The homotopy deformation of a differential graded algebra (DGA), which classically gives It\^o calculus, can be applied to noncommutative calculus. As a specific example of the deformation, we use the standard Podle\'s noncommutative sphere $S^2_q$ and its differential calculus. We then study the resulting operator $ \Delta$ on the noncommutative sphere, and its Hodge theory giving the eigenvalues and eigenforms on $ \Omega S^{2}_{q}$.
In this paper we present our results on calculating the Monodromy group and Galois group of 320 conics lying on certain quartic K3 surface. Part of the poster will show a toy example to explain these concepts, the other part will list our results. A final part of the poster will outline various deductions we can make using the previous results.
DonaldsonThomas theory arose as a way to attach integer invariants to CalabiYau 3folds. Subsequent evolutions have tried migrating techniques and results beyond the original geometric context, and one of the most significant example concerns quiver with potential, a tool developed in theoretical physics to investigate supersymmetry.
Loosely speaking, classical deformation theory is the study of functors from Artinian local rings to sets. Derived deformation theory studies extensions of these functors (often called formal moduli problems) to derived Artinian rings: this extension is analogous to taking the derived category of an abelian category. The starting point for this (ongoing) work is the observation that formal moduli problems satisfy a higher categorical sheaf condition with respect to the topology on derived Artinian rings in which every morphism is a covering. We hypothesise that formal moduli problems are actually characterised by this property, together with some extra descent conditions for hypercoverings. As evidence, we remark that a linearised version of the soughtafter result follows directly from Goodwillie's calculus of functors.
Moduli spaces of polarised irreducible symplectic manifolds are related to locally symmetric varieties known as 'Orthogonal modular varieties'. Many of these moduli spaces are conjectured to be of general type; this, of course, follows if the associated modular variety is also of general type.
One typically shows that an orthogonal modular variety is of general type by exhibiting spaces of orthogonal modular forms satisfying certain conditions determined by the geometry of a compactification of the modular variety. In particular, one needs to pay careful attention to the singularities and the singular locus.
My results here concern the singularities of certain orthogonal modular varieties associated with the moduli of a type of irreducible symplectic manifold called a 'deformation generalised Kummer variety'.
An important result of ChenDonaldsonSun and Tian relates the existence of KählerEinstein metrics on Fano varieties to an algebrogeometric notion called Kstability. Kstability is however understood in very few cases. We show that certain finite covers of Kstable Fano varieties are Kstable.
We show how to link the Hodge Structure of a smooth projective variety X to the infinitesimal deformation module of the affine cone over X, in the spirit of a generalization of the famous Griffiths's Residue theory. We then discuss some new application of these techniques, for example to Mukai varieties.
The restriction of LazarsfeldMukai bundles to curves on K3 surface have been used before as counterexample for Mercat's conjecture for rank 3 and 4. In this poster, we explain how wallcrossing with respect to Bridgeland stability conditions helps to prove slope stability of restriction of LazarsfeldMukai bundle. As a result, we find many new counterexamples to Mercat's conjecture for rank greater than two.
We give an some methods to understand the integral cohomology of the generalised Kummer varieties via the morphism to the Hilbert scheme. In the case of the generalised Kummer fourfold we are able to give a complete description. This is joint work with Grégoire Menet.
The moduli space of stable locally free sheaves with fixed determinant on a smooth projective curve is a Fano variety. The C_1 conjecture predicts that if the curve is defined over a C_1 field, then this variety has a rational point. We will give a sketch of the proof of this result .
Maximum likelihood (ML) estimation is a fundamental computational task in statistical inference, when one wishes to select the statistical model of the process that generates given data. The ML estimation problem is often reduced to solving a system of polynomial equations. Recently techniques have been developed for computing ML estimates using software developed for research in algebraic geometry, an area of mathematics concerned with solutions of a set of polynomial equations and their geometric properties. The idea is that statistical models correspond to algebraic varieties and one can apply techniques from algebraic geometry to study their properties. For some problems the use of such methods can have significant advantages over approximation methods commonly used in statistics, making the computation of ML estimates faster and more accurate. Initially, these algorithms were developed for smooth varieties, however the statistically important models are rarely smooth. In my research I have calculated the unique Maximum Likelihood Estimate for cubic and quartic toric Del Pezzo surfaces with Du Val singularities.
We consider rationality of rationally connected Mori fiber spaces of dimension three. There are three cases to consider: Fano varieties, conic bundles and del Pezzo fibrations. A lot is known in the first 2 cases, but the latter one has been only considered for smooth varieties. It is acceptable answer if smooth varieties are dence in the moduli spaces and it is for degree 3 and higher. Unfortunately this is not the case for del Pezzo fibrations of degree 1 and 2. I prove birational rigidity, and in particular rationality, of del Pezzo fibrations of degree 2 with mild singularities. As an application I consider finite subgroups of Cremona group of rank three and prove that there aren't many geometric models for embeddings of PSL(2,7) into Cremona group which are del Pezzo fibrations.
A Riemannian manifold $(M, g)$ is called hyperk\"ahler if it admits a triple of a complex structures $I, J, K$ satisfying quaternionic relations and K\"ahler with a respect to $g$. There are known two infinite series of simple hyperk\"ahler manifolds  the Hilbert Schemes of points on K3 and the generalized Kummer varieties, and two sporadic examples of O'Grady in dimension six and ten. An important conjecture of Beauville states that there exists only a finite number of simple hyperkähler manifolds in each dimension up to deformation. I will explain boundedness conditions for the second Betti number and inequalities involving Betti numbers which follow from RozanskyWitten invariants and an so(4, 22)action on the cohomology of hyperkähler manifolds. And the second question is existence of absolutely trianalytic subvarieties in known hyperk\"ahler manifolds. In particular, I prove that there are no absolutely trianalytic tori in the generalized Kummer variety.
I am interested in relative canonical models of the threefold cyclic quotient singularities. My poster will show an easy algorithm involving HirzebruchJung continued fractions for computing these models for some smaller class of cyclic group actions on \CC^3 and how these relate to computations of relative canonical rings.
I would like to present recent developments on the bounded negativity conjecture. In particular, I will present a somehow surprising relation between bounds on negative selfintersection of curves and coefficients of Zariski decompositions of pseudoeffective divisors.
Many of the results of the log minimal model program are now known in full for 3folds in characteristic $p>5$. We discuss these results and some techniques used in their proof.
Due to the absence of the KawamataViehweg vanishing theorem, the classification of algebraic varieties in positive characteristic, as of very recently, has been seen as an insurmountable task. Recent progress in the field has been inspired by the discovery of Frobeniussplit varieties. In my poster, I will discuss connections between the geometry of projective varieties and properties of the Frobenius action, focusing particularly on surfaces.
There are two classes of cyclic quotient singularities that are of especial interest in certain styles of mirror symmetry: Tsingularities and residual singularities, characterised by being the smoothable and rigid singularities respectively in this class. My poster will recap how they interplay with polytope mutation and outline some insights they offer into the structure of Hilbert series of del Pezzo surfaces with only cyclic quotient singularities. These insights can be applied to provide nonexistence results for models of Hilbert series, as well as explain period collapse in Ehrhart series of rational polygons.

2nd BrAG meeting >