Department of Mathematics |

*The seminar is taking place either in room A32 or in room A31. All talks are on Friday at 3:00 p.m. EEST (Athens).
*

**June 16, 2023**

Georgios Domazakis (University of Sussex)
* Rigidity of extremals for perimeter inequality under Schwarz symmetrisation*

**Abstract.** This talk will discuss the rigidity of extremals for perimeter inequality
under Schwarz symmetrisation. The term rigidity refers to the situation in which the
equality cases are only obtained by translations of the Schwarz symmetric set. We
will present sufficient and necessary conditions for rigidity under this framework.
Our analysis will based on the properties of the corresponding distribution
function, and on a careful study of the transformations that can be applied on the
symmetric set, without creating any perimeter contribution.

Webex link (3:00 PM - 4:00 PM, Friday, June 2, 2023)

**June 2, 2023**

Michael Sigal (University of Toronto)
*Ginzburg-Landau equation on non-compact Riemann surfaces*

**Abstract.** In this talk I will consider Ginzburg-Landau equations (GLE) on
non-compact surfaces, more precisely on line bundles over such surfaces.
GLE is a system of two equations for section and connection on these
bundles yielding a crude model for a quantum engine where electrons are
coming from infinity through one of the cusps and then departing through
another. In this talk I will report on recent results with Nick Ercolani
and Jingxuan Zhang on existence of energy minimizing solutions for such
equations. In the talk I will introduce all necessary definitions beyond
differential forms and exterior derivatives. In particular, no preliminary
knowledge of Riemann surfaces and line bundles will be requiref.

Webex link (3:00 PM - 4:00 PM, Friday, June 2, 2023)

**May 19, 2023**

Filippo Santambroggio (Institut Camille Jordan, Université Claude Bernard -
Lyon 1)
*Optimal trajectories in L1 and under L1 penalizations*

**Abstract.** I will present a recent work (just posted online ) in collaboration with my student Annette
Dumas. Motivated by a MFG model where the trajectories of the agents are
piecewise constants and agents pay for the number of jumps, we study a
variational problem for curves of measures where the cost includes the
length of the curve measures with the L1 distance, as well as other,
non-autonomous, cost terms arising from congestion effects. We prove
several regularity results (first in time, then in space) on the solution,
based on suitable approximation and maximum principle techniques. Modern
algorithms in non-smooth convex optimization allow to numerically simulate
such solutions.

Webex link (3:00 PM - 4:00 PM, Friday, May 19, 2023)

**May 5, 2023**

Dimitrios Gazoulis (University of Crete)
* P-functions and their Applications to Nonlinear Equations*

**Abstract.** We introduce the notion of P-function for fully-nonlinear
equations and study some applications. P-functions can be thought as
quantities of a function u and its higher order derivatives that are
related to a differential equation and have the property that satisfy the
maximum principle. We study P-functions for a class of quasilinear elliptic equations and provide some
general criterion for obtaining such quantities. In addition, we prove
some gradient bounds and De Giorgi-type properties of entire solutions. As
a special case, we obtain a gradient bound for the Allen-Cahn equations
that differs from the Modica inequality.

Webex link (3:00 PM - 4:00 PM, Friday, May 5, 2023)

**March 31, 2023**

Ioannis Anapolitanos (Karlsruher Institut für Technologie)
* How molecules get in shape*

**Abstract.** Chemical properties of a molecule are determined by its shape. For
example, the energy needed to initiate chemical reactions depends highly
on the shape and structure of the involved molecules. It has also been
known for a long time that the biological properties of protein molecules
are determined by their structure. This naturally leads to several deep
questions: How to determine the
structure of molecules, is there a way to optimize the energy needed for
chemical reactions, and so on.
I will explain how to mathematically formulate this type of question for a
specific group of chemical reactions, which are called isomerizations.
This involves a single molecule, which changes its shape. Often, the
initial and final shape of a molecule are given and one is seeking a path
in the configuration space of the molecule, which describes how it changes
its shape. The calculation of such reaction paths is very standard in
quantum chemistry. While in quantum chemistry it is tentatively assumed
that an optimal reaction path exists, this is far from obvious from a
mathematical point of view. In fact, it is still an open problem!
I will discuss recent results which provide a first step in this
direction. In particular, I will describe some fundamental properties of
molecules, concerning their structure and the forces between them. While
our personal experience of the world involves mostly classical terms, the
properties of molecules and matter at a microscopic scale necessarily
involve quantum mechanics. However, for the purpose of this talk, I will
not assume any previous knowledge of quantum mechanics. The talk is based
on two recent papers, one with Mathieu Lewin, which is published, the
other one being work in arxiv with Marco Olivieri and Sylvain Zalc.

Webex link (3:00 PM - 4:00 PM, Friday, March 31, 2023)

**December 15, 2022**

Konstantinos Gkikas (University of Athens)
*Poisson problems involving fractional Hardy operators and measures*

**Abstract.** We study the Poisson problem involving a fractional Hardy operator and a measure source. The complex interplay between the nonlocal nature of
the operator, the peculiar effect of the singular potential and the measure source induces several new fundamental difficulties in comparison with the local case.
To overcome these difficulties, we perform a careful analysis of the dual operator in the weighted distributional sense and establish
fine properties of the associated function spaces, which in turn allow us to formulate the Poisson problem in an appropriate framework.
In light of the close connection between the Poisson problem and its dual problem, we are able to establish various aspects of the theory for the Poisson problem
including the solvability, a priori estimates, variants of Kato's inequality and regularity results (Joint work with H. Chen and P.T. Nguyen). K. G. acknowledges the support
by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “2nd Call for H.F.R.I. Research Projects
to support Post-Doctoral Researchers” (Project Number: 59)

**December 8, 2022**

Zhiyuan Geng (Basque Center for Applied Mathematics, Spain)
*On the triple junction problem without symmetry hypotheses*

**Abstract.** In this talk, I will present some joint work with Nicholas
Alikakos on the vector Allen-Cahn equation with a triple-well potential.
Without assuming any symmetry hypothesis on the solution, we construct an
entire minimizing solution that possesses a weak triple junction
structure. A tight energy lower bound is established, which is crucial for
estimates of the location and size of the diffuse interface. Other key
tools used in our analysis include the 1D heteroclinic connection,
Caffarelli-Córdoba density estimates and a blow-up analysis near the
diffuse interface.

**December 1, 2022**

Giorgio Fusco (University of L'Aquila, Italy)*On the existence of triple junctions in bounded domains*

**Abstract.** We consider a smooth nonnegative potential with exactly three
nondegenerate zeros. A model of the bulk energy of a material which can
exist in three different phases. By a variational approach we show that
the connectivity of the phases is a sufficient condition for the existence
of a triple junction solution of the vector Allen-Cahn equation in bounded
two-dimensional domains. If time allows we also remark on the existence of
triple junction solutions defined on the entire plane.

**November 24, 2022**

Dimitrios Gazoulis (University of Athens, Greece)*On the Γ-convergence of the Allen-Cahn functional with boundary conditions*

**Abstract.** We will briefly introduce the notion of the Γ-convergence, its applications to scalar phase transition type problems and
how this is related to the theory of minimal surfaces. Then, we will describe our setting, that is, the study of the ε-energy functional for the
vector Allen-Cahn equations and establish
the Γ-limit of the functional with boundary conditions. Finally, we will determine some properties of the minimizers of the limiting functional.

**November 10, 2022**

Grigorios Fournodavlos (University of Crete, Greece)*On the nature of the Big Bang singularity*

**Abstract.** 100 years ago, Friedmann and Kasner discovered the
first exact cosmological solutions to Einstein’s equations, revealing the
presence of a striking new phenomenon, namely, the Big Bang singularity.
Since then, it has been the object of study in a great deal of research on
general relativity. However, the nature of the ‘generic’ Big Bang
singularity remains a mystery. Rivaling scenarios are abound
(monotonicity, chaos, spikes) that make the classification of all
solutions a very intricate problem. I will give a historic overview of the
subject and describe recent progress that confirms a small part of the
conjectural picture.

**October 20, 2022**

Manousos Maridakis (Aristotle University of Thessaloniki, Greece)*The Concentration Principle for Dirac operators*

**Abstract.** We study the spectrum and the properties of perturbed Dirac
operators of the form D_{s} = D + sA: Γ(E)→Γ(F) on compact
Riemannian manifolds (X,g), with vector bundles Ε ⊕ F → X and
symbol c and perturbated terms, linear operators A: E→F with s >> 0. A
simple relation between c and A, guarantees that the spectrum of the
perturbated operator D_{s} is distinguished into slow and fast eigenvalues
as s→∞. The eigenfunctions of slow eigenvalues are concentrated
as s→∞ around the singular set Z_{A}⊂ X of A. We illustrate
many interesting examples.

**October 6, 2022**

Vasiliki Bitsouni (University of Athens, Greece)*On the quasi-steady-state assumption in enzyme kinetics:
rigorous analysis*

**Abstract.** We present, from a purely quantitative point of view, the
quasi-steady-state assumption for the
fundamental mathematical model of the general enzymatic reaction. In
particular, we define the
two parts of the assumption in a quantitative fashion, we employ a simple
algorithm for the proper
scaling of the corresponding problem which naturally provides us with the
necessary and sufficient
information, and we comment, among other issues, on a dispensable third
part of the assumption.
Joint work with Nikolaos Gialelis (National and Kapodistrian University of
Athens) and Ioannis
G. Stratis (National and Kapodistrian University of Athens).

**September 29, 2022**

Phuoc-Tai Nguyen (Masaryk University, Czech Republic)*Nonlinear Schrödinger equations with a singular potential: global existence versus
finite time blowup*

**Abstract.** In this talk, I will discuss the focusing nonlinear Schrödinger (NLS) equation with an inverse-square potential. The presence of the potential
yields substantial difficulties and requires a fine analysis. I will show
the existence ofa minimizer of Hardy-Gagliardo-Nirenberg inequality and
the uniqueness of the ground state solution to the NLS equation. As a
consequence, I will show that any minimizer can be represented by the
ground state solution. Then I will establish global existence versus
finite time blowup for the focusing NLS equation.

**September 23, 2022**

Matteo Rizzi (University of Giessen, Germany)

*Symmetry results for the Cahn-Hilliard equation*

**Abstract.** In the talk we will give a classification of the entire solutions
to the Cahn-Hilliard equation according to their symmetry properties, such as
radial and cylindrical symmetry. We will see that this classification is
related to the theory of embedded constant mean curvature surfaces.