L6

We investigate the dynamics of a class of tumor growth

models known as mixed models. The key characteristic of these type of

tumor growth models is that the different populations of cells are

continuously present everywhere in the tumor at all times. In this

work we focus on the evolution of tumor growth in the presence of

proliferating, quiescent and dead cells as well as a nutrient.

The system is given by a multi-phase flow model and the tumor is

described as a growing continuum such that both the domain occupied by the tumor as well as its boundary evolve in time. Global-in-time weak solutions

are obtained using an approach based on penalization of the boundary

behavior, diffusion and viscosity in the weak formulation.

Further extensions will be discussed.

This is joint work with D. Donatelli.

We study liquid crystal point defects in 2D domains. We employ Landau-de

Gennes theory and provide a simplified description of global minimizers

of Landau- de Gennes energy under homeothropic boundary conditions. We

also provide explicit solutions describing defects of various strength

under Lyuksutov's constraint.

Steiner symmetrization is a very useful tool in the study of isoperimetric inequality. This is also due to the fact that the perimeter of a set is less or equal than the perimeter of its Steiner symmetral. In the same way, in the Gaussian setting,

it is well known that Ehrhard symmetrization does not increase the Gaussian perimeter. We will show characterization results for equality cases in both Steiner and Ehrhard perimeter inequalities. We will also characterize rigidity of equality cases. By rigidity, we mean the situation when all equality cases are trivially obtained by a translation of the Steiner symmetral (or, in the Gaussian setting, by a reflection of the Ehrhard symmetral). We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness, and through a fine analysis of the barycenter function

for a special class of sets. These results are obtained in collaboration with Maria Colombo, Guido De Philippis, and Francesco Maggi.

In this talk we will show the existence  of a regular "small" weak solution to the flow of the higher dimensional H-systems with initial-boundary conditions. We also analyze its time asymptotic bahavior and we give a stability result.

This talk will focus on extremizers for

a family of Fourier restriction inequalities on planar curves. It turns

out that, depending on whether or not a certain geometric condition

related to the curvature is satisfied, extremizing sequences of

nonnegative functions may or may not have a subsequence which converges

to an extremizer. We hope to describe the method of proof, which is of

concentration compactness flavor, in some detail. Tools include bilinear

estimates, a variational calculation, a modification of the usual

method of stationary phase and several explicit computations.

We discuss some questions related to coloring the edge/vertices of randomgraphs. In particular we look at
(i) The game chromatic number;
(ii) Rainbow Matchings and Hamilton cycles;
(iii) Rainbow Connection;
(iv) Zebraic Colorings.

Several objects can be associated to a list of vectors with integer coordinates: among others, a family of tori called toric arrangement, a convex polytope called zonotope, a function called vector partition function; these objects have been described in a recent book by De Concini and Procesi. The linear algebra of the list of vectors is axiomatized by the combinatorial notion of a matroid; but several properties of the objects above depend also on the arithmetics of the list. This can be encoded by the notion of a "matroid over Z". Similarly, applications to tropical geometry suggest the introduction of matroids over a discrete valuation ring.Motivated by the examples above, we introduce the more general notion of a "matroid over a commutative ring R". Such a matroid arises for example from a list of elements in a R-module. When R is a Dedekind domain, we can extend the usual properties and operations holding for matroids (e.g., duality). We can also compute the Tutte-Grothendieck ring of matroids over R; the class of a matroid in such a ring specializes to several invariants, such as the Tutte polynomial and the Tutte quasipolynomial. We will also outline other possible applications and open problems. (Joint work with Alex Fink).

A Steiner Triple System on a set X is a collection T of 3-element subsets of X such that every pair of elements of X is contained in exactly one of the triples in T. An example considered by Plücker in 1835 is the affine plane of order three, which consists of 12 triples on a set of 9 points. Plücker observed that a necessary condition for the existence of a Steiner Triple System on a set with n elements is that n be congruent to 1 or 3 mod 6. In 1846, Kirkman showed that this necessary condition is also sufficient.

In 1853, Steiner posed the natural generalisation of the question: given integers q and r, for which n is it possible to choose a collection Q of q-element subsets of an n-element set X such that any r elements of X are contained in exactly one of the sets in Q? There are some natural necessary divisibility conditions generalising the necessary conditions for Steiner Triple Systems. The Existence Conjecture states that for all but finitely many n these divisibility conditions are also sufficient for the existence of general Steiner systems (and more generally designs).

We prove the Existence Conjecture, and more generally, we show that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition.

We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees.

Joint work with Christian Borgs, Jennifer T. Chayes, and Henry Cohn.