Past

The set $M_n(\mathbb{R})$ of all $n \times n$ matrices over the real numbers is an example of an algebraic structure called a $C^*$-algebra. The subalgebra $D$ of diagonal matrices has special properties and is called a \emph{Cartan subalgebra} of $M_n(\mathbb{R})$. Given an arbitrary $C^*$-algebra, it can be very hard (but also very rewarding) to find a Cartan subalgebra, if one exists at all. However, if the $C^*$-algebra is generated by a cocycle $c$ and a group (or groupoid) $G$, then it is natural to look within $G$ for a subgroup (or subgroupoid) $S$ that may give rise to a Cartan subalgebra. In this talk, we identify sufficient conditions on $S$ and $c$ so that the subalgebra generated by $(S,c)$ is indeed a Cartan subalgebra of the $C^*$-algebra generated by $(G,c)$. We then apply our theorem to $C^*$-algebras generated by $k$-graphs, which are directed graphs in higher dimensions. This is joint work with J. Briones Torres, A. Duwenig, L. Gallagher, E. Gillaspy, S. Reznikoff, H. Vu, and S. Wright.

A self-similar k-graph is a pair consisting of a (discrete countable) group and a k-graph, such that the group acts on the k-graph self-similarly. For such a pair, one can associate it with a universal C*-algebra, called the self-similar k-graph C*-algebra. This class of C*-algebras embraces many important and interesting C*-algebras,  such as the higher rank graph C*-algebras of Kumjian-Pask, the Katsura algebras,  the Nekrashevych algebras constructed from self-similar groups, and the Exel-Pardo algebra. 

In this talk, we will survey some results on self-similar k-graph C*-algebras. 

Dimensions of mapping class groups of orientable and non-orientable surfaces

1:30pm

Luis Jorge Sánchez Saldaña (UNAM)

Mapping class groups have been studied extensively for several decades. Still in these days these groups keep being studied from several point of views. In this talk I will talk about several notions of dimension that have been computed (and some that are not yet known) for mapping class groups of both orientable and non-orientable manifolds. Among the dimensions that I will mention are the virtual cohomological dimension, the proper geometric dimension, the virtually cyclic dimension and the virtually abelian dimension. Some of the results presented are in collaboration with several colleagues: Trujillo-Negrete, Hidber, León Álvarez and Jimaénez Rolland.

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Diffeomorphisms of reducible 3-manifolds

2:45pm

Rachael Boyd (Glasgow)

I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space B Diff(M), for M a compact, connected, reducible 3-manifold. We prove that when M is orientable and has non-empty boundary, B Diff(M rel ∂M) has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where M is irreducible by Hatcher and McCullough.

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Nonunique ergodicity in strata of geodesic laminations and the boundary of Outer space

4:00pm

Mladen Bestvina (Utah)

It follows from the work of Gabai and Lenzhen-Masur that the maximal number of projectively distinct ergodic transverse measures on a filling geodesic lamination on a hyperbolic surface is equal to the number of curves in a pants decomposition. In a joint work with Jon Chaika and Sebastian Hensel, we answer the analogous question when the lamination is restricted to have specified polygons as complementary components. If there is enough time, I will also talk about the joint work with Elizabeth Field and Sanghoon Kwak where we consider the question of the maximal number of projectively distinct ergodic length functions on a given arational tree on the boundary of Culler-Vogtmann's Outer space of a free group.
 

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.

In this talk, I will describe some of the ways in which mathematical modelling contributed to the Covid-19 pandemic response in New Zealand. New Zealand adopted an elimination strategy at the beginning of the pandemic and used a combination of public health measures and border restrictions to keep incidence of Covid-19 low until high vaccination rates were achieved. The low or zero prevalence for first 18 months of the pandemic called for a different set of modelling tools compared to high-prevalence settings. It also generated some unique data that can give valuable insights into epidemiological characteristics and dynamics. As well as describing some of the modelling approaches used, I will reflect on the value modelling can add to decision making and some of the challenges and opportunities in working with stakeholders in government and public health.        

The goal of Post-Quantum Cryptography (PQC) is to design cryptosystems which are secure against classical and quantum adversaries. A topic of fundamental research for decades, the status of PQC drastically changed with the NIST PQC standardization process. Recently there have been AI attacks on some of the proposed systems to PQC. In this talk, we will give an overview of the progress of quantum computing and how it will affect the security landscape. 

Group-based cryptography is a relatively new family in post-quantum cryptography, with high potential. I will give a general survey of the status of post-quantum group-based cryptography and present some recent results.

In the second part of my talk, I speak about Post-quantum hash functions using special linear groups with implication to post-quantum blockchain technologies.

The goal of Post-Quantum Cryptography (PQC) is to design cryptosystems which are secure against classical and quantum adversaries. A topic of fundamental research for decades, the status of PQC drastically changed with the NIST PQC standardization process. Recently there have been AI attacks on some of the proposed systems to PQC. In this talk, we will give an overview of the progress of quantum computing and how it will affect the security landscape. 

Group-based cryptography is a relatively new family in post-quantum cryptography, with high potential. I will give a general survey of the status of post-quantum group-based cryptography and present some recent results.

In the second part of my talk, I speak about Post-quantum hash functions using special linear groups with implication to post-quantum blockchain technologies.

To neural activity one may associate a space of correlations and a space of population vectors. These can provide complementary information. Assume the goal is to infer properties of a covariate space, represented by ochestrated activity of the recorded neurons. Then the correlation space is better suited if multiple neural modules are present, while the population vector space is preferable if neurons have non-convex receptive fields. In this talk I will explain how to coherently combine both pieces of information in a bifiltration using Dowker complexes and their total weights. The construction motivates an interesting extension of Dowker’s duality theorem to simplicial categories associated with two composable relations, I will explain the basic idea behind it’s proof.

In this presentation, we will review how the field of Mechanics of Materials is generally framed and see how it can benefit from and be of benefit to the current progress in AI. We will approach this problematic in the particular context of Brain mechanics with an application to traumatic brain injury in police investigations. Finally we will briefly show how our group is currently applying the same methodology to a range of engineering challenges.

It is well-known that the set of irreducible (finite-dimensional) representations of a semisimiple complex Lie algebra g can be indexed by the dominant weights. The Borel-Weil theorem asserts that they can be seen geometrically as the global sections of line bundles over the flag variety. The Borel-Weil-Bott theorem computes the higher sheaf cohomology groups. There are several ways to prove the Borel-Weil-Bott theorem, which we will discuss. The classical idea is to study how the Casimir operator acts on the sheaf of sections of line bundles. Instead of this, the geometric idea is trying to compute the Doubeault cohomology, transferring the sheaf cohomology to the Lie algebra cohomology. The algebraic idea is to realize that the sheaf cohomology group can be computed by the derived functor of the induction, by using the Peter-Weyl the Borel-Weil theorem can be shown immediately.

I will exposit some recent joint work with Paul Larson and Grigor Sargsyan that uses higher models of the Axiom of Determinacy---models with nontrivial structure above $\Theta$, the least ordinal which is not the surjective image of the reals---to show that instances of the fundamental problem of inner model theory, the iterability conjecture, consistently fail.

Let $S$ be a set of rational places of odd cardinality containing infinity and a rational prime $p$. We can associate to $S$ a Shimura curve $X$ defined over $\mathbb{Q}$. The Gross--Kohnen--Zagier theorem states that certain generating series of Heegner points of $X$ are modular forms of weight $3/2$ valued in the Jacobian of $X$. We will state this theorem and outline a new approach to proving it using the theory of $p$-adic uniformization and $p$-adic families of modular forms of half-integral weight. This is joint work with Lea Beneish, Henri Darmon, and Lennart Gehrmann.

We extend stochastic optimal transport to path-dependent settings. The problem is to find a semimartingale measure that satisfies general path-dependent constraints, while minimising a cost function on the drift and diffusion coefficients. Duality is established and expressed via non-linear path-dependent partial differential equations (PPDEs). The technique has applications in volatility calibration, including the calibration of path-dependent derivatives, LSV models, and joint SPX-VIX models. It produces a non-parametric volatility model that localises to the features of the derivatives. Another application is in the robust pricing and hedging of American options in continuous time. This is achieved by establishing duality in a space enlarged by the stopping decisions, and showing that the extremal points of martingale measures on the enlarged space are in fact martingale measures on the original space coupled with stopping times.

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 study of counting point-hyperplane incidences in the $d$-dimensional space was initiated in the 1990's by Chazelle and became one of the central problems in discrete geometry. It has interesting connections to many other topics, such as additive combinatorics and theoretical computer science. Assuming a standard non-degeneracy condition, i.e., that no $s$ points are contained in the intersection of $s$ hyperplanes, the currently best known upper bound on the number of incidences of $m$ points and $n$ hyperplanes in $\mathbb{R}^d$ is $O((mn)^{1-1/(d+1)}+m+n)$. This bound by Apfelbaum and Sharir is based on geometrical space partitioning techniques, which apply only over the real numbers.

In this talk, we discuss a novel combinatorial approach to study such incidence problems over arbitrary fields. Perhaps surprisingly, this approach matches the best known bounds for point-hyperplane incidences in $\mathbb{R}^d$ for many interesting values of $m, n, d$. Moreover, in finite fields our bounds are sharp as a function of $m$ and $n$ in every dimension. This approach can also be used to study point-variety incidences and unit-distance problem in finite fields, giving tight bounds for both problems under a similar non-degeneracy assumption. Joint work with A. Milojevic and I. Tomon.

This seminar will introduce 2D-MoSub, a derivative-free optimization method based on the subspace method and quadratic models, specifically tackling large-scale derivative-free problems. 2D-MoSub combines 2-dimensional quadratic interpolation models and trust-region techniques to update the points and explore the 2-dimensional subspace iteratively. Its framework includes constructing the interpolation set, building the quadratic interpolation model, performing trust-region trial steps, and updating the trust-region radius and subspace. Computation details and theoretical properties will be discussed. Numerical results demonstrate the advantage of 2D-MoSub.

Short Bio:
Pengcheng Xie, PhD (Chinese Academy of Sciences), is joining Lawrence Berkeley National Laboratory as a postdoctoral scholar specializing in mathematical optimization and numerical analysis. He has developed optimization methods, including 2D-MoSub and SUSD-TR. Pengcheng has published in major journals and presented at ISMP 2024 (upcoming), ICIAM 2023, and CSIAM 2022. He received the Hua Loo-keng scholarship in 2019 and the CAS-AMSS Presidential scholarship in 2023.
 

Even though most of us tie our shoelaces "wrongly," knots in ropes and filaments have been used as functional structures for millennia, from sailing and climbing to dewing and surgery. However, knowledge of the mechanics of physical knots is largely empirical, and there is much need for physics-based predictive models. Tight knots exhibit highly nonlinear and coupled behavior due to their intricate 3D geometry, large deformations, self-contact, friction, and even elasto-plasticity. Additionally, tight knots do not show separation of the relevant length scales, preventing the use of centerline-based rod models. In this talk, I will present an overview of recent work from our research group, combining precision experiments, Finite Element simulations, and theoretical analyses. First, we study the mechanics of two elastic fibers in frictional contact. Second, we explore several different knotted structures, including the overhand, figure-8, clove-hitch, and bowline knots. These knots serve various functions in practical settings, from shoelaces to climbing and sailing. Lastly, we focus on surgical knots, with a particularly high risk of failure in clinical settings, including complications such as massive bleeding or the unraveling of high-tension closures. Our research reveals a striking and robust power law, with a general exponent, between the mechanical strength of surgical knots, the applied pre-tension, and the number of throws, providing new insights into their operational and safety limits. These findings could have potential applications in the training of surgeons and enhanced control of robotic-assisted surgical devices.

Solovay defined the inner model $L(\mathbb{R}, \mu)$ in the context of $\mathsf{AD}_{\mathbb{R}}$ by using it to define the supercompactness measure $\mu$ on $\mathcal{P}_{\omega_1}(\mathbb{R})$ naturally given by $\mathsf{AD}_{\mathbb{R}}$. Solovay speculated that stronger versions of this inner model should exist, corresponding to stronger versions of the measure $\mu$. Woodin, in his unpublished work, defined $\mu_{\infty}$ which is arguably the ultimate version of the supercompactness measure $\mu$ that Solovay had defined. I will talk about $\mu_{\infty}$ in the context of $\mathsf{AD}^+$ and the axiom $\mathsf{V} = \mathsf{Ultimate\ L}$.

https://woloszyn.org/

If \(F\) is a free group and \(F/N\) is a presentation of a group \(G\), there is a natural way to turn the abelianisation of \(N\) into a \(\mathbb ZG\)-module, known as the relation module of the presentation. The images of normal generators for \(N\) yield \(\mathbb ZG\)-module generators of the relation module, but 'lifting' \(\mathbb ZG\)-generators to normal generators cannot always be done by a result of Dunwoody. Nevertheless, it is an open problem, known as the relation gap problem, whether the relation module can have strictly fewer \(\mathbb ZG\)-module generators than \(N\) can have normal generators when \(G\) is finitely presented. In this talk I will survey what is known and what is not known about this problem and its variations and discuss some recent progress for groups with a cyclic relation module.

A central theme in the 40-year-old Ribe program is the quest for metric invariants that characterize local properties of Banach spaces. These invariants are usually closely related to the geometry of certain sequences of finite graphs (Hamming cubes, binary trees, diamond graphs...) and provide quantitative bounds on the bi-Lipschitz distortion of those graphs.

A more recent program, deeply influenced by the late Nigel Kalton, has a similar goal but for asymptotic properties instead. In this talk, we will motivate the (asymptotic) notions of infrasup umbel convexity (introduced in collaboration with Chris Gartland (UC San Diego)) and bicone convexity. These asymptotic notions are inspired by the profound work of Lee, Mendel, Naor, and Peres on the (local) notion of Markov convexity and of Eskenazis, Mendel, and Naor on the (local) notion of diamond convexity. 

All these metric invariants share the common feature of being derived from point-configuration inequalities which generalize curvature inequalities.

If time permits we will discuss the values of these invariants for Heisenberg groups.

I will give an overview and introduction to the most important decision problems in combinatorial semigroup theory, including the word problem, and describe attempts to solve a problem that has been open since 1914: the word problem in one-relation semigroups. I will link it with some of my results from formal language theory, as well as recent joint work with I. Foniqi and R. D. Gray (East Anglia) on proving undecidability of certain harder problems, proved by way of passing via one-relator groups.

An interesting twist on classical subgraph containment problems in graph theory is the following: given a graph $H$ and a collection $\{G_1, \dots , G_m\}$ of graphs on a common vertex set $[n]$, what conditions on $G_i$ guarantee a copy of $H$ using at most one edge from each $G_i$? Such a subgraph is called transversal, and the above problem is closely related to the study of temporal graphs in Network Theory. In 2020 Joos and Kim showed that if $\delta(G_i)\geq n/2$, the collection contains a transversal Hamilton cycle. We improve on their result by showing that it actually contains every transversal Hamilton cycle if $\delta(G_i)\geq (1/2+o(1))n$. That is, for every function $\chi:[n]\to[m]$, there is a Hamilton cycle whose $i$-th edge belongs to $G_{\chi(i)}$.

This is joint work with Candida Bowtell, Patrick Morris and Katherine Staden.

The objective of the talk is to present elements of Euclidean Quantum Field Theory and of the Symanzik's polymer representation for a model which includes an interaction with a magnetic field. We will explain how the problem of constructing such an EQFT can be translated into the problem of renormalising the Amperean area of a planar Brownian motion, an object that we will introduce during the talk. No prerequisite knowledge of the topic is expected.

Based on http://perso.ens-lyon.fr/isao.sauzedde/square_field3_3.pdf 

Khintchine's Theorem is one of the cornerstones in metric Diophantine approximation. The question of removing the monotonicity condition on the approximation function in Khintchine's Theorem led to the recently proved Duffin-Schaeffer conjecture. Gallagher showed an analogue of Khintchine's Theorem for multiplicative Diophantine approximation, again assuming monotonicity. In this talk, I will discuss my joint work with L. Frühwirth about a Duffin-Schaeffer version for Gallagher's Theorem. Furthermore, I will give a broader overview on various questions in metric Diophantine approximation and demonstrate the deep connection to both analytic and combinatorial number theory that is hidden inside the proof of these statements.

 A commensuration of a group G is an isomorphism between finite-index subgroups of G. Equivalence classes of such maps form a group, whose importance first emerged in the work of Margulis on the rigidity and arithmeticity of lattices in semisimple Lie groups. Drawing motivation from this classical setting and from the study of mapping class groups of surfaces, I shall explain why, when N is at least 3, the group of automorphisms of the free group of rank N is its own abstract commensurator. Similar results hold for certain subgroups of Aut(F_N). These results are the outcome of a long-running project with Ric Wade. An important element in the proof is a non-abelian analogue of the Fundamental Theorem of Projective Geometry in which projective subspaces are replaced by the free factors of a free group; this is the content of a long-running project with Mladen Bestvina.
 

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. An intriguing property of their model is a conjectured phase transition between models that converge to growing disks, and 'turbulent' non-disk like models. In this talk I will describe a natural generalisation of the Hastings-Levitov family in which the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. In recent joint work with Norris and Silvestri, we show that when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. This phase transition in fluctuations can be interpreted as the beginnings of a macroscopic phase transition from disks to non-disks analogous to that present in the Hastings-Levitov family.

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.