Past

I will discuss recent progress on the study of homological duality properties of complex algebraic manifolds, with a view towards the projective Singer-Hopf conjecture. (Joint work with Y. Liu and B. Wang.)

A vast class of 4d SCFTs can be engineered by wrapping a stack of M5-branes on a Riemann surface. These SCFTs can exhibit a variety of global symmetries, continuous or discrete, including both ordinary (0-form) symmetries, as well as generalized (higher-form) symmetries. In this talk, I will focus on discrete and higher-form symmetries in setups with M5-branes on a smooth Riemann surface. Adopting a holographic point of view, a crucial role is played by topological mass terms in 5d supergravity (similar to BF terms in four dimensions). I will discuss how the global symmetries of the boundary 4d theory are inferred from the 5d topological terms, and how one can compute 4d ‘t Hooft anomalies involving discrete and/or higher-form symmetries.

Martin Gallauer (North): "Algebraic algebraic geometry"
If a space is described by algebraic equations, its algebraic invariants are endowed with additional structure. I will illustrate this with some simple examples, and speculate on the meaning of the title of my talk.

Zhaohe Dai (South): "Two-dimensional material bubbles"
Two-dimensional (2D) materials are a relatively new class of thin sheets consisting of a single layer of covalently bonded atoms and have shown a host of unique electronic properties. In 2D material electronic devices, however, bubbles often form spontaneously due to the trapping of air or ambient contaminants (such as water molecules and hydrocarbons) at sheet-substrate interfaces. Though they have been considered to be a nuisance, I will discuss that bubbles can be used to characterize 2D materials' bending rigidity after the pressure inside being well controlled. I will then focus on bubbles of relatively large deformations so that the elastic tension could drive the radial slippage of the sheet on its substrate. Finally, I will discuss that the consideration of such slippage is vital to characterize the sheet's stretching stiffness and gives new opportunities to understand the adhesive and frictional interactions between the sheet and various substrates that it contacts.
 

In this talk I will present an efficient algorithm to produce a provably dense sample of a smooth compact algebraic variety. The procedure is partly based on computing bottlenecks of the variety. Using geometric information such as the bottlenecks and the local reach we also provide bounds on the density of the sample needed in order to guarantee that the homology of the variety can be recovered from the sample.

Motivated by string dualities we propose topological gravity as the early phase of our universe.  The topological nature of this phase naturally leads to the explanation of many of the puzzles of early universe cosmology.  A concrete realization of this scenario using Witten's four dimensional topological gravity is considered.  This model leads to the power spectrum of CMB fluctuations which is controlled by the conformal anomaly coefficients $a,c$.  In particular the strength of the fluctuation is controlled by $1/a$ and its tilt by $c g^2$ where $g$ is the coupling constant of topological gravity.  The positivity of $c$, a consequence of unitarity, leads automatically to an IR tilt for the power spectrum.   In contrast with standard inflationary models, this scenario predicts $\mathcal{O}(1)$ non-Gaussianities for four- and higher-point correlators and the absence of tensor modes in the CMB fluctuations.

Snow densification and meltwater refreezing store water in alpine regions and transform snow into ice on the surface of glaciers. Despite their importance in determining snow-water equivalent and glacier-induced sea level rise, we still lack a complete understanding of the physical mechanisms underlying snow compaction and the infiltration of meltwater into snowpacks. Here we (i) analyze snow compaction experiments as a promising direction for determining the rheology of snow though its many stages of densification and (ii) solve for the motion of refreezing fronts and for the temperature increase due to the release of latent heat, which we compare to temperature observations from the Greenland Ice Sheet (Humphrey et al., 2012). In the first part, we derive a mixture theory for compaction and air flow through the porous snow (cf. Hewitt et al. 2016) to compare against laboratory data (Wang and Baker, 2013). We find that a plastic compaction law explains experimental results. Taking standard forms for the permeability and effective pressure as functions of the porosity, we show that this compaction mode persists for a range of densities and overburden loads (Meyer et al., 2020). We motivate the second part of the talk by the observed melting at high elevations on the Greenland Ice Sheet, which causes the refreezing layers that are observed in ice cores. Our analysis shows that as surface temperatures increase, the capacity for meltwater storage in snow decreases and surface runoff increases leading to sea level rise (Meyer and Hewitt, 2017). Together these studies provide a holistic picture for how snow changes through compaction and the role of meltwater percolation in altering the temperature and density structure of surface snow.

Multi-modal data sets are growing rapidly in single cell genomics, as well as other fields in science and engineering. We introduce MultiMAP, an approach for dimensionality reduction and integration of multiple datasets. MultiMAP embeds multiple datasets into a shared space so as to preserve both the manifold structure of each dataset independently, in addition to the manifold structure in shared feature spaces. MultiMAP is based on the rich mathematical foundation of UMAP, generalizing it to the setting of more than one data manifold. MultiMAP can be used for visualization of multiple datasets as well as an integration approach that enables subsequent joint analyses. Compared to other integration for single cell data, MultiMAP is not restricted to a linear transformation, is extremely fast, and is able to leverage features that may not be present in all datasets. We apply MultiMAP to the integration of a variety of single-cell transcriptomics, chromatin accessibility, methylation, and spatial data, and show that it outperforms current approaches in run time, label transfer, and label consistency. On a newly generated single cell ATAC-seq and RNA-seq dataset of the human thymus, we use MultiMAP to integrate cells across pseudotime. This enables the study of chromatin accessibility and TF binding over the course of T cell differentiation.

Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Through examples I will discuss how this allows one to compute the K-theory of the stable algebra, S, and the stable Ruelle algebra, S\rtimes Z. These computations involve writing S as a stationary inductive limit and S\rtimes Z as a Cuntz-Pimsner algebra. These constructions reemphasize the view point that Smale space C*-algebras are higher dimensional generalizations of Cuntz-Krieger algebras. The main results are joint work with Magnus Goffeng and Allan Yashinski.

We show how traders use immediate execution limit orders (IELOs) to liquidate a position when the time between a trade attempt and the outcome of the attempt is random, i.e., there is latency in the marketplace and latency is random. We frame our model as a delayed impulse control problem in which the trader controls the times and the price limit of the IELOs she sends to the exchange. The contribution of the paper is twofold: (i) Our paper is the first to study an optimal liquidation problem that accounts for random delays, price impact, and transaction costs. (ii) We introduce a new type of impulse control problem with stochastic delay, not previously studied in the literature. We characterise the value functions as the solution to a coupled system of a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI) and a partial differential equation. We use a Feynman-Kac type representation to reduce the system of coupled value functions to a non-standard HJBQVI, and we prove existence and uniqueness of this HJBQVI in a viscosity sense. Finally, we implement the latency-optimal strategy and compare it with three benchmarks:  (i)  optimal execution with deterministic latency, (ii) optimal execution with zero latency, (iii) time-weighted average price strategy. We show that when trading in the EUR/USD currency pair, the latency-optimal strategy outperforms the benchmarks between ten USD per million EUR traded and ninety USD per million EUR traded.

There are many examples of thin-film flows in fluid dynamics, and in many cases similarity solutions are possible. In the typical, well-known case the thin-film shape is described by a nonlinear partial differential equation in two independent variables (say x and t), which upon recognition of a similarity variable, reduces the problem to a nonlinear ODE. In this talk I describe work we have done on 1) Marangoni-driven spreading on pre-wetted films, where the thickness of the pre-wetted film affects the dynamics, and 2) the drainage of a film on a vertical substrate of finite width. In the latter case we find experimentally a structure to the film shape near the edge, which is a function of time and two space variables. Analysis of the corresponding thin-film equation shows that there is a similarity solution, collapsing three independent variables to one similarity variable, so that the PDE becomes an ODE. The solution is in excellent agreement with the experimental measurements.

We have various ways of describing the extent to which two countably infinite groups are "the same." Are they isomorphic? If not, are they commensurable? Measure equivalent? Quasi-isometric? Orbit equivalent? W*-equivalent? Von Neumann equivalent? In this expository talk, we will define these notions of equivalence, discuss the known relationships between them, and work out some examples. Along the way, we will describe recent joint work with Ishan Ishan and Jesse Peterson.

In this talk, we introduce a new preconditioner for the large, structured systems appearing in implicit Runge–Kutta time integration of parabolic partial differential equations. This preconditioner is based on a block LDU factorization with algebraic multigrid subsolves for scalability.

We compare our preconditioner in condition number and eigenvalue distribution, and through numerical experiments, with others in the literature. In experiments run with implicit Runge–Kutta stages up to s = 7, we find that the new preconditioner outperforms the others, with the improvement becoming more pronounced as the spatial discretization is refined and as temporal order is increased.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

Any binary classifier (or score-function) can be used to define a dissimilarity between two distributions of points with positive and negative labels. Actually, many well-known distribution-dissimilarities are classifier-based dissimilarities: the total variation, the KL- or JS-divergence, the Hellinger distance, etc. And many recent popular generative modelling algorithms compute or approximate these distribution-dissimilarities by explicitly training a classifier: eg GANs and their variants. After a brief introduction to these classifier-based dissimilarities, I will focus on the influence of the classifier's capacity. I will start with some theoretical considerations illustrated on maximum mean discrepancies --a weak form of total variation that has grown popular in machine learning-- and then focus on deep feed-forward networks and their vulnerability to adversarial examples. We will see that this vulnerability is already rooted in the design and capacity of our current networks, and will discuss ideas to tackle this vulnerability in future.

I will present a nonlinear version of the open mapping principle which applies to constant-coefficient PDEs which are both homogeneous and weak* stable. An example of such a PDE is the Jacobian equation. I will discuss the consequences of such a result for the Jacobian and its relevance towards an answer to a long-standing problem due to Coifman, Lions, Meyer and Semmes. This is based on joint work with Lukas Koch and Sauli Lindberg.

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I present a general nonlinear open mapping principle suited to applications to scale-invariant PDEs in regularity regimes where the equations are stable under weak* convergence. As an application I show that, for any $p < \infty$, the set of initial data for which there are dissipative weak solutions in $L^p_t L^2_x$ is meagre in the space of solenoidal L^2 fields. This is based on joint work with A. Guerra (Oxford) and S. Lindberg (Aalto).

The Zilber-Pink conjecture predicts how large the intersection of a d-dimensional subvariety of an abelian variety/algebraic torus/Shimura variety/... with the union of special subvarieties of codimension > d can be (where the definition of "special" depends on the setting). In joint work with Fabrizio Barroero, we have reduced this conjecture for complex abelian varieties to the same conjecture for abelian varieties defined over the algebraic numbers. In work in progress, we introduce the notion of a distinguished category, which contains both connected commutative algebraic groups and connected mixed Shimura varieties. In any distinguished category, special subvarieties can be defined and a Zilber-Pink statement can be formulated. We show that any distinguished category satisfies the defect condition, introduced as a useful technical tool by Habegger and Pila. Under an additional assumption, which makes the category "very distinguished", we show furthermore that the Zilber-Pink statement in general follows from the case where the subvariety is defined over the algebraic closure of the field of definition of the distinguished variety. The proof closely follows our proof in the case of abelian varieties and leads also to unconditional results in the moduli space of principally polarized abelian surfaces as well as in fibered powers of the Legendre family of elliptic curves.

I plan to survey known facts and open questions about ultrafilters on omega generating (or not generating) ultrafilters in forcing extensions.

Primitive elements are elements that are part of a basis for a free group. We present the classical Whitehead algorithm for the recognition of such elements, and discuss the ideas behind the proof. We also present a second algorithm, more recent and completely different in the approach.

I will describe a general method for comparing the counting functions of determinantal point processes in terms of trace class norm distances between their kernels (and review what all of those words mean). Then I will outline joint work with Elizabeth Meckes using this method to prove a version of a self-similarity property of eigenvalues of Haar-distributed unitary matrices conjectured by Coram and Diaconis.  Finally, I will discuss ongoing work by my PhD student Kyle Taljan, bounding the rate of convergence for counting functions of GUE eigenvalues to the Sine or Airy process counting functions.

Smectic A liquid crystals are of great interest in physics for their striking defect structures, including curvature walls and focal conics. However, the mathematical modeling of smectic liquid crystals has not been extensively studied. This work takes a step forward in understanding these fascinating topological defects from both mathematical and numerical viewpoints. In this talk, we propose a new (two- and three-dimensional) mathematical continuum model for the transition between the smectic A and nematic phases, based on a real-valued smectic order parameter for the density perturbation and a tensor-valued nematic order parameter for the orientation. Our work expands on an idea mentioned by Ball & Bedford (2015). By doing so, physical head-to-tail symmetry in half charge defects is respected, which is not possible with vector-valued nematic order parameter.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

I will explain how the representation theory of rational Cherednik algebras interacts with the commutative algebra of certain subspace arrangements arising from the reflection arrangement of a complex reflection group. Potentially, the representation theory allows one to study both qualitative questions (e.g., is the arrangement Cohen-Macaulay or not?) and quantitative questions (e.g., what is the Hilbert series of the ideal of the arrangement, or even, what are its graded Betti numbers?), by applying the tools (such as orthogonal polynomials, Kazhdan-Lusztig characters, and Dirac cohomology) that representation theory provides. This talk is partly based on joint work with Susanna Fishel and Elizabeth Manosalva.

Motivated by the advent of machine learning, the last few years saw the return of hardware-supported low-precision computing. Computations with fewer digits are faster and more memory and energy efficient, but can be extremely susceptible to rounding errors. An application that can largely benefit from the advantages of low-precision computing is the numerical solution of partial differential equations (PDEs), but a careful implementation and rounding error analysis are required to ensure that sensible results can still be obtained. In this talk we study the accumulation of rounding errors in the solution of the heat equation, a proxy for parabolic PDEs, via Runge-Kutta finite difference methods using round-to-nearest (RtN) and stochastic rounding (SR). We demonstrate how to implement the numerical scheme to reduce rounding errors and we present \emph{a priori} estimates for local and global rounding errors. Let $u$ be the roundoff unit. While the worst-case local errors are $O(u)$ with respect to the discretization parameters, the RtN and SR error behaviour is substantially different. We show that the RtN solution is discretization, initial condition and precision dependent, and always stagnates for small enough $\Delta t$. Until stagnation, the global error grows like $O(u\Delta t^{-1})$. In contrast, the leading order errors introduced by SR are zero-mean, independent in space and mean-independent in time, making SR resilient to stagnation and rounding error accumulation. In fact, we prove that for SR the global rounding errors are only $O(u\Delta t^{-1/4})$ in 1D and are essentially bounded (up to logarithmic factors) in higher dimensions.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.