Past

Let $S$ be a smooth projective surface with $p_g>0$ and $H^1(S,{\mathbb Z})=0$. 
We consider the moduli spaces $M=M_S^H(r,c_1,c_2)$ of $H$-semistable sheaves on $S$ of rank $r$ and 
with Chern classes $c_1,c_2$. Associated a suitable class $v$ the Grothendieck group of vector bundles
on $S$ there is a deteminant line bundle $\lambda(v)\in Pic(M)$, and also a tautological sheaf $\tau(v)$ on $M$.

In this talk we derive a conjectural generating function for the virtual Verlinde numbers, i.e. the virtual holomorphic 
Euler characteristics of all determinant bundles $\lambda(v)$ on M, and for Segre invariants associated to $\tau(v)$ . 
The argument is based on conjectural blowup formulas and a virtual version of Le Potier's strange duality. 
Time permitting we also sketch a common refinement of these two conjectures, and their proof for Hilbert schemes of points.
 

The response of the Greenland and Antarctic ice sheets to a changing climate is one of the largest sources of uncertainty in future sea level predictions.  The behaviour of the subglacial environment, where ice meets hard rock or soft sediment, is a key determinant in the flux of ice towards the ocean, and hence the loss of ice over time.  Predicting how ice sheets respond on a range of timescales brings together mathematical models of the elastic and viscous response of the ice, subglacial sediment and water and is a rich playground where the simplified models of the contact between ice, rock and ocean can shed light on very large scale questions.  In this talk we’ll see how these simplified models can make sense of a variety of field and laboratory data in order to understand the dynamical phenomena controlling the transient response of large ice sheets.

 

CANCELLED DUE TO ILLNESS

The response of the Greenland and Antarctic ice sheets to a changing climate is one of the largest sources of uncertainty in future sea level predictions.  The behaviour of the subglacial environment, where ice meets hard rock or soft sediment, is a key determinant in the flux of ice towards the ocean, and hence the loss of ice over time.  Predicting how ice sheets respond on a range of timescales brings together mathematical models of the elastic and viscous response of the ice, subglacial sediment and water and is a rich playground where the simplified models of the contact between ice, rock and ocean can shed light on very large scale questions.  In this talk we’ll see how these simplified models can make sense of a variety of field and laboratory data in order to understand the dynamical phenomena controlling the transient response of large ice sheets.

Discrete Morse theory serves as a combinatorial tool for simplifying the structure of a given (regular) CW-complex up to homotopy equivalence, in terms of the critical cells of discrete Morse functions. In this talk, I will introduce a refinement of this theory that not only ensures homotopy equivalence with the simplified CW-complex but also guarantees a Whitehead simple homotopy equivalence. Furthermore, it offers an explicit description of the construction of the simplified Morse complex and provides bounds on the dimension of the complexes involved in the Whitehead deformation.
This refined approach establishes a suitable theoretical framework for addressing various problems in combinatorial group theory and topological data analysis. I will show applications of this technique to the Andrews-Curtis conjecture and computational methods for inferring the fundamental group of point clouds.

This talk is based on the article: Fernandez, X. Morse theory for group presentations. Trans. Amer. Math. Soc. 377 (2024), 2495-2523.

Global symmetries greatly enrich the phase diagram of quantum many-body systems. As well as symmetry-breaking phases, symmetry-protected topological (SPT) phases have symmetric ground states that cannot be connected to a trivial state without a phase transition. There can also be symmetry-enriched critical points between these phases of matter. I will demonstrate these phenomena in phase diagrams constructed using the N-state chiral clock family of spin chains.  [Based on joint work with Paul Fendley and Abhishodh Prakash.]

Choose your favourite connected graph $\Delta$ and shade a subset $J$ of its vertices. The intersection arrangement associated to the data $(\Delta, J)$ is a collection of real hyperplanes in dimension $|Jc|$, first defined by Iyama and Wemyss. This construction involves taking the classical Coxeter arrangement coming from $\Delta$ and then setting all variables indexed by $J$ to be zero. It turns out that for many choices of $J$ the chambers of the intersection arrangement admit a nice combinatorial description, along with a wall crossing rule to pass between them. I will start by making all this precise before discussing my work to classify tilings of the hyperbolic plane arising as intersection arrangements. This has applications to local notions of stability conditions using the tilting theory of contracted preprojective algebras.

Frontiers in Quantitative Finance is brought to you by the Oxford Mathematical and Computational Finance Group and sponsored by CitiGroup and Mosaic SmartData.

Abstract
This paper parsimoniously generalizes the VIX variance index by constructing model-free factor portfolios that replicate skewness and higher moments. It then develops an infinite series to replicate option payoffs in terms of the stock, bond, and factor returns. The truncated series offers new formulas that generalize the Black-Scholes formula to hedge variance and skewness risk.


About the speaker
Steve Heston is Professor of Finance at the University of Maryland. He is known for his pioneering work on the pricing of options with stochastic volatility.
Steve graduated with a double major in Mathematics and Economics from the University of Maryland, College Park in 1983, an MBA in 1985 followed by a PhD in Finance in 1990. He has held previous faculty positions at Yale, Columbia, Washington University, and the University of Auckland in New Zealand and worked in the private sector with Goldman Sachs in Fixed Income Arbitrage and in Asset Management Quantitative Equities.

Does the limit construction for inverse systems of first-order structures preserve elementary equivalence? I will give sufficient conditions for when this is the case. Using Karp's theorem, we explain the connection between a syntactic and formal-semantic approach to inverse limits of structures. We use this to give a simple proof of van den Dries' AKE theorem (in ZFC), a general AKE theorem for mixed characteristic henselian valued fields with no assumptions on ramification. We also recall a seemingly forgotten result of Feferman, that can be interpreted as a "saturated" AKE theorem in positive characteristic: given two elementarily equivalent $\aleph_1$-saturated fields $k$ and $k'$, the formal power series rings $k[[t]]$ and $k'[[t]]$ are elementarily equivalent as well. We thus hope to popularise some ideas from categorical logic.

The 12th of Hilbert's 23 problems posed in 1900 asks for an explicit description of abelian extensions of a given base field. Over the rationals, this is given by the exponential function, and over imaginary quadratic fields, by meromorphic functions on the complex upper half plane.  Darmon and Vonk's theory of rigid meromorphic cocycles, or "RM theory", includes conjectures giving a $p$-adic solution over real quadratic fields. These turn out to be closely linked to purely topological questions about intersections of geodesics in the upper half plane, and to $p$-adic deformations of Hilbert modular forms. I will explain an extension of results of Darmon, Pozzi and Vonk proving some of these conjectures, and some ongoing work concerning analogous results on Shimura curves.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

The classical finite element method uses piecewise-polynomial function spaces satisfying continuity and boundary conditions. Hybrid finite element methods, by contrast, drop these continuity and boundary conditions from the function spaces and instead enforce them weakly using Lagrange multipliers. The hybrid approach has several numerical and implementational advantages, which have been studied over the last few decades.

In this talk, we show how the hybrid perspective has yielded new insights—and new methods—in structure-preserving numerical PDEs. These include multisymplectic methods for Hamiltonian PDEs, charge-conserving methods for the Maxwell and Yang-Mills equations, and hybrid methods in finite element exterior calculus.

In geophysical and astrophysical flows, we are often interested in understanding the impact of internal waves on the non-wavelike flow. For example, oceanic internal waves generated at the surface and the seafloor transfer energy from the large scale flow to dissipative scales, thereby influencing the global ocean state. A primary challenge in the study of wave-flow interactions is how to separate these processes – since waves and non-wavelike flows can vary on similar spatial and temporal scales in the Eulerian frame. However, in a Lagrangian flow-following frame, temporal filtering offers a convenient way to isolate waves. Here, I will discuss a recently developed method for evolving Lagrangian mean fields alongside the governing equations in a numerical simulation, and extend this theory to allow effective filtering of waves from non-wavelike processes.

Given a pair of metric tensors g_j ≥ g₀ on a Riemannian manifold, M, it is well known that Vol_j(M)≥Vol₀(M). Furthermore, the volumes are equal if and only if the metric tensors are the same, g_j=g₀. Here we prove that if for a sequence g_j, we have g_j≥g₀, Vol_j(M)→Vol₀(M) and diam(M_j) ≤ D then (M,g_j) converges to (M,g₀) in the volume preserving intrinsic flat sense. The previous result will then be applied to prove stability of a class of tori.
 

This talk is based on joint works of myself with: Allen and Sormani (https://arxiv.org/abs/2003.01172), and Cabrera Pacheco and Ketterer (https://arxiv.org/abs/1902.03458).

The Hilbert scheme of d-points on a smooth surface is a well-studied object that still enjoys relatively large interest. We generalize Aldo Conca's Canonical Hilbert-Burch matrices and obtain explicit families of d-points. We show that such descriptions give us Białynicki-Birula cells of the Hilbert scheme for any choice of one-dimensional torus, thus describing the punctual component. This can be potentially applied to the study of singularities of the nested Hilbert scheme of points.

In 2006, Jan Dymara introduced the concept of weighted \(\ell^2\) Betti numbers as a method of computing regular \(\ell^2\) Betti numbers of buildings. This notion of dimension is measured by using Hecke algebras associated to the relevant Coxeter groups. I will briefly introduce buildings and then give a comparison between the regular \(\ell^2\) Betti numbers and the weighted ones.

The organizing principle of Ramsey theory is that in large mathematical structures, there are relatively large substructures which are homogeneous. This is quantified in combinatorics by the notion of Ramsey numbers $r(s,t)$, which denote the minimum $N$ such that in any red-blue coloring of the edges of the complete graph on $N$ vertices, there exists a red complete graph on $s$ vertices or a blue complete graph on $t$ vertices.

While the study of Ramsey numbers goes back almost one hundred years, to early papers of Ramsey and Erdős and Szekeres, the long-standing conjecture of Erdős that $r(s,t)$ has order of magnitude close to $t^{s-1}$ as $t \to \infty$ remains open in general. It took roughly sixty years before the order of magnitude of $r(3,t)$ was determined by Jeong Han Kim, who showed $r(3,t)$ has order of magnitude $t^2/\log t$ as $t \to \infty$. In this talk, we discuss a variety of new techniques, including the modern method of containers, which lead to a proof of the conjecture of Erdős that $r(4,t)$ is of order close to $t^3$.

One of the salient philosophies in our approach is that good Ramsey graphs hide inside pseudorandom graphs, and the long-standing emphasis of tackling Ramsey theory from the point of view of purely random graphs is superseded by pseudorandom graphs. Via these methods, we also come close to determining the well-studied related quantities known as Erdős-Rogers functions and discuss related hypergraph coloring problems and applications.

Joint work in part with Sam Mattheus, Dhruv Mubayi and David Conlon.

The space of measured laminations on a hyperbolic surface is a generalisation of the set of weighted multi curves. The action of the mapping class group on this space is an important tool in the study of the geometry of the surface. 
For orientable surfaces, orbit closures are now well-understood and were classified by Lindenstrauss and Mirzakhani. In particular, it is one of the pillars of Mirzakhani’s curve counting theorems. 
For non-orientable surfaces, the behaviour of this action is very different and the classification fails. In this talk I will review some of these differences and describe mapping class group orbit closures of (projective) measured laminations for non-orientable surfaces. This is joint work with Erlandsson, Gendulphe and Souto.

Dispersive nonlinear partial differential equations can be used to describe a range of physical systems, from water waves to spin states in ferromagnetism. The numerical approximation of solutions with limited differentiability (low-regularity) is crucial for simulating fascinating phenomena arising in these systems including emerging structures in random wave fields and dynamics of domain wall states, but it poses a significant challenge to classical algorithms. Recent years have seen the development of tailored low-regularity integrators to address this challenge. Inherited from their description of physicals systems many such dispersive nonlinear equations possess a rich geometric structure, such as a Hamiltonian formulation and conservation laws. To ensure that numerical schemes lead to meaningful results, it is vital to preserve this structure in numerical approximations. This, however, results in an interesting dichotomy: the rich theory of existent structure-preserving algorithms is typically limited to classical integrators that cannot reliably treat low-regularity phenomena, while most prior designs of low-regularity integrators break geometric structure in the equation. In this talk, we will outline recent advances incorporating structure-preserving properties into low-regularity integrators. Starting from simple discussions on the nonlinear Schrödinger and the Korteweg–de Vries equation we will discuss the construction of such schemes for a general class of dispersive equations before demonstrating an application to the simulation of low-regularity vortex filaments. This is joint work with Yvonne Alama Bronsard, Valeria Banica, Yvain Bruned and Katharina Schratz.

Models for networks that evolve and change over time are ubiquitous in a host of domains including modeling social networks, understanding the evolution of systems in proteomics, the study of the growth and spread of epidemics etc.

This talk will give a brief summary of three recent findings in this area where stochastic approximation techniques play an important role:

1. Understanding the effect and detectability of change point in the evolution of the system dynamics.
2. Reconstructing the initial "seed" that gave rise to the current network, sometimes referred to as Network Archeology.
3. The disparity in the behavior of different centrality measures such as degree and page rank centrality for measuring popularity in settings where there are vertices of different types such as majorities and minorities as well as insight analyzing such problems give for at first sight unrelated issues such as sampling rare groups within the network.

The main goal will to be convey unexpected findings in each of these three areas and in particular the "unreasonable effectiveness" of continuous time branching processes.

I'll give an introduction to a recent theme in the Langlands program over number fields and mixed characteristic local fields (with a much older history over function fields). This is enhancing the traditional 'set-theoretic' Langlands correspondence into something with a more geometric flavour. For example, relating (categories of) representations of p-adic groups to sheaves on moduli spaces of Galois representations. No number theory or 'Langlands' background will be assumed!