# Sala Seminari (Dip. Matematica)

## Chemical reaction networks: deterministic and stochastic models

Chemical reaction networks are mathematical models used in

biochemistry, as well as in other fields. Specifically, the time

evolution of a system of biochemical reactions are modelled either

deterministically, by means of a system of ordinary differential

equations, or stochastically, by means of a continuous time Markov

chain. It is natural to wonder whether the dynamics of the two modelling

regimes are linked, and whether properties of one model can shed light

on the behavior of the other one. In this talk some connections will be

## Introduction to interacting particle systems

Interacting particle systems is a recently developed field in

the theory of Markov processes with many applications: particle systems

have been used to model phenomena ranging from traffic behaviour to spread

of infection and tumour growth. We introduce this field through the study

of the simple exclusion process. We will construct the generator of this

process and we will give a convergence result of the spatial particle

density to the solution of the heat equation. We will also discuss a

## Averaging along irregular curves and regularization of ODEs

Paths of some stochastic processes such as fractional Brownian Motion have some amazing regularization properties. It is well known that in order to have uniqueness in differential systems such as

dy_t = b(y_t) dt,

## Kodaira fibrations with small signature

ABSTRACT Kodaira fibrations are surfaces of general type with a non-isotrivial fibration,

which are differentiable fibre bundles. First examples were constructed by

Kodaira and Atiyah to show that the signature need not be multiplicative in

fibre bundles.

We review approaches to construct such surfaces with focus on controlling the

signature. From this we obtain new examples with signature 16. (Joint work

with Ju A Lee and Michael Lönne)

## Kodaira fibrations with small signature

ABSTRACT:

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration,

which are differentiable fibre bundles. First examples were constructed by

Kodaira and Atiyah to show that the signature need not be multiplicative in

fibre bundles.

We review approaches to construct such surfaces with focus on controlling the

signature. From this we obtain new examples with signature 16. (Joint work

with Ju A Lee and Michael Lönne)

## A particle system approach to cellular aggregation model

## Uniqueness and persistence of minimal Lagrangian submanifolds

I will discuss the "hows and whys" of the following recent results (joint

with J.Lotay, UCL):

1) in a negative Kaehler–Einstein manifold M, compact minimal Lagrangian

submanifolds L are locally unique;

2) for any small Kaehler–Einstein perturbation of M there corresponds a

deformation of L which is minimal Lagrangian with respect to the new

structure.

These results are also available on arXiv:1704.08226

<https://arxiv.org/abs/1704.08226>

## On the spectrum of minimal submanifolds in space forms

Let $\varphi : M^m \to N^n$ be an immersed minimal submanifold in

Euclidean or hyperbolic space. In this talk, I survey on some recent

results obtained in collaboration with various colleagues from Brazil, to

ensure that the Laplace-Beltrami operator of $M$ has purely discrete

(respectively, purely essential) spectrum. In the last case, we also give

an explicit description of the spectrum. Our criteria apply to many

examples of minimal submanifolds constructed in the literature, and answer

## Constantin and Iyer's representation formula for the Navier--Stokes equations on manifolds

In this talk, we will present a probabilistic representation

formula for the Navier-Stokes equations on compact Riemannian manifolds.

Such a formula has been provided by Constantin and Iyer in the flat

case. On a Riemannian manifold, there are several different choices of

Laplacian operators acting on vector fields. We shall use the de

Rham-Hodge Laplacian operator which seems more relevant to the

probabilistic setting, and adopt Elworthy-Le Jan-Li's idea to decompose

it as a sum of the square of Lie derivatives. This is a joint work with