in modern theoretical physics

**Kenichi Konishi** – *Advent of non–Abelian vortices*

Generalizing the well–known Abrikosov–Nielsen–Olesen vortex solution in the Abelian Higgs model to non–Abelian gauge
theories, vortex solutions carrying continuous zeromodes (vortex moduli space) were constructed in the year 2003 inspired by supersymmetric
SU(N) gauge theories, triggering an intense research activity. The vast set of new results obtained since then cover the issues of
non–Abelian monopoles, vortex dynamics and their relation to the gauge dynamics in 4 dimensions, study of higher–winding
solutions and their moduli space, semi–local vortex solutions in theories with larger number of matter multiplets, stability of
non–BPS vortex solutions, vortex solutions in gauge theories based on a more general gauge groups such as SO(N), USp(2N), etc., and
the fractional vortex and sigma–model lump. Some of the most salient features of this exciting development will be discussed in these
lectures.

**Paul Michael Sutcliffe** – *Topological solitons and nuclei*

Skyrmions are topological soliton solutions of a generalized harmonic map equation from 3–dimensional Euclidean space
into SU(2). Physically the model describes a nonlinear theory of pions in which the soliton is interpreted as the nucleon.
Nuclei are then modelled by multi-soliton solutions, with an identification between the number of solitons and nucleons.
Results will be present on multi-soliton solutions and their symmetries: which play an important role in quantization.
Rational maps between Riemann spheres will be used to provide some understanding of these results. If time permits,
a connection between Skyrmions and Yang–Mills instantons will be discussed.

**Stanley Alama** – *Anisotropy and the Ginzburg–Landau model*

Anisotropic superconductors form an important class of superconducting materials, including many high–temperature superconductors,
which exhibit a layered structure. I will present recent analysis of two models of layered superconductors (the anisotropic Ginzburg–Landau
and the Lawrence–Doniach functionals) which illustrate how anisotropy of the material affects the orientation of the induced magnetic
field and vortex lattice compared to the direction of the externally applied magnetic field.

This talk represents joint work with L. Bronsard and E. Sandier.

**Daniele Bartolucci** – *Sharp existence/non existence results for a critical mean field equation with singular data*

We discuss a recent result obtained in collaboration with C.S. Lin concerning the Dirichlet problem for a critical mean field equation with singular data on two-dimensional simply connected domains. Necessary and sufficient conditions for the existence of solutions are obtained in terms of Riemann maps relative to the domain and the strenghts and position of the singularities.

**Emanuele Caglioti** – *The Boltzmann–Grad limit of the periodic Lorentz Gas in two space dimensions*

**Francesco Fucito** – *Intersecting branes and enumerative geometry*

Systems of parallel D(-1)D3 branes can reproduce the effect of gauge instantons in the low energy limit. We review how this is possible and extend these notions to intersecting branes systems giving interesting extensions of gauge instantons to dimensions higher than four.

**Robert Hardt** – *A Mass Decreasing Flow for Rectifiable Chains in some Metric Spaces*

In 1993, F. Almgren, J. Taylor, and L. Wang and independently X. Cheng constructed mass–reducing flows for rectifiable currents in Euclidean space. We combine some of these arguments with recent joint work with T. DePauw on rectifiability and compactness properties for chains in a metric space having coefficients in a complete normed abelian group. There are interesting questions about the spatial slices of such flows.

**Gianluca Panati** – *Dynamics of electrons in perturbed periodic media*

**Adriano Pisante** – *Symmetry of local minimizers for the three dimensional Ginzburg–Landau functional*

**Lesley Sibner** – *On hyperbolic multi–monopoles*

Proof of existence of hyperbolic multi–monopoles with arbitrary positive mass at infinity is discussed. The method involves the Taubes
gluing construction in which an approximate monopole is constructed from explicit well separated charge one monopoles. The analysis is
more complicated than the Euclidean case due to the different behavior of the spectrum of the Laplacian on one forms. Weighted spaces are
used to circumvent this difficulty and the arguments are somewhat more delicate because of this. The relationship between these monopoles
and instantons with holonomy is briefly discussed. These objects appear to be of interest in Theoretical Physics where hyperbolic space
occurs in models involving anti-de Sitter space. Recent interest in calorons is also somewhat related to this research.

This is a joint work with Robert Sibner.

**Daniel Spirn** – *Vortex motion in thin micromagnetic materials*

A simplified model for the energy of the magnetization of a thin ferromagnetic film gives rise to a version of the theory of
Ginzburg–Landau vortices for sphere–valued maps. The dynamics of the magnetization is ruled by the Landau–Lifshitz–Gilbert
equation, which combines characteristic properties of a nonlinear Schrödinger equation and a gradient flow. I will discuss the motion
of the vortex centers under this evolution equation.

This is joint work with M. Kurzke, C. Melcher, and R. Moser.

**Peter Sternberg** – *When does a vortex have a reason to live?*

Within Ginzburg–Landau theory, which models the behavior of a superconductor subjected perhaps to magnetic or electric fields,
the central object of study is the vortex, that is, a zero of the complex–valued order parameter carrying nontrivial
degree. Analyses of minimizers of the GL energy and of solutions of the time–dependent GL evolution focus on locating and tracking
the vortices. In this talk, I will survey a wide array of settings and results by various authors in which the mechanism leading to the
appearance of vortices is expected and relatively well understood, others in which vortex appearance is somewhat surprising and/or not
well understood, as well as some open questions about whether or not stable vortices can make any appearance at all.

**Takashi Suzuki** – *Smoluchowski–Poisson equation and its relatives – method of the scaling weak limit*

**Yisong Yang** – *Electrically Charged Solitons in Gauge Field Theory and Calculus of Variations*

Monopoles and vortices are well known magnetically charged soliton solutions of gauge field equations. Extending the idea of Dirac on
monopoles, Schwinger pioneered the concept of solitons carrying both electric and magnetic charges, called dyons, which are useful in
modeling elementary particles. Mathematically, the existence o dyons presents interesting variational PDE problems, subject to topological
constraints. This talk is a survey on recent progress in the study of dyons.

**Veronica Felli** – *Local asymptotics at singularities for many–particle Schringer operators*

We discuss asymptotics of solutions to Schringer equations with singular homogeneous potentials. Through an Almgren type monotonicity formula and separation of variables, we describe the exact behavior near the singularity of solutions to at most critical semilinear elliptic equations with cylindrical and quantum multi–body singular potentials.

**Annibale Magni** – *On Perelman's Dilaton*

We will describe a framework to describe the gradient–like structure of the Ricci flow.

**Peter Mason** – *Two–component Bose–Einstein condensates under rotation*

Bose–Einstein condensation is a macroscopic quantum phenomenon that occurs in very cold dilute alkali gases and was first experimentally realised in 1995. It is the condensation of atoms into the lowest available energy state, the ground state. The dynamics within a Bose–Einstein condensate (BEC) are governed by a non–linear differential equation in terms of a single macroscopic wavefunction, the Gross–Pitaevskii (GP) equation, that permits a wide variety of physical phenomena. When a mixture of different BEC's is considered (be it different isotopes of the same atom, different atoms or atoms in different spin states) the physical phenomena becomes even richer.

Here I will consider a two–component BEC placed into rotation where all interactions are repulsive. The dynamics are governed by a coupled GP equation and exhibit various phenomena such as coreless vortices that form triangular or square lattices or might even be isolated. Other features of interest are symmetry breaking states and the appearance of giant coreless vortices. A detailed phase diagram of the various features will be shown.

**Dimitri Mugnai** – *A Schrödinger–Poisson system with positive potential*

Some results on the existence of finite energy solutions for a Schrödinger–Poisson system will be presented. The main novelty is the
presence of a positive potential in the Schrödinger equation.

**Yuko Nagase** – *Analysis of interface and its curvature*

In the two–phase problem, in the variational problem of the Modica–Mortola type energy under volume constraint, it is well–known that the energy converges to the surface area of the minimal surface. For the curvature, Luckhaus and Modica showed Lagrange multiplier which comes from the volume constraint in the Euler–Lagrange equation converges to constant the mean curvature of the limit interface.

In this talk, we consider the generalization of this problem to the non homogeneous anisotropic energy. We also present an application to the stochastic Allen–Cahn equation.

**Alessio Pomponio** – *On the Schrödinger–Maxwell equations in presence of a general nonlinear term*

Abstract.

**Ryo Takahashi** – *Critical exponents of a semilinear elliptic equation with constraints in higher dimensional space*

We are concerned with a semilinear elliptic equation with constraints in higher dimensional space. The problem has the properties close to the Liouville equation in two dimensional space. Classification of entire solutions with bounded energy condition, existence of a $\sup + \inf$ type inequality, quantized blowup mechanism are shown, and then, simplicity of the blowup points, their locations, and classification of the singular limits are studied for the free boundary value problem.

**Gianmaria Verzini** – *Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition*

Abstract.

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**Important Notices**