Past

My goal for the talk is to give a "from the ground-up" introduction to symplectic topology. We will cover the Darboux lemma, pseudo-holomorphic curves, Gromov-Witten invariants, quantum cohomology and Floer cohomology.

We present a novel approach to the solution of time-dependent PDEs via the so-called monolithic or all-at-once formulation.

This approach will be explained for simple parabolic problems and its utility in the context of PDE constrained optimization problems will be elucidated.

The underlying linear algebra includes circulant matrix approximations of Toeplitz-structured matrices and allows for effective parallel implementation. Simple computational results will be shown for the heat equation and the wave equation which indicate the potential as a parallel-in-time method.

This is joint work with Elle McDonald (CSIRO, Australia), Jennifer Pestana (Strathclyde University, UK) and Anthony Goddard (Durham University, UK)

According to the Harish-Chandra philosophy, cuspidal representations are the basic building blocks in the representation theory of finite reductive groups.  Similarly for supercuspidal representations of p-adic groups.  Self-dual representations play a special role in the study of parabolic induction.  Thus, it is of interest to know whether self-dual (super)cuspidal representations exist.  With a few exceptions involving some small fields, I will show precisely when a finite reductive group has irreducible cuspidal representations that are self-dual, of Deligne-Lusztig type, or both.  Then I will look at implications for the existence of irreducible, self-dual supercuspidal representations of p-adic groups.  This is joint work with Manish Mishra.

The Einstein equations in wave coordinates are an example of a system 
which does not obey the "null condition". This leads to many 
difficulties, most famously when attempting to prove global existence, 
otherwise known as the "nonlinear stability of Minkowski space". 
Previous approaches to overcoming these problems suffer from a lack of 
generalisability - among other things, they make the a priori assumption 
that the space is approximately scale-invariant. Given the current 
interest in studying the stability of black holes and other related 
problems, removing this assumption is of great importance.

The p-weighted energy method of Dafermos and Rodnianski promises to 
overcome this difficulty by providing a flexible and robust tool to 
prove decay. However, so far it has mainly been used to treat linear 
equations. In this talk I will explain how to modify this method so that 
it can be applied to nonlinear systems which only obey the "weak null 
condition" - a large class of systems that includes, as a special case, 
the Einstein equations. This involves combining the p-weighted energy 
method with many of the geometric methods originally used by 
Christodoulou and Klainerman. Among other things, this allows us to 
enlarge the class of wave equations which are known to admit small-data 
global solutions, it gives a new proof of the stability of Minkowski 
space, and it also yields detailed asymptotics. In particular, in some 
situations we can understand the geometric origin of the slow decay 
towards null infinity exhibited by some of these systems: it is due to 
the formation of "shocks at infinity".

In the spirit of the Ax-Kochen-Ershov principle, one could conjecture that the imaginaries in equicharacteristic zero Henselian fields can be entirely classified in terms of the Haskell-Hrushovski-Macpherson geometric imaginaries, residue field imaginaries and value group imaginaries. However, the situation is more complicated than that. My goal in this talk will be to present what we believe to be an optimal conjecture and give elements of a proof.

It is often fruitful to study an infinite discrete group via its finite quotients.  For this reason, conditions that guarantee many finite quotients can be useful.  One such notion is residual finiteness.
A group is residually finite if for any non-identity element g there is a homomorphism onto a finite group, which doesn’t map g to e. I will mention how this relates to topology, present an argument why the surface groups are residually finite and I’ll show that in this case it is enough to consider homomorphisms onto alternating groups.

This talk is a brief introduction to the analysis of Donaldson theory, a branch of gauge theory. Roughly, this is an area of differential topology that aims to extract smooth structure invariants from the geometry of the space of solutions (moduli space) to a system of partial differential equations: the Yang-Mills equations.

I will start by discussing the differential geometric background required to talk about Yang-Mills connections. This will involve introducing the concepts of principal fibre bundles, connections and curvature. In the second half of the talk I will attempt to convey the flavour of the mathematics used to address technical issues in gauge theory. I plan to do this by presenting a sketch of the proof of Uhlenbeck's compactness theorem, the main technical tool involved in the compactification of the moduli space.

The quantum unipotent coordinate ring has a cluster algebra structure. On the other hand, this ring is isomorphic to the Grothendieck ring of the module category of quiver Hecke algebras (QHA). We can prove that cluster monomials of the quantum unipotent coordinate ring correspondi to real simple modules. This is a joint work with Seok-Jin Kang, Myungho Kim and Se-jin Oh.

Numerical simulations indicate that deep artificial neural networks (DNNs) seem to be able to overcome the curse of dimensionality in many computational  problems in the sense that the number of real parameters used to describe the DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy and the dimension of the function which the DNN aims to approximate. However, there are only a few special situations where results in the literature can rigorously explain the success of DNNs when approximating high-dimensional functions.

In this talk it is revealed that DNNs do indeed overcome the curse of dimensionality in the numerical approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift coefficients. A crucial ingredient in our proof of this result is the fact that the artificial neural network used to approximate the PDE solution really is a deep artificial neural network with a large number of hidden layers.

For $G$ connected, reductive algebraic group defined over $\mathbb{C}$ the Springer Correspondence gives a bijection between the irreducible representations of the Weyl group $W$ of $G$ and certain pairs comprising a $G$-orbit on the nilpotent cone of the Lie algebra of $G$ and an irreducible local system attached to that $G$-orbit. These irreducible representations can be concretely realised as a W-action on the top degree homology of the fibres of the Springer resolution. These Springer fibres are geometrically very rich and provide interesting Weyl group combinatorics: for instance, the irreducible components of these Springer fibres form a basis for the corresponding irreducible representation of $W$. In this talk, I'll give a general survey of the Springer Correspondence and then discuss recent joint projects with Daniele Rosso, Vinoth Nandakumar and Arik Wilbert on Kato's Exotic Springer correspondence.

Graph sampling with noise is a fundamental problem in graph signal processing (GSP). A popular biased scheme using graph Laplacian regularization (GLR) solves a system of linear equations for its reconstruction. Assuming this GLR-based reconstruction scheme, we propose a fast sampling strategy to maximize the numerical stability of the linear system--i.e., minimize the condition number of the coefficient matrix. Specifically, we maximize the eigenvalue lower bounds of the matrix that are left-ends of Gershgorin discs of the coefficient matrix, without eigen-decomposition. We propose an iterative algorithm to traverse the graph nodes via Breadth First Search (BFS) and align the left-ends of all corresponding Gershgorin discs at lower-bound threshold T using two basic operations: disc shifting and scaling. We then perform binary search to maximize T given a sample budget K. Experiments on real graph data show that the proposed algorithm can effectively promote large eigenvalue lower bounds, and the reconstruction MSE is the same or smaller than existing sampling methods for different budget K at much lower complexity.

Integrable models provide simplified examples of quantum field theories with self-interaction. As often in relativistic quantum theory, their local observables are difficult to control mathematically. One either tries to construct pointlike local quantum fields, leading to possibly divergent series expansions, or one defines the local observables indirectly via wedge-local quantities, losing control over their explicit form.

We propose a new, hybrid approach: We aim to describe local quantum fields; but rather than exhibiting their n-point functions and verifying the Wightman axioms, we establish them as closed operators affiliated with a net of von Neumann algebras. This is shown to work at least in the Ising model.

George Soules [1] introduced a set of vectors $r_1,...,r_N$ with the remarkable property that for any set of ordered numbers $\lambda_1\geq\dots\geq\lambda_N$, the matrix $\sum_n \lambda_nr_nr_n^T$ has nonnegative off-diagonal entries. Later, it was found [2] that there exists a whole class of such vectors - Soules vectors - which are intimately connected to binary rooted trees. In this talk I will describe the construction of Soules vectors starting from a binary rooted tree, and introduce some basic properties. I will also cover a number of applications: the inverse eigenvalue problem, equitable partitions in Laplacian matrices and the eigendecomposition of the Clauset-Moore-Newman hierarchical random graph model.

[1] Soules (1983), Constructing Symmetric Nonnegative Matrices
[2] Elsner, Nabben and Neumann (1998), Orthogonal bases that lead to symmetric nonnegative matrices

In joint work with Peter Topping we introduce pyramid Ricci flows, defined throughout uniform regions of spacetime that are not simply parabolic cylinders, and enjoying curvature estimates that are not required to remain spatially constant throughout the domain of definition. This weakened notion of Ricci flow may be run in situations ill-suited to the classical theory. As an application of pyramid Ricci flows, we obtain global regularity results for three-dimensional Ricci limit spaces (extending results of Miles Simon and Peter Topping) and for higher dimensional PIC1 limit spaces (extending not only the results of Richard Bamler, Esther Cabezas-Rivas and Burkhard Wilking, but also the subsequent refinements by Yi Lai).
 

Many singular stochastic PDEs are expected to be universal objects that govern a wide range of microscopic models in different universality classes. Two notable examples are KPZ and \Phi^4_3. In these cases, one usually finds a parameter in the system, and tunes according to the space-time scale in such a way that the system rescales to the SPDE in the large-scale limit. We justify this belief for a large class of continuous microscopic growth models (for KPZ) and phase co-existence models (for Phi^4_3), allowing microscopic nonlinear mechanisms far beyond polynomials. Aside from the framework of regularity structures, the main new ingredient is a moment bound for general nonlinear functionals of Gaussians. This essentially allows one to reduce the problem of a general function to that of a polynomial. Based on a joint work with Martin Hairer, and another joint work in progress with Chenjie Fan and Jiawei Li. 

In a recent paper, Basterra, Bobkova, Ponto, Tillmann and Yeakel defined
topological operads with homological stability (OHS) and proved that the
group completion of an algebra over an OHS is weakly equivalent to an
infinite loop space.

In this talk, I shall outline a construction which to an algebra A over
an OHS associates a new infinite loop space. Under mild conditions on
the operad, this space is equivalent as an infinite loop space to the
group completion of A. This generalises a result of Wahl on the
equivalence of the two infinite loop space structures constructed by
Tillmann on the classifying space of the stable mapping class group. I
shall also talk about an application of this construction to stable
moduli spaces of high-dimensional manifolds in thesense of Galatius and
Randal-Williams.

We will start by presenting two basic probabilistic effects for questions concerning the regularity of functions and nonlinear operations on functions. We will then overview well-posedenss results for the nonlinear wave equation, the nonlinear Schr\"odinger equation and the nonlinear heat equation, in the presence of singular randomness.

In this talk we discuss new effective methods to test pairwise containment of arbitrary (possibly singular) subvarieties of any smooth projective toric variety and to determine algebraic multiplicity without working in local rings. These methods may be implemented without using Gröbner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used. The methods arise from techniques developed to compute the Segre class s(X,Y) of X in Y for X and Y arbitrary subschemes of some smooth projective toric variety T. In particular, this work also gives an explicit method to compute these Segre classes and other associated objects such as the Fulton-MacPherson intersection product of projective varieties.
These algorithms are implemented in Macaulay2 and have been found to be effective on a variety of examples. This is joint work with Corey Harris (University of Oslo).