Past

We study dynamic backward problems, with the computation of conditional expectations as a special objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the Functional Ito calculus to our more general framework. In particular, unlike in the Functional Ito calculus, where one needs only to consider stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths.  We then derive the path dependent PDEs for the backward problems. Finally, an application to option pricing and hedging in a financial market with rough volatility is presented.

Joint work with JianFeng Zhang (USC).

In some ways the theory of mapping class groups of 4-manifolds is in 2020 at the same place where the theory of mapping class groups of 2-manifolds was in 1973, before Thurston changed everything.  In this talk I will describe some first steps in an ongoing joint project with Eduard Looijenga where we are trying to understand mapping class groups of certain algebraic surfaces (e.g. rational elliptic surfaces, and also K3 surfaces) from the Thurstonian point of view.

TBA

Given a log Calabi-Yau surface Y with maximal boundary D, I'll explain how to construct a mirror Landau-Ginzburg model, and sketch a proof of homological mirror symmetry for these pairs when (Y,D) is distinguished within its deformation class (this is mirror to an exact manifold). I'll explain how to relate this to the total space of the SYZ fibration predicted by Gross--Hacking--Keel, and, time permitting, explain ties with earlier work of Auroux--Katzarkov--Orlov and Abouzaid. Joint work with Paul Hacking.

I will revisit the question of what is the holographic dual of N=6 supersymmetric Chern-Simons theories.

Eigenvalue-eigenvector pairs of combinatorial graph Laplacians are extensively used in graph theory and network analysis. It is well known that the spectrum of the Laplacian L of a given graph G encodes aspects of the geometry of G  - the multiplicity of the eigenvalue 0 counts the number of connected components while the second smallest eigenvalue (called the Fiedler eigenvalue) quantifies the well-connectedness of G . In network analysis, one uses Laplacian eigenvectors associated with small eigenvalues to perform spectral clustering. In graph signal processing, graph Fourier transforms are defined in terms of an orthonormal eigenbasis of L. Eigenvectors of L also play a central role in graph neural networks.

Motivated by this we study eigenvalue-eigenvector pairs of Laplacians of random graphs and their potential use in TDA. I will present simulation results on what persistent homology barcodes of Bernoulli random graphs G(n, p) look like when we use Laplacian eigenvectors as filter functions. Also, I will discuss the conjectures made from the simulations as well as the challenges that arise when trying to prove them. This is work in progress.
 

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints.  We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we prove that LASSO leads to parameter shrinkage, propose measures to quantify robustness of neural networks to adversarial examples and compute sensitivities of optimised certainty equivalents in finance. We also propose extensions of this framework to a multiperiod setting. This talk is based on joint work with Daniel Bartl, Samuel Drapeau and Jan Obloj.

The many facets of community detection on networks 

Community detection, the decomposition of a graph into essential building blocks, has been a core research topic in network science over the past years. Since a precise notion of what consti- tutes a community has remained evasive, community detection algorithms have often been com- pared on benchmark graphs with a particular form of assortative community structure and classified based on the mathematical techniques they employ. However, this comparison can be misleading because apparent similarities in their mathematical machinery can disguise different goals and rea- sons for why we want to employ community detection in the first place. Here we provide a focused review of these different motivations that underpin community detection. This problem-driven classification is useful in applied network science, where it is important to select an appropriate algorithm for the given purpose. Moreover, highlighting the different facets of community detection also delineates the many lines of research and points out open directions and avenues for future research.

Given a curve , what is the surface  that has smallest area among all surfaces spanning ? This classical problem and its generalizations are called Plateau's problem. In this talk we consider area minimizers among the class of integral currents, or roughly speaking, orientable manifolds. Since the 1960s a lot of work has been done by De Giorgi, Almgren, et al to study the interior regularity of these minimizers. Much less is known about the boundary regularity, in the case of codimension greater than 1. I will speak about some recent progress in this direction.

We discuss two recent extensions of work on Reynolds-robust preconditioners for the Navier-Stokes equations, to non-Newtonian fluids and to the equations of magnetohydrodynamics.  We model non-Newtonian fluids by means of an implicit constitutive relation between stress and strain. This framework is broadly applicable and allows for proofs of convergence under quite general assumptions. Since the stress cannot in general be solved for in terms of the strain, a three-field stress-velocity-pressure formulation is adopted. By combining the augmented Lagrangian approach with a kernel-capturing space decomposition, we derive a preconditioner that is observed to be robust to variations in rheological parameters in both two and three dimensions.  In the case of magnetohydrodynamics, we consider the stationary incompressible resistive Newtonian equations, and solve a four-field formulation for the velocity, pressure, magnetic field and electric field. A structure-preserving discretisation is employed that enforces both div(u) = 0 and div(B) = 0 pointwise. The basic idea of the solver is to split the fluid and electromagnetic parts and to employ our existing Navier-Stokes solver in the Schur complement. We present results in two dimensions that exhibit robustness with respect to both the fluids and magnetic Reynolds numbers, and describe ongoing work to extend the solver to three dimensions.

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Recently in a joint work with J. Dobrowolski and N. Ramsey it is shown that in any NSOP1 theory with existence,
Kim-independence satisfies all the basic axioms over sets (except base monotonicity) that hold in simple theories with forking-independence. This is an extension of the earlier work by I. Kaplan and N. Ramsey that such hold over models in any NSOP1 theory. All simple theories; unbounded PAC fields; vector spaces over ACF with bilinear maps; the model companion of the empty theory in any language are typical NSOP1 examples.

   An important issue now is to know the existence of canonical bases. In stable and simple theories well-behaving notion of canonical bases for types over models exists, which is used in almost all the advanced studies. But there are a couple of crucial obstacles in finding canonical bases in NSOP1 theories. In this talk I will report a partial success/limit of the project. Namely, a type of a certain Morley sequence over a model has the weak canonical base. In my talk I will try to explain all the related notions.

https://arxiv.org/abs/2003.02844

Leibniz’s principle of identity of indiscernibles at first sight appears completely unrelated to set theory, but Mycielski (1995) formulated a set-theoretic axiom nowadays referred to as LM (for Leibniz-Mycielski) which captures the spirit of Leibniz’s dictum in the following sense:  LM holds in a model M of ZF iff M is elementarily equivalent to a model M* in which there is no pair of indiscernibles.  LM was further investigated in a 2004  paper of mine, which includes a proof that LM is equivalent to the global form of the Kinna-Wagner selection principle in set theory.  On the other hand, one can formulate a strong negation of Leibniz’s principle by first adding a unary predicate I(x) to the usual language of set theory, and then augmenting ZF with a scheme that ensures that I(x) describes a proper class of indiscernibles, thus giving rise to an extension ZFI of ZF that I showed (2005) to be intimately related to Mahlo cardinals of finite order. In this talk I will give an expository account of the above and related results that attest to a lively interaction between set theory and Leibniz’s principle of identity of indiscernibles.

Much progress in the study of 3-manifolds has been made by considering the geometric structures they admit. This is nowhere more true than for 3-manifolds which admit a hyperbolic structure. However, in the land of algorithms a more combinatorial approach is necessary, replacing our charts and isometries with finite simplicial complexes that are defined by a finite amount of data. 

In this talk we'll have a look at how in fact one can combine the two approaches, using the geometry of hyperbolic 3-manifolds to assist in this more combinatorial approach. To do so we'll combine tools from Hyperbolic Geometry, Triangulations, and perhaps suprisingly Polynomial Algebra to find explicit bounds on the runtime of an algorithm for comparing Hyperbolic manifolds.

We study the following question at the intersection of extremal and structural graph theory. Given a fixed graph $H$ that embeds in a fixed surface $\Sigma$, what is the maximum number of copies of $H$ in an $n$-vertex graph that embeds in $\Sigma$? Exact answers to this question are known for specific graphs $H$ when $\Sigma$ is the sphere. We aim for more general, albeit less precise, results. We show that the answer to the above question is $\Theta(nf(H))$, where $f(H)$ is a graph invariant called the `flap-number' of $H$, which is independent of $\Sigma$. This simultaneously answers two open problems posed by Eppstein (1993). When $H$ is a complete graph we give more precise answers. This is joint work with Tony Huynh and Gwenaël Joret [https://arxiv.org/abs/2003.13777]

The small subgraph conditioning method is an analysis of variance technique which was introduced by Robinson and Wormald in 1992, in their proof that almost all cubic graphs are Hamiltonian. The method has been used to prove many structural results about random regular graphs, mostly to show that a certain substructure is present with high probability. I will discuss some applications of the small subgraph conditioning method to hypergraphs, and describe a subtle issue which is absent in the graph setting.

Abstract: 

In this talk I will discuss regularization by noise from a pathwise perspective using non-linear Young integration, and discuss the relations with occupation measures and local times. This methodology of pathwise regularization by noise was originally proposed by Gubinelli and Catellier (2016), who use the concept of averaging operators and non-linear Young integration to give meaning to certain ill posed SDEs. 

In a recent work together with   Nicolas Perkowski we show that there exists a class of paths with exceptional regularizing effects on ODEs, using the framework of Gubinelli and Catellier. In particular we prove existence and uniqueness of ODEs perturbed by such a path, even when the drift is given as a Scwartz distribution. Moreover, the flow associated to such ODEs are proven to be infinitely differentiable. Our analysis can be seen as purely pathwise, and is only depending on the existence of a sufficiently regular occupation measure associated to the path added to the ODE. 

As an example, we show that a certain type of Gaussian processes has infinitely differentiable local times, whose paths then can be used to obtain the infinitely regularizing effect on ODEs. This gives insight into the powerful effect that noise may have on certain equations. I will also discuss an ongoing extension of these results towards regularization of certain PDE/SPDEs by noise.​

I will discuss some ongoing work on three-dimensional supersymmetric gauge theories and their relationship to (equivariant) quantum K-theory. I will emphasise the interplay between the physical and mathematical motivations and approaches, and attempt to build a dictionary between the two.  As an interesting example, I will discuss the quantum K-theory of flag manifolds. The QK ring will be related to the vacuum structure of a gauge theory with Chern-Simons interactions, and the (genus-0) K-theoretic invariants will be computed in terms of explicit residue formulas that can be derived from the relevant supersymmetric path integrals.

We discuss hyperkahler implosion spaces, their relevance to group actions and why they should fit into the symplectic duality picture. For certain groups we present candidates for the symplectic duals of the associated implosion spaces and provide computational evidence. This is joint work with Amihay Hanany and Frances Kirwan.