Past

The course will aim to provide an introduction to stochastic PDEs from the classical perspective, that being a mixture of stochastic analysis and PDE analysis. We will focus in particular on the variational approach to semi-linear parabolic problems, `a  la  Lions. There will also be comments on  other models and approaches.

  Suggested Pre-requisites: The course is broadly aimed at graduate students with some knowledge of PDE theory and/or stochastic  analysis. Familiarity with measure theory and functional analysis will be useful.

Lecture 1:  Introduction and Preliminaries

• Regularity of solutions
• Ergodicity
• Pathwise approach to SPDE

Literature: [Hai09, DKM⁺09, DPZ96, Hai14, GIP15]

References

[DKM⁺09] Robert Dalang, Davar Khoshnevisan, Carl Mueller, David Nualart, and Yimin Xiao. A minicourse on stochastic partial differential equations, vol- ume 1962 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2009.

[DPZ96] G. Da Prato and J. Zabczyk. Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Note Series. Cambridge University Press, 1996.

[DPZ14] Giuseppe Da Prato and Jerzy Zabczyk. Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2 edition, 2014.

[Eva10] Lawrence Craig Evans. Partial Differential Equations. American Mathe- matical Society, 2010.

[GIP15] Massimiliano Gubinelli, Peter Imkeller, and Nicolas Perkowski. Paracon- trolled distributions and singular PDEs. Forum Math. Pi, 3:75, 2015.

[Hai09]  Martin Hairer.  An Introduction to Stochastic PDEs.  Technical  report, The University of Warwick / Courant Institute, 2009. Available at: http://hairer.org/notes/SPDEs.pdf

[Hai14] M. Hairer. A theory of regularity structures. Inventiones mathematicae, 198(2):269–504, 2014.

[Par07] Etienne  Pardoux. Stochastic  partial  differential  equations.  https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.405.4805&rep=rep1&type=pdf  2007.

[PR07] Claudia Pr´evˆot and Michael R¨ockner. A concise course on stochastic partial differential equations. Springer, 2007.

This talk will be the first in a spin-off series on the Lawrence-Venkatesh approach to showing that every hyperbolic curve$/K$ has finitely many $K$-points. In this talk, we will give the overall outline of the approach and prove several of  the preliminary results, such as Faltings' finiteness theorem for semisimple Galois representations.

We study several matrix diffusion processes constructed from a unitary Brownian motion. In particular, we use the Stiefel fibration to lift the Brownian motion of the complex Grass- mannian to the complex Stiefel manifold and deduce a skew-product decomposition of the Stiefel Brownian motion. As an application, we prove asymptotic laws for the determinants of the block entries of the unitary Brownian motion.

Grothendieck's Quot schemes — moduli spaces of quotient sheaves — are fundamental objects in algebraic geometry, but we know very little about them. This talk will focus on a relatively simple special case: the Quot scheme Quotˡ(E) of length l quotients of a vector bundle E of rank r on a smooth surface S. The scheme Quotˡ(E) is a cross of the Hilbert scheme of points of S (E=O) and the projectivisation of E (l=1); it carries a virtual fundamental class, and if l and r are at least 2, then Quotˡ(E) is singular. I will explain how the ADHM description of Quotˡ(E) provides a conjectural description of the singularities, and show how they can be resolved in the l=2 case. Furthermore, I will describe the relation between Quotˡ(E) and Quotˡ of a quotient of E, prove a functoriality result for the virtual fundamental class, and use it to compute certain tautological integrals over Quotˡ(E).

We will discuss several versions of Ito’s formula in the case where the function is path dependent and only concave or C1 in the sense of Dupire. In particular, we will show that it can be used to solve (super) hedging problems, in the context of market impact or under volatility uncertainty. 

I will outline a new algorithm for unknot recognition that runs in quasi-polynomial time. The input is a diagram of a knot with n crossings, and the running time is n^{O(log n)}. The algorithm uses hierarchies, normal surfaces and Heegaard splittings.

I will review recent work on N=(2,2) boundary conditions of 3d
N=4 theories which mimic isolated massive vacua at infinity. Subsets of
local operators supported on these boundary conditions form lowest
weight Verma modules over the quantised bulk Higgs and Coulomb branch
chiral rings. The equivariant supersymmetric Casimir energy is shown to
encode the boundary ’t Hooft anomaly, and plays the role of lowest
weights in these modules. I will focus on a key observable associated to
these boundary conditions; the hemisphere partition function, and apply
them to the holomorphic factorisation of closed 3-manifold partition
functions and indices. This yields new “IR formulae” for partition
functions on closed 3-manifolds in terms of Verma characters. I will
also discuss ongoing work on connections to enumerative geometry, and
the construction of elliptic stable envelopes of Aganagic and Okounkov,
in particular their physical manifestation via mirror duality
interfaces.

This talk is based on 2010.09741 and ongoing work with Mathew Bullimore
and Samuel Crew.

Speaker: Elena Gal (4pm)

Title: Associativity and Geometry

Abstract: An operation # that satisfies a#(b#c)=(a#b)#c is called "associative". Associativity is "common" - if we are asked to give an example of operation we are more likely to come up with one that has this property. However if we dig a bit deeper we encounter in geometry, topology and modern physics many operations that are not associative "on the nose" but rather up to an equivalence. We will talk about how to describe and work with this higher associativity notion.

Speaker: Alexandre Bovet (4:30pm)

Title: Investigating disinformation in social media with network science

Abstract:
While disinformation and propaganda have existed since ancient times, their importance and influence in the age of
social media is still not clear.  We investigate the spread of disinformation and traditional misinformation in Twitter in the context of the 2016 and 2020 US presidential elections. We analyse the information diffusion networks by reconstructing the retweet networks corresponding to each type of news and the top news spreaders of each network are identified. Our investigation provides new insights into the dynamics of news diffusion in Twitter, namely our results suggests that disinformation is governed by a different diffusion mechanism than traditional centre and left-leaning news. Centre and left leaning traditional news diffusion is driven by a small number of influential users, mainly journalists, and follow a diffusion cascade in a network with heterogeneous degree distribution which is typical of diffusion in social networks, while the diffusion of disinformation seems to not be controlled by a small set of users but rather to take place in tightly connected clusters of users that do not influence the rest of Twitter activity. We also investigate how the situation evolved between 2016 and 2020 and how the top news spreaders from the different news categories have driven the polarization of the Twitter ideological landscape during this time.

Reducing a chain complex (whilst preserving its homotopy-type) using algebraic Morse theory ([1, 2, 3]) gives the same end-result as Gaussian elimination, but AMT does it only on certain rows/columns and with several pivots (in all matrices simultaneously). Crucially, instead of doing costly row/column operations on a sparse matrix, it computes traversals of a bipartite digraph. This significantly reduces the running time and memory load (smaller fill-in and coefficient growth of the matrices). However, computing with AMT requires the construction of a valid set of pivots (called a Morse matching).

In [4], we discover a family of Morse matchings on any chain complex of free modules of finite rank. We show that every acyclic matching is a subset of some member of our family, so all maximal Morse matchings are of this type.

Both the input and output of AMT are chain complexes, so the procedure can be used iteratively. When working over a field or a local PID, this process ends in a chain complex with zero matrices, which produces homology. However, even over more general rings, the process often reveals homology, or at least reduces the complex so much that other algorithms can finish the job. Moreover, it also returns homotopy equivalences to the reduced complexes, which reveal the generators of homology and the induced maps $H_{*}(\varphi)$. 

We design a new algorithm for reducing a chain complex and implement it in MATHEMATICA. We test that it outperforms other CASs. As a special case, given a sparse matrix over any field, the algorithm offers a new way of computing the rank and a sparse basis of the kernel (or null space), cokernel (or quotient space, or complementary subspace), image, preimage, sum and intersection subspace. It outperforms built-in algorithms in other CASs.

References

[1]  M. Jöllenbeck, Algebraic Discrete Morse Theory and Applications to Commutative Algebra, Thesis, (2005).

[2]  D.N. Kozlov, Discrete Morse theory for free chain complexes, C. R. Math. 340 (2005), no. 12, 867–872.

[3]  E. Sköldberg, Morse theory from an algebraic viewpoint, Trans. Amer. Math. Soc. 358 (2006), no. 1, 115–129.

[4]  L. Lampret, Chain complex reduction via fast digraph traversal, arXiv:1903.00783.

There has been a surge of interest in machine learning in the past few years, and deep learning techniques are more and more integrated into
the way we do quantitative science. A particularly exciting case for deep learning is molecular physics, where some of the "superpowers" of
machine learning can make a real difference in addressing hard and fundamental computational problems - on the other hand the rigorous
physical footing of these problems guides us in how to pose the learning problem and making the design decisions for the learning architecture.
In this lecture I will review some of our recent contributions in marrying deep learning with statistical mechanics, rare-event sampling
and quantum mechanics.

Introduced by Fomin and Zelevinsky in 2002, cluster algebras have become ubiquitous in algebra, combinatorics and geometry. In this talk, I'll introduce the notion of a cluster algebra and present the approach of Kang-Kashiwara-Kim-Oh to categorify a large class of them arising from quantum groups. Time allowing, I will explain some recent developments related to the coherent Satake category.

Learning a representation from data that disentangles different factors of variation is hypothesized to be a critical ingredient for unsupervised learning. Defining disentangling is challenging - a "symmetry-based" definition was provided by Higgins et al. (2018), but no prescription was given for how to learn such a representation. We present a novel nonparametric algorithm, the Geometric Manifold Component Estimator (GEOMANCER), which partially answers the question of how to implement symmetry-based disentangling. We show that fully unsupervised factorization of a data manifold is possible if the true metric of the manifold is known and each factor manifold has nontrivial holonomy – for example, rotation in 3D. Our algorithm works by estimating the subspaces that are invariant under random walk diffusion, giving an approximation to the de Rham decomposition from differential geometry. We demonstrate the efficacy of GEOMANCER on several complex synthetic manifolds. Our work reduces the question of whether unsupervised disentangling is possible to the question of whether unsupervised metric learning is possible, providing a unifying insight into the geometric nature of representation learning.

It is well known that expected signatures can be used as the “moments” of the law of stochastic processes. Inspired by this fact, we introduced higher rank expected signatures to capture the essences of the weak topologies of adapted processes, and characterize the information evolution pattern associated with stochastic processes. This approach provides an alternative perspective on a recent important work by Backhoff–Veraguas, Bartl, Beiglbock and Eder regarding adapted topologies and causal Wasserstein metrics.

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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 contact @email.

Biological systems provide an inspiration for creating a new paradigm
for materials synthesis. What would it take to enable inanimate material
to acquire the properties of living things? A key difference between
living and synthetic materials is that the former are programmed to
behave as they do, through interactions, energy consumption and so
forth. The nature of the program is the result of billions of years of
evolution. Understanding and emulating this program in materials that
are synthesizable in the lab is a grand challenge. At its core is an
optimization problem: how do we choose the properties of material
components that we can create in the lab to carry out complex reactions?
I will discuss our (not-yet-terribly-successful efforts)  to date to
address this problem, by designing both equiliibrium and kinetic 
properties of materials, using a combination of statistical mechanics,
kinetic modeling and ideas from machine learning.

I will discuss regularity properties for solutions of linear second order non-uniformly elliptic equations in divergence form. Assuming certain integrability conditions on the coefficient field, we obtain local boundedness and validity of Harnack inequality. The assumed integrability assumptions are sharp and improve upon classical results due to Trudinger from the 1970s.

As an application of the local boundedness result, we deduce a quenched invariance principle for random walks among random degenerate conductances. If time permits I will discuss further regularity results for nonlinear non-uniformly elliptic variational problems.

I will discuss the multi-sorted structure of analytic covers H -> Y(N), where H is the upper half-plane and Y(N) are the N-level modular curves, all N, in a certain language, weaker than the language applied by Adam Harris and Chris Daw.  We define a certain locally modular reduct of the structure which is called "pure" structure - an extension of the structure of special subvarieties.  
The problem of non-elementary categorical axiomatisation for this structure is closely related to the theory of "canonical models for Shimura curves", in particular, the description of Gal_Q action on the CM-points of the Y(N). This problem for the case of curves is basically solved (J.Milne) and allows the beautiful interpretation in our setting:  the abstract automorphisms of the pure structure on CM-points are exactly the automorphisms induced by Gal_Q.  Using this fact and earlier theorem of Daw and Harris we prove categoricity of a natural axiomatisation of the pseudo-analytic structure.
If time permits I will also discuss a problem which naturally extends the above:  a categoricity statement for the structure of unramified analytic covers H -> X, where X runs over all smooth curves over a given number field.  

Let us say that a theory $T$ in the language of set theory is $\beta$-consistent at $\alpha$ if there is a transitive model of $T$ of height $\alpha$, and let us say that it is $\beta$-categorical at $\alpha$ iff there is at most one transitive model of $T$ of height $\alpha$. Let us also assume, for ease of formulation, that there are arbitrarily large $\alpha$ such that $\mathrm{ZFC}$ is $\beta$-consistent at $\alpha$.

The sentence $\mathrm{VEL}$ has the feature that $\mathrm{ZFC}+\mathrm{VEL}$ is $\beta$-categorical at $\alpha$, for every $\alpha$. If we assume in addition that $\mathrm{ZFC}+\mathrm{VEL}$ is $\beta$-consistent at $\alpha$, then the uniquely determined model is $L_\alpha$, and the minimal such model, $L_{\alpha_0}$, is model of determined by the $\beta$-categorical theory $\mathrm{ZFC}+\mathrm{VEL}+M$, where $M$ is the statement "There does not exist a transitive model of $\mathrm{ZFC}$."

It is natural to ask whether $\mathrm{VEL}$ is the only sentence that can be $\beta$-categorical at $\alpha$; that is, whether, there can be a sentence $\phi$ such that $\mathrm{ZFC}+\phi$ is $\beta$-categorical at $\alpha$, $\beta$-consistent at $\alpha$, and where the unique model is not $L_\alpha$.  In the early 1970s Harvey Friedman proved a partial result in this direction. For a given ordinal $\alpha$, let $n(\alpha)$ be the next admissible ordinal above $\alpha$, and, for the purposes of this discussion, let us say that an ordinal $\alpha$ is minimal iff a bounded subset of $\alpha$ appears in $L_{n(\alpha)}\setminus L_\alpha$. [Note that $\alpha_0$ is minimal (indeed a new subset of $\omega$ appears as soon as possible, namely, in a $\Sigma_1$-definable manner over $L_{\alpha_0+1}$) and an ordinal $\alpha$ is non-minimal iff $L_{n(\alpha)}$ satisfies that $\alpha$ is a cardinal.] Friedman showed that for all $\alpha$ which are non-minimal, $\mathrm{VEL}$ is the only sentence that is $\beta$-categorical at $\alpha$. The question of whether this is also true for $\alpha$ which are minimal has remained open.

In this talk I will describe some joint work with Hugh Woodin that bears on this question. In general, when approaching a "lightface" question (such as the one under consideration) it is easier to first address the "boldface" analogue of the question by shifting from the context of $L$ to the context of $L[x]$, where $x$ is a real. In this new setting everything is relativized to the real $x$: For an ordinal $\alpha$, we let $n_x(\alpha)$ be the first $x$-admissible ordinal above $\alpha$, and we say that $\alpha$ is $x$-minimal iff a bounded subset of $\alpha$ appears in $L_{n_x(\alpha)}[x]\setminus L_{\alpha}[x]$.

Theorem. Assume $M_1^\#$ exists and is fully iterable. There is a sentence $\phi$ in the language of set theory with two additional constants, \r{c} and \r{d}, such that for a Turing cone of $x$, interpreting \r{c} by $x$, for all $a$

1. if $L_\alpha[x]\vDash\mathrm{ZFC}$ then there is an interpretation of \r{d}  by something in $L_\alpha[x]$ such that there is a $\beta$-model of $\mathrm{ZFC}+\phi$ of height $\alpha$ and not equal to $L_\alpha[x]$, and
2. if, in addition, $\alpha$ is $x$-minimal, then there is a unique $\beta$-model of $\mathrm{ZFC}+\phi$ of height $\alpha$ and not equal to $L_\alpha[x]$.

The sentence $\phi$ asserts the existence of an object which is external to $L_\alpha[x]$ and which, in the case where $\alpha$ is minimal, is canonical. The object is a branch $b$ through a certain tree in $L_\alpha[x]$, and the construction uses techniques from the HOD analysis of models of determinacy.

In this talk I will sketch the proof, describe some additional features of the singleton, and say a few words about why the lightface version looks difficult.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

In the world of finitely generated groups, presentations are a blessing and a curse. They are versatile and compact, but in general tell you very little about the group. Tietze transformations offer much (but deliver little) in terms of understanding the possible presentations of a group. I will introduce a different way of transforming presentations of a group called a Nielsen transformation, and show how topological methods can be used to study Nielsen transformations.