Past

Homophily can put minority groups at a disadvantage by restricting their ability to establish connections with majority groups or to access novel information. In this talk, I show how this phenomenon is manifested in a variety of online and face-to-face social networks and what societal consequences it has on the visibility and ranking of minorities. I propose a network model with tunable homophily and group sizes and demonstrate how the ranking of nodes is affected by homophilic
behavior. I will discuss the implications of this research on algorithms and perception biases.

In this talk we will consider the extremal values of the stationary distribution of the sparse directed configuration model. Under the assumption of linear $(2+\eta)$-moments on the in-degrees and of bounded out-degrees, we obtain tight comparisons between the maximum value of the stationary distribution and the maximum in-degree. Under the further assumption that the order statistics of the in-degrees have power-law behavior, we show that the upper tail of the stationary distribution also has power-law behavior with the same index. Moreover, these results extend to the PageRank scores of the model, thus confirming a version of the so-called power-law hypothesis. Joint work with Xing Shi Cai, Pietro Caputo and Matteo Quattropani.

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.

We study the minimization of functionals of the form $$ u  \mapsto \int_\Omega  f(\nabla u) \, dx $$

with a convex integrand $f$ of linear growth (such as the area integrand), among all functions in the Sobolev space W$^{1,1}$ with prescribed boundary values. Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and might in fact fail, as it is well-known already for the non-parametric minimal surface problem. In such cases, the functional is extended suitably to the space BV of functions of bounded variation via relaxation, and for the relaxed functional one can in turn guarantee the existence of minimizers. However, in contrast to the original minimization problem, these BV minimizers might in principle have interior jump discontinuities or not attain the prescribed boundary values.

After a short introduction to the problem I want to focus on the question of regularity of BV minimizers. In past years, Sobolev regularity was established provided that the lack of ellipticity -- which is always inherent for such linear growth integrands -- is mild, while, in general, only some structure results seems to be within reach. In this regard, I will review several results which were obtained in cooperation with Miroslav Bulíček (Prague), Franz Gmeineder (Bonn), Erika Maringová (Vienna), and Thomas Schmidt (Hamburg).

In my talk, I will introduce a family of human-machine interaction (HMI) models in optimal portfolio construction (robo-advising). Modeling difficulties stem from the limited ability to quantify the human’s risk preferences and describe their evolution, but also from the fact that the stochastic environment, in which the machine optimizes, adapts to real-time incoming information that is exogenous to the human. Furthermore, the human’s risk preferences and the machine’s states may evolve at different scales. This interaction creates an adaptive cooperative game with both asymmetric and incomplete information exchange between the two parties.

As a result, challenging questions arise on, among others, how frequently the two parties should communicate, what information can the machine accurately detect, infer and predict, how the human reacts to exogenous events, how to improve the inter-linked reliability between the human and the machine, and others. Such HMI models give rise to new, non-standard optimization problems that combine adaptive stochastic control, stochastic differential games, optimal stopping, multi-scales and learning.

I will briefly introduce the Chabauty-Kim argument for effective finiteness results on "topologically rich enough" curves. I will then introduce the Fontaine-Mazur conjecture and show how it provides an effective proof of Faltings' Theorem.

In the case of non-CM elliptic curves minus a point, following work of Federico Amadio Guidi, I'll show how the relevant input for effective finiteness is provided by the vanishing of adjoint Selmer groups proven by Newton and Thorne.

C*-algebras provide non commutative analogues of locally compact Hausdorff spaces. In this talk I’ll provide a survey of the large scale project to classify simple amenable C*-algebras, indicating the role played by non commutative versions of topological ideas. No prior knowledge of C*-algebras will be assumed.

I'll be presenting my PhD work, in which I define two new algebraic structures on the equivariant symplectic cohomology of a convex symplectic manifold. The first is a collection of shift operators which generalise the shift operators on equivariant quantum cohomology in algebraic geometry. That is, given a Hamiltonian action of the torus T, we assign to a cocharacter of T an endomorphism of (S1 × T)-equivariant Floer cohomology based on the equivariant Floer Seidel map. The second is a connection which is a multivariate version of Seidel’s q-connection on S1 -equivariant Floer cohomology and generalises the Dubrovin connection on equivariant quantum cohomology.

The periods of a Calabi-Yau manifold are of interest both to number theorists and to physicists. To a number theorist the primary object of interest is the zeta function. I will explain what this is, and why this is of interest also to physicists. For applications it is important to be able to calculate the local zeta function for many primes p. I will set out a method, adapted from a procedure proposed by Alan Lauder that makes the computation of the zeta function practical, in this sense, and comment on the form of the results. This talk is based largely on the recent paper hepth 2104.07816 and presents joint work with Xenia de la Ossa and Duco van Straten.

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.

Path signature has unique advantages on extracting high-order differential features of sequential data. Our team has been studying the path signature theory and actively applied it to various applications, including infant cognitive score prediction, human motion recognition, hand-written character recognition, hand-written text line recognition and writer identification etc. In this talk, I will share our most recent works on infant cognitive score prediction using deep path signature. The cognitive score can reveal individual’s abilities on intelligence, motion, language abilities. Recent research discovered that the cognitive ability is closely related with individual’s cortical structure and its development. We have proposed two frameworks to predict the cognitive score with different path signature features. For the first framework, we construct the temporal path signature along the age growth and extract signature features of developmental infant cortical features. By incorporating the cortical path signature into the multi-stream deep learning model, the individual cognitive score can be predicted with missing data issues. For the second framework, we propose deep path signature algorithm to compute the developmental feature and obtain the developmental connectivity matrix. Then we have designed the graph convolutional network for the score prediction. These two frameworks have been tested on two in-house cognitive data sets and reached the state-of-the-art results.

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.