\, Sobolev\, Besov\,…) the question\, wh
ether the corresponding Dirichlet problem is well-posed\, is inseparably t
ied to the question\, whether there is a compatible Poisson semigroup on X
. \n\n\nOn a "semigroup space" the infinitesimal generator has (almost?) e
very operator theoretic property that one can dream of and these can be us
ed to prove well-posedness. But it turns out that there are genuinely more
"well-posedness spaces" than "semigroup spaces". For example\, up to boun
dary dimension n=4 there is a well-posed BMO-Dirichlet problem\, whose uni
que solution has no reason to keep its tangential regularity in the interi
or of the domain. \n\n\nI’ll give an introduction to the general theme a
nd discuss some new results\, all based on a recent monograph jointly writ
ten with Pascal Auscher.\n
LOCATION:https://researchseminars.org/talk/anpdews/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Murat Akman (University of Essex)
DTSTART:20210428T160000Z
DTEND:20210428T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/60
DESCRIPTION:Title: A Minkowski-type problem for measure associated to A-harmonic PDEs\nb
y Murat Akman (University of Essex) as part of HA-GMT-PDE Seminar\n\n\nAbs
tract\nThe classical Minkowski problem consists in finding a convex polyhe
dron from data consisting of normals to their faces and their surface area
s. In the smooth case\, the corresponding problem for convex bodies is to
find the convex body given the Gauss curvature of its boundary\, as a func
tion of the unit normal. The proof consists of three parts: existence\, un
iqueness\, and regularity. \n\n\nIn this talk\, we study a Minkowski probl
em for certain measure associated with a compact convex set E with nonempt
y interior and its A-harmonic capacitary function in the complement of E.
Here A-harmonic PDE is a non-linear elliptic PDE whose structure is modell
ed on the p-Laplace equation. If \\mu_E denotes this measure\, then the M
inkowski problem we consider in this setting is that\; for a given finite
Borel measure \\mu on S^(n-1)\, find necessary and sufficient conditions f
or which there exists E as above with \\mu_E =\\mu. We will discuss the ex
istence\, uniqueness\, and regularity of this problem in this setting.\n
LOCATION:https://researchseminars.org/talk/anpdews/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cole Jeznach (University of Minnesota)
DTSTART:20210510T160000Z
DTEND:20210510T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/61
DESCRIPTION:Title: Regularized Distances and Geometry of Measures\nby Cole Jeznach (Univ
ersity of Minnesota) as part of HA-GMT-PDE Seminar\n\n\nAbstract\nI will p
resent joint work with Max Engelstein and Svitlana Mayboroda where we gene
ralize the notion of the regularized distance function \n$$\nD_{\\mu\,\\al
pha}(x)= \\left(\\int |x-y|^{d-\\alpha}\\\, d\\mu(y)\\right)^{1/\\alpha}\,
\n$$\n \n\nto functions with more general integrands. We provide a large c
lass of integrands for which the corresponding distance functions contain
geometric information about $\\mu$. In particular\, we produce examples th
at are in some sense far from the original kernel $|x-y|^{d-\\alpha}$ but
still characterize the geometry of $\\mu$ since they have nice symmetries
with respect to flat sets. In co-dimension 1\, these examples are explici
t\, but in higher co-dimensions\, our proof of existence of such examples
is non-constructive\, and thus we have no additional information about the
ir structure.\n
LOCATION:https://researchseminars.org/talk/anpdews/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linhan Li (University of Minnesota)
DTSTART:20210517T160000Z
DTEND:20210517T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/62
DESCRIPTION:Title: Carleson measure estimates for the Green function\nby Linhan Li (Univ
ersity of Minnesota) as part of HA-GMT-PDE Seminar\n\n\nAbstract\nWe are i
nterested in the relations between an elliptic operator on a domain\, the
geometry of the domain\, and the boundary behavior of the Green function.
In joint work with Guy David and Svitlana Mayboroda\, we show that if the
coefficients of the operator satisfy a quadratic Carleson condition\, then
the Green function on the half-space is almost affine\, in the sense that
the normalized difference between the Green function with a sufficiently
far away pole and a suitable affine function at every scale satisfies a Ca
rleson measure estimate. We demonstrate with counterexamples that our resu
lts are optimal\, in the sense that the class of the operators considered
are essentially the best possible.\n\nThis work is motivated mainly by fin
ding PDE characterizations of uniform rectifiable sets with higher co-dime
nsion. I’ll talk about this motivation and backgrounds\, our recent resu
lts\, as well as possible directions in the future.\n
LOCATION:https://researchseminars.org/talk/anpdews/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Erisa Hasani (Florida Institute of Technology)
DTSTART:20210503T160000Z
DTEND:20210503T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/63
DESCRIPTION:Title: On the compactness threshold in the critical Kirchhoff equation\nby E
risa Hasani (Florida Institute of Technology) as part of HA-GMT-PDE Semina
r\n\n\nAbstract\nWe study a class of critical Kirchhoff problems with a ge
neral nonlocal term. The main difficulty here is the absence of a closed-f
orm formula for the compactness threshold. First we obtain a variational c
haracterization of this threshold level. Then we prove a series of existen
ce and multiplicity results based on this variational characterization.\n
LOCATION:https://researchseminars.org/talk/anpdews/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefan Steinerberger (University of Washington)
DTSTART:20210525T160000Z
DTEND:20210525T165000Z
DTSTAMP:20260422T212936Z
UID:anpdews/64
DESCRIPTION:Title: Mean-Value Inequalities for Harmonic Functions\nby Stefan Steinerberg
er (University of Washington) as part of HA-GMT-PDE Seminar\n\n\nAbstract\
nThe mean-value theorem for harmonic functions says that we can bound the
integral of a harmonic function in a ball by the average value on the boun
dary (and\, in fact\, there is equality). What happens if we replace the
ball by a general convex or even non-convex set? As it turns out\, this s
imple question has connections to classical potential theory\, probability
theory\, PDEs and even mechanics: one of the arising questions dates back
to Saint Venant (1856). There are some fascinating new isoperimetric pro
blems: for example\, the worst case convex domain in the plane seems to lo
ok a lot like the letter "D" but we cannot prove it. I will discuss some r
ecent results and many open problems.\n
LOCATION:https://researchseminars.org/talk/anpdews/64/
END:VEVENT
END:VCALENDAR