Yang–Mills solutions and dyons on cylinders over coset spaces with Sasakian structure

We present solutions of the Yang–Mills equation on cylinders R×G/HR×G/H over coset spaces of odd dimension 2m+12m+1 with Sasakian structure. The gauge potential is assumed to be SU(m)SU(m)-equivariant, parameterized by two real, scalar-valued functions. Yang–Mills theory with torsion in this setup reduces to the Newtonian mechanics of a point particle moving in R2R2 under the influence of an inverted potential. We analyze the critical points of this potential and present an analytic as well as several numerical finite-action solutions. Apart from the Yang–Mills solutions that constitute SU(m)SU(m)-equivariant instanton configurations, we construct periodic sphaleron solutions on S1×G/HS1×G/H and dyon solutions on iR×G/HiR×G/H.

Instantons on Calabi–Yau cones

The Hermitian Yang–Mills equations on certain vector bundles over Calabi–Yau cones can be reduced to a set of matrix equations; in fact, these are Nahm-type equations. The latter can be analysed further by generalising arguments of Donaldson and Kronheimer used in the study of the original Nahm equations. Starting from certain equivariant connections, we show that the full set of instanton equations reduce, with a unique gauge transformation, to the holomorphicity condition alone.

Truncated projective spaces, Brown–Gitler spectra and indecomposable A(1)-modules

MSC 19L41; 55S10.

Automorphisms of O'Grady's sixfolds

We study automorphisms of irreducible holomorphic symplectic (IHS) manifolds deformation equivalent to the O’Grady’s sixfold. We classify non-symplectic and symplectic automorphisms using lattice theoretic criterions related to the lattice structure of the second integral cohomology. Moreover we introduce the concept of induced automorphisms. There are two birational models for O'Grady's sixfolds, the first one introduced by O'Grady, which is the resolution of singularities of the Albanese fiber of a moduli space of sheaves on an abelian surface, the second one which concerns in the quotient of an Hilbert cube by a symplectic involution. We find criterions to know when an automorphism is induced with respect to these two different models, i.e. it comes from an automorphism of the abelian surface or of the Hilbert cube.

GENEOs, compactifications, and graphs

Our objective in this thesis is to study the pseudo-metric and topological structure of the space of group equivariant non-expansive operators (GENEOs). We introduce the notions of compactification of a perception pair, collectionwise surjectivity, and compactification of a space of GENEOs. We obtain some compactification results for perception pairs and the space of GENEOs. We show that when the data spaces are totally bounded and endow the common domains with metric structures, the perception pairs and every collectionwise surjective space of GENEOs can be embedded isometrically into the compact ones through compatible embeddings. An important part of the study of topology of the space of GENEOs is to populate it in a rich manner. We introduce the notion of a generalized permutant and show that this concept too, like that of a permutant, is useful in defining new GENEOs. We define the analogues of some of the aforementioned concepts in a graph theoretic setting, enabling us to use the power of the theory of GENEOs for the study of graphs in an efficient way. We define the notions of a graph perception pair, graph permutant, and a graph GENEO. We develop two models for the theory of graph GENEOs. The first model addresses the case of graphs having weights assigned to their vertices, while the second one addresses weighted on the edges. We prove some new results in the proposed theory of graph GENEOs and exhibit the power of our models by describing their applications to the structural study of simple graphs. We introduce the concept of a graph permutant and show that this concept can be used to define new graph GENEOs between distinct graph perception pairs, thereby enabling us to populate the space of graph GENEOs in a rich manner and shed more light on its structure.

A cohomological approach to Ruelle-Pollicott resonances and speed of mixing of Anosov diffeomorphisms

Given a transitive Anosov diffeomorphism on a closed manifold it is known that, for smooth enough observables, the system is mixing w.r.t. the measure of maximal entropy. Therefore, it makes sense to investigate the speed of decay of correlations and to look for the so-called Ruelle-Pollicott resonances, in order to determine a complete asymptotics for the decay of correlations. In this thesis we are able to find the first terms of that asymptotics and to prove an estimate for the speed of decaying of correlations. The proof is based on a surprising connection between the action of a transfer operator on suitable anisotropic Banach spaces of currents and the action induced by the Anosov map on the de Rham cohomology.

The combinatorics of Hilbert-Poincaré series of matroids

In this thesis we explore the combinatorial properties of several polynomials arising in matroid theory. Our main motivation comes from the problem of computing them in an efficient way and from a collection of conjectures, mainly the real-rootedness and the monotonicity of their coefficients with respect to weak maps. Most of these polynomials can be interpreted as Hilbert--Poincaré series of graded vector spaces associated to a matroid and thus some combinatorial properties can be inferred via combinatorial algebraic geometry (non-negativity, palindromicity, unimodality); one of our goals is also to provide purely combinatorial interpretations of these properties, for example by redefining these polynomials as poset invariants (via the incidence algebra of the lattice of flats); moreover, by exploiting the bases polytopes and the valuativity of these invariants with respect to matroid decompositions, we are able to produce efficient closed formulas for every paving matroid, a class that is conjectured to be predominant among all matroids. One last goal is to extend part of our results to a higher categorical level, by proving analogous results on the original graded vector spaces via abelian categorification or on equivariant versions of these polynomials.

Automorphisms of irreducible holomorphic symplectic manifolds and related problems

We study automorphisms and the mapping class group of irreducible holomorphic symplectic (IHS) manifolds. We produce two examples of manifolds of K3[2] type with a symplectic action of the alternating group A7. Our examples are realized as double EPW-sextics, the large cardinality of the group allows us to prove the irrationality of the associated families of Gushel-Mukai threefolds. We describe the group of automorphisms of double EPW-cubes. We give an answer to the Nielsen realization problem for IHS manifolds in analogy to the case of K3 surfaces, determining when a finite group of mapping classes fixes an Einstein (or Kähler-Einstein) metric. We describe, for some deformation classes, the mapping class group and its representation in second cohomology. We classify non-symplectic involutions of manifolds of OG10 type determining the possible invariant and coinvariant lattices. We study non-symplectic involutions on LSV manifolds that are geometrically induced from non-symplectic involutions on cubic fourfolds.