Past

Measure equivalence was introduced by Gromov as a measure-theoretic analogue to quasi-isometry between finitely generated groups. In this talk I will present measure equivalence classification results for right-angled Artin groups, and more generally graph products. This is based on joint works with Jingyin Huang and with Amandine Escalier. 

The saturation number of a graph is a famous and well-studied counterpoint to the Turán number, and the rainbow saturation number is a generalisation of the saturation number to the setting of coloured graphs. Specifically, for a given graph $F$, an edge-coloured graph is $F$-rainbow saturated if it does not contain a rainbow copy of $F$, but the addition of any non-edge in any colour creates a rainbow copy of $F$. The rainbow saturation number of $F$ is the minimum number of edges in an $F$-rainbow saturated graph on $n$ vertices. Girão, Lewis, and Popielarz conjectured that, like the saturation number, for all $F$ the rainbow saturation number is linear in $n$. I will present our attractive and elementary proof of this conjecture, and finish with a discussion of related results and open questions.

One of the oldest open problems in representation theory is to classify the irreducible unitary representations of a semisimple Lie group G_R. Such representations play a fundamental role in harmonic analysis and the Langlands program and arise in physics as the state space of quantum mechanical systems in the presence of G_R-symmetry. Most unitary representations of G_R are realized, via some kind of induction, from unitary representations of proper Levi subgroups. Thus, the major obstacle to understanding the unitary dual of G_R is identifying the "non-induced" unitary representations of G_R. In previous joint work with Losev and Matvieievskyi, we have proposed a general construction of these non-induced representations, which we call "unipotent" representations of G_R. Unfortunately, the methods we employ do not provide a proof that these representations are unitary. In this talk, I will explain how one can apply Saito's theory of mixed Hodge modules to overcome this difficulty, giving a uniform proof of the unitarity of all unipotent representations. This is joint work in progress with Dougal Davis

We study the stochastic quantisation for the fractional $\varphi^4$ theory. The model has been studied by Brydges, Mitter and Scopola in 2003 as a natural extension of $\phi^4$ theories to fractional sub-critical dimensions. The stochastic quantisation equation is given by the (formal) SPDE 

\[

(\partial_t + (-\Delta)^{s}) \varphi = - \lambda \varphi^3 + \xi\]

where $\xi$ is a space-time white noise over the three dimensional torus. The equation is sub-critical for $s > \frac{3}{4}$.

We derive a priori estimates in the full sub-critical regime $s>\frac{3}{4}$. These estimates rule out explosion in finite time and they imply the existence of an invariant measure with a standard Krylov-Bogoliubov argument. 

Our proof is based on the strategy developed for the parabolic case $s=1$ in [Chandra, Moinat, Weber, ARMA 2023]. In order to implement this strategy here, a new Schauder estimate for the fractional heat operator is developed. Additionally, several algebraic arguments from [Chandra, Moinat, Weber, ARMA 2023] are streamlined significantly. 

This is joint work with Hendrik Weber (Münster). 

It is a classical fact of rational homotopy theory that the $E_\infty$-algebra of rational cochains on a sphere is formal, i.e., quasi-isomorphic to the cohomology of the sphere. In other words, this algebra is square-zero. This statement fails with integer or mod p coefficients. We show, however, that the cochains of the n-sphere are still $E_n$-trivial with coefficients in arbitrary cohomology theories. This is a consequence of a more general statement on (iterated) loops and suspensions of $E_n$-algebras, closely related to Koszul duality for the $E_n$-operads. We will also see that these results are essentially sharp: if the R-valued cochains of $S^n$ have square-zero $E_{n+1}$-structure (for some rather general ring spectrum R), then R must be rational. This is joint work with Markus Land.

A cap set is a subset of $\mathbb{F}_3^n$ with no solutions to $x + y + z = 0$ other than when $x = y = z$, or equivalently no non-trivial $3$-term arithmetic progressions. The cap set problem asks how large a cap set can be, and is an important problem in additive combinatorics and combinatorial number theory. In this talk, I will introduce the problem, give some background and motivation, and describe how I was able to provide the first progress in 20 years on the lower bound for the size of a maximal cap set. Building on a construction of Edel, we use improved computational methods and new theoretical ideas to show that, for large enough $n$, there is always a cap set in $\mathbb{F}_3^n$ of size at least $2.218^n$. I will then also discuss recent developments, including an extension of this result by Google DeepMind.

We consider the sharp interface limit problem for 1D stochastic Allen-Cahn equation, and extend a classic result by Funaki to the full small noise regime. One interesting point is that the notion of "small noise" turns out to depend on the topology one uses. The main new idea in the proof is the construction of a series of functional correctors, which are designed to recursively cancel out potential divergences. At a technical level, in order to show these correctors are well behaved, we also develop a systematic decomposition of functional derivatives of the deterministic Allen-Cahn flow of all orders, which might have its own interest.
Based on a joint work with Wenhao Zhao (EPFL) and Shuhan Zhou (PKU).

Deep Gaussian processes have proved remarkably successful as a tool for various statistical inference tasks. This success relates in part to the flexibility of these processes and their ability to capture complex, non-stationary behaviours. 

In this talk, we will introduce the general framework of deep Gaussian processes, in which many examples can be constructed, and demonstrate their superiority in inverse problems including computational imaging and regression.

 We will discuss recent algorithmic developments for efficient sampling, as well as recent theoretical results which give crucial insight into the behaviour of the methodology.

 

From genetic studies and astrophysics simulations to statistical modelling and AI, scientific computing has enabled amazing discoveries and there is no doubt it will continue to do so. However, the corresponding environmental impact is a growing concern in light of the urgency of the climate crisis, so what can we all do about it? Tackling this issue and making it easier for scientists to engage with sustainable computing is what motivated the Green Algorithms project. Through the prism of the GREENER principles for environmentally sustainable science, we will discuss what we learned along the way, how to estimate the impact of our work and what levers scientists and institutions have to make their research more sustainable. We will also debate what hurdles exist and what is still needed moving forward.

The Junior Algebra and Representation Theory Seminar will kick-off the start of Trinity term with a social event in the common room. Come to catch up with your fellow students and maybe play a board game or two. Afterwards we'll have lunch together.

We study supersymmetric Berry connections of 2d N = (2,2) gauged linear sigma models (GLSMs) quantized on a circle, which are periodic monopoles, with the aim to provide a fruitful physical arena for recent mathematical constructions related to the latter. These are difference modules encoding monopole solutions via a Hitchin-Kobayashi correspondence established by Mochizuki. We demonstrate how the difference modules arises naturally by studying the ground states as the cohomology of a one-parameter family of supercharges. In particular, we show how they are related to one kind of monopole spectral data, a deformation of the Cherkis–Kapustin spectral curve, and relate them to the physics of the GLSM. By considering states generated by D-branes and leveraging the difference modules, we derive novel difference equations for brane amplitudes. We then show that in the conformal limit, these degenerate into novel difference equations for hemisphere partition functions, which are exactly calculable. When the GLSM flows to a nonlinear sigma model with Kähler target X, we show that the difference modules are related to deformations of the equivariant quantum cohomology of X.

We prove that any adjoint Chevalley group over an arbitrary commutative ring is regularly bi-interpretable with this ring. The same results hold for central quotients of arbitrary Chevalley groups and for Chevalley groups with bounded generation.
Also, we show that the corresponding classes of Chevalley groups (or their central quotients) are elementarily definable and even finitely axiomatizable.

We consider the reinforcement learning problem of controlling an unknown dynamical system to maximise the long-term average reward along a single trajectory. Most of the literature considers system interactions that occur in discrete time and discrete state-action spaces. Although this standpoint is suitable for games, it is often inadequate for systems in which interactions occur at a high frequency, if not in continuous time, or those whose state spaces are large if not inherently continuous. Perhaps the only exception is the linear quadratic framework for which results exist both in discrete and continuous time. However, its ability to handle continuous states comes with the drawback of a rigid dynamic and reward structure.

        This work aims to overcome these shortcomings by modelling interaction times with a Poisson clock of frequency $\varepsilon^{-1}$ which captures arbitrary time scales from discrete ($\varepsilon=1$) to continuous time ($\varepsilon\downarrow0$). In addition, we consider a generic reward function and model the state dynamics according to a jump process with an arbitrary transition kernel on $\mathbb{R}^d$. We show that the celebrated optimism protocol applies when the sub-tasks (learning and planning) can be performed effectively. We tackle learning by extending the eluder dimension framework and propose an approximate planning method based on a diffusive limit ($\varepsilon\downarrow0$) approximation of the jump process.

        Overall, our algorithm enjoys a regret of order $\tilde{\mathcal{O}}(\sqrt{T})$ or $\tilde{\mathcal{O}}(\varepsilon^{1/2} T+\sqrt{T})$ with the approximate planning. As the frequency of interactions blows up, the approximation error $\varepsilon^{1/2} T$ vanishes, showing that $\tilde{\mathcal{O}}(\sqrt{T})$ is attainable in near-continuous time.

A conjecture of Malle predicts an asymptotic formula for the number of number fields with given Galois group and bounded discriminant. Malle conjectured the shape of the formula but not the leading constant. We present a new conjecture on the leading constant motivated by a version for algebraic stacks of Peyre's constant from Manin's conjecture. This is joint work with Tim Santens.

We introduce a new method - ESPIRA (Estimation of Signal Parameters via Iterative Rational Approximation) \cite{DP22,  DPP21} - for the recovery of complex exponential  sums
$$
f(t)=\sum_{j=1}^{M} \gamma_j \mathrm{e}^{\lambda_j t},
$$
that are determined by a finite number of parameters: the order $M$, weights $\gamma_j \in \mathbb{C} \setminus \{0\}$ and nodes  $\mathrm{e}^{\lambda_j} \in \mathbb{C}$ for $j=1,...,M$.  Our new recovery procedure is based on the observation that Fourier coefficients (or DFT coefficients) of exponential sums have a special rational structure.  To  reconstruct this structure in a stable way we use the AAA algorithm  proposed by Nakatsukasa et al.   We show that ESPIRA can be interpreted as a matrix pencil method applied to Loewner matrices. 

During the talk we will demonstrate that ESPIRA outperforms Prony-like methods such as ESPRIT and MPM for noisy data and for signal approximation by short exponential sums.  

Bibliography
N. Derevianko,  G.  Plonka, 
Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients, Anal.  Appl.,  20(3),  2022,  543-577.

N. Derevianko,  G. Plonka,  M. Petz, 
From ESPRIT to ESPIRA: Estimation of signal parameters by iterative rational approximation,   IMA J. Numer. Anal.,  43(2),  2023, 789--827.  

Y. Nakatsukasa, O. Sète,   L.N. Trefethen,  The AAA algorithm for rational approximation.
SIAM J. Sci. Comput., 40(3),   2018,  A1494–A1522.  

We develop a comprehensive geometrically-exact theory for an end-loaded elastic rod constrained to deform on a cylindrical surface. By viewing the rod-cylinder system as a special case of an elastic braid, we are able to obtain all forces and moments imparted by the deforming rod to the cylinder as well as all contact reactions. This framework allows us to give a complete treatment of static friction consistent with force and moment balance. In addition to the commonly considered model of hard frictionless contact, we analyse two friction models in which the rod, possibly with intrinsic curvature, experiences either lateral or tangential friction. As applications of the theory we study buckling of the constrained rod under compressive and torsional loads, finding critical loads to depend on Coulomb-like friction parameters, as well as the tendency of the rod to lift off the cylinder under further loading. The cylinder can also have arbitrary orientation relative to the direction of gravity. The cases of a horizontal and vertical cylinder, with gravity having only a lateral or axial component, are amenable to exact analysis, while numerical results map out the transition in buckling mechanism between the two extremes. Weight has a stabilising effect for near-horizontal cylinders, while for near-vertical cylinders it introduces the possibility of buckling purely due to self-weight. Our results are relevant for many engineering and medical applications in which a slender structure winds inside or outside a cylindrical boundary.

We will decide on speakers for Trinity term 2024.

It is a standard result that mapping class groups of high genus do not surject the integers. This is easily shown by computing the abelianization of the mapping class group using a presentation. Once we pass to finite index subgroups, this becomes a conjecture of Ivanov. More generally, we can ask which groups admit epimorphisms from finite index subgroups of the mapping class group. In this talk, I will present a geometric approach to this question, using harmonic maps, and explain some recent results.

Product systems represent powerful contemporary tools in the study of mathematical structures. A major success in the theory came from Katsura (2007), who provided a complete description of the gauge-invariant ideals of many important C*-algebras arising from product systems over Z+. This result recaptures existing results from the literature, illustrating the versatility of product system theory. The question now becomes whether or not Katsura's result can be bolstered to product systems over semigroups other than Z+ and, if so, what applications do we obtain? An answer has been elusive, owing to the more pathological nature of product systems over general semigroups. However, recent strides by Dor-On and Kakariadis (2018) supply a more tractable subclass of product systems that still includes the important cases of C*-dynamics, row-finite higher-rank graphs, and regular product systems. 

In this talk we will build a parametrisation of the gauge-invariant ideals, starting from first principles and gradually increasing in complexity. We will pay particular attention to the higher-rank subtleties that are not witnessed in Katsura's theorem, and comment on the applications.
 

Approximate lattices are aperiodic generalisations of lattices in locally compact groups. They were first introduced in abelian groups by Yves Meyer before being studied as mathematical models for quasi-crystals. Since then their structure has been thoroughly investigated in both abelian and non-abelian settings.

In this talk I will survey what is known of the structure of approximate lattices. I will highlight some objects - such as a notion of cohomology sitting between group cohomology and bounded cohomology - that appear in their study. I will also formulate open problems and conjectures related to approximate lattices. 

The main component of flow machines is the rotor; however, there may also be stationary parts surrounding the rotor, which are the diffuser blades. In order to consider these two parts simultaneously, the most intuitive approach is to perform a transient flow simulation; however, the computational cost is relatively high. Therefore, one possible approach is the Multiple Reference Frame (MRF) approach, which considers two directly coupled zones: one for the rotating reference frame (for the rotor blades) and one for the stationary reference frame (for the diffuser blades). When taking into account topology optimisation, some changes are required in order to take both rotating and stationary parts simultaneously in the design, which also leads to changes in the composition of the multi-objective function. Therefore, the topology optimisation method is formulated for MRF while also proposing this new multi-objective function. An integer variable-based optimisation algorithm is considered, with some adjustments for the MRF case. Some numerical examples are presented.

We show that there exists $C>1$, such that if $A$ is a subset of a non-alternating finite simple group $G$ of density $|A|/|G|= \alpha$, then $AA^{-1}AA^{-1}$ contains a subgroup of density at least $\alpha^{C}$. We will also give a corresponding (slightly weaker) statement for alternating groups.

To prove our results we introduce new hypercontractive inequalities for simple groups. These allow us to show that the (non-abelian) Fourier spectrum of indicators of 'global' sets are concentrated on the high-dimensional irreducible representations. Here globalness is a pseudorandomness notion reminiscent of the notion of spreadness.

The talk is based on joint works with David Ellis, Shai Evra, Guy Kindler, Nathan Lindzey, and Peter Keevash, and Dor Minzer. No prior knowledge of representation theory will be assumed.

Our research explores the application of reinforcement learning (RL) strategies to solve complex combinatorial research problems, specifically the Job-shop Scheduling Problem (JSP) and the Stochastic Vehicle Routing Problem with Time Windows (SVRP). For JSP, we utilize Curriculum Learning (CL) to enhance the performance of dispatching policies. This approach addresses the significant optimality gap in existing end-to-end solutions by structuring the training process into a sequence of increasingly complex tasks, thus facilitating the handling of larger, more intricate instances. Our study introduces a size-agnostic model and a novel strategy, the Reinforced Adaptive Staircase Curriculum Learning (RASCL), which dynamically adjusts difficulty levels during training, focusing on the most challenging instances. Experimental results on Taillard and Demirkol datasets show that our approach reduces the average optimality gap to 10.46% and 18.85%, respectively.

For SVRP, we propose an end-to-end framework employing an attention-based neural network trained through RL to minimize routing costs while addressing uncertain travel costs and demands, alongside specific customer delivery time windows. This model outperforms the state-of-the-art Ant-Colony Optimization algorithm by achieving a 1.73% reduction in travel costs and demonstrates robustness across diverse environmental settings, making it a valuable baseline for future research. Both studies mark advancements in the application of machine learning techniques to operational research.

We will use Representation Theory to calculate systematically and efficiently the topological invariants of compact Lie groups and homogeneous spaces.
 

Most of the talk is covered by our second paper on ArXiv with John Jones and Adam Thomas, who are both at Warwick. The paper is part of the ongoing project to study the topological invariants of the four exceptional Rosenfeld projective planes.

The curvature-dimension condition, CD(K,N) for short, and the (weaker) measure contraction property, or MCP(K,N), are two synthetic notions for a metric measure space to have Ricci curvature bounded from below by K and dimension bounded from above by N. In this talk, we investigate the validity of these conditions in sub-Finsler geometry, which is a wide generalization of Finsler and sub-Riemannian geometry. Firstly, we show that sub-Finsler manifolds equipped with a smooth strongly convex norm and with a positive smooth measure can not satisfy the CD(K,N) condition for any K and N. Secondly, we focus on the sub-Finsler Heisenberg group, where we show that, on the one hand, the CD(K,N) condition can not hold for any reference norm and, on the other hand, the MCP(K,N) may hold or fail depending on the regularity of the reference norm. 

In this talk, we will study the problem in additive combinatorics of determining for a finite Abelian group $G$ the size of its smallest subset $A\subset G$ that has no unique sum, meaning that for every two $a_1,a_2\in A$ we can write $a_1+a_2=a’_1+a’_2$ for different $a’_1,a’_2\in A$. We begin by using classical rectification methods to obtain the previous best lower bounds of the form $|A|\gg \log p(G)$, which stood for 50 years. Our main aim is to outline the proof of a recent improvement and discuss some of its key notions such as additive dimension and the density increment method. This talk is based on Bedert, B. On Unique Sums in Abelian Groups. Combinatorica (2023).

I will discuss three different constructions of smooth tori in S^4 whose complements have fundamental group Z: turned 1-twist-spun tori due to Boyle, the union of a ribbon disc with a genus one Seifert surface constructed by Cochran and Davis, and certain tori with four critical points. They are all topologically unknotted, but it is not known whether they are smoothly standard, except for tori with four critical points whose middle level set is a split link. The branched double cover of S^4 along any of these surfaces is a potentially exotic copy of S^2 x S^2, though, in the case of Boyle's example, it cannot be distinguished from the standard S^2 x S^2 using Seiberg-Witten invariants. This is joint work with Mark Powell.

We will introduce the Quintic Ornstein-Uhlenbeck model that jointly calibrates SPX-VIX options with a particular focus on its mathematical tractability namely for fast pricing SPX options using Fourier techniques. Then, we will consider the more general class of  stochastic volatility models where the dynamics of the volatility are given by a possibly infinite linear combination of the elements of the time extended signature of a Brownian motion. First, we show that the model is remarkably universal, as it includes, but is not limited to, the celebrated Stein-Stein, Bergomi, and Heston models, together with some path-dependent variants. Second, we derive the joint characteristic functional of the log-price and integrated variance provided that some infinite-dimensional extended tensor algebra valued Riccati equation admits a solution. This allows us to price and (quadratically) hedge certain European and path-dependent options using Fourier inversion techniques. We highlight the efficiency and accuracy of these Fourier techniques in a comprehensive numerical study.

We introduce a notion of refined Harder-Narasimhan filtration, defined abstractly for algebraic stacks satisfying natural conditions. Examples include moduli stacks of objects at the heart of a Bridgeland stability condition, moduli stacks of K-semistable Fano varieties, moduli of principal bundles on a curve, and quotient stacks. We will explain how refined Harder-Narasimhan filtrations are closely related both to stratifications and to the asymptotics of certain analytic flows, relating and expanding work of Kirwan and Haiden-Katzarkov-Kontsevich-Pandit, respectively. In the case of quotient stacks by the action of a torus, the refined Harder-Narasimhan filtration can be computed in terms of convex geometry.

Graph signals provide a natural representation of data in many applications, such as social networks, web information analysis, sensor networks, and machine learning. Graph signal & data processing is currently an active field of mathematical research that aims to extend the well-developed tools for analyzing conventional signals to signals on graphs while exploiting the underlying connectivity. A key challenge in this context is the problem of quantization, that is, finding efficient ways of representing the values of graph signals with only a finite number of bits.

In this talk, we address the problem of quantizing bandlimited graph signals. We introduce two classes of noise-shaping algorithms for graph signals that differ in their sampling methodologies. We demonstrate that these algorithms can efficiently construct quantized representatives of bandlimited graph-based signals with bounded amplitude.

Inspired by the results of Zhang et al. in 2022, we provide theoretical guarantees on the relative error between the true signal and its quantized representative for one of the algorithms.
As will be discussed, the incoherence of the underlying graph plays an important role in the quantization process. Namely, bandlimited signals supported on graphs of lower incoherence allow for smaller relative errors. We support our findings with various numerical experiments showcasing the performance of the proposed quantization algorithms for bandlimited signals defined on graphs with different degrees of incoherence.

This is joint work with Felix Krahmer (Technical University of Munich), He Lyu (Meta), Rayan Saab (University of California San Diego), and Rongrong Wang (Michigan State University).

Join us for our first event of term to discuss those topics which are slightly taboo. We’ll be talking about periods, pregnancy, chronic illness, gender identity... This event is open to all but we will be taking extra steps to make sure it is a safe space for everyone. 

Transportation cost spaces are of high theoretical interest,  and they also are fundamental in applications in many areas of applied mathematics, engineering, physics, computer science, finance, and social sciences. 

Obtaining low distortion embeddings of transportation cost spaces into L_1 became important in the problem of finding nearest points, an important research subject in theoretical computer science. After introducing

these spaces we will present some results on upper  and lower estimates of the distortion of embeddings of Transportation Cost Spaces into L_1

Investors fear that surging volumes in short-term, especially same-day expiry (0DTE), options can destabilize markets by propagating large price jumps. Contrary to the intuition that 0DTE sellers predominantly generate delta-hedging flows that aggravate market moves, high open interest gamma in 0DTEs does not propagate past volatility. 0DTEs and underlying markets have become more integrated over time, leading to a marginally stronger link between the index volatility and 0DTE trading. Nonetheless, intraday 0DTE trading volume shocks do not amplify recent past index returns, inconsistent with the view that 0DTEs market growth intensifies market fragility.

About the speaker
Grigory Vilkov, Professor of Finance at the Frankfurt School of Finance and Management, holds an MBA from the University of Rochester and a Ph.D. from INSEAD, with further qualifications from Goethe University Frankfurt. He has been a professor at both Goethe University and the University of Mannheim.
His academic work focused on improving long-term portfolio strategies by building better expectations of risks, returns, and their dynamics. He is known for practical innovations in finance, such as developing forward-looking betas marketed by IvyDB OptionMetrics, establishing implied skewness and generalized lower bounds as cross-sectional stock characteristics, and creating measures for climate change exposure from earnings calls. His current research encompasses factor dispersions, factor and sector rotation, asset allocation with implied data, and machine learning in options analysis. 

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While the typical behaviors of stochastic systems are often deceptively oblivious to the tail distributions of the underlying uncertainties, the ways rare events arise are vastly different depending on whether the underlying tail distributions are light-tailed or heavy-tailed. Roughly speaking, in light-tailed settings, a system-wide rare event arises because everything goes wrong a little bit as if the entire system has conspired up to provoke the rare event (conspiracy principle), whereas, in heavy-tailed settings, a system-wide rare event arises because a small number of components fail catastrophically (catastrophe principle). In the first part of this talk, I will introduce the recent developments in the theory of large deviations for heavy-tailed stochastic processes at the sample path level and rigorously characterize the catastrophe principle for such processes. 
The empirical success of deep learning is often attributed to the mysterious ability of stochastic gradient descents (SGDs) to avoid sharp local minima in the loss landscape, as sharp minima are believed to lead to poor generalization. To unravel this mystery and potentially further enhance such capability of SGDs, it is imperative to go beyond the traditional local convergence analysis and obtain a comprehensive understanding of SGDs' global dynamics within complex non-convex loss landscapes. In the second part of this talk, I will characterize the global dynamics of SGDs building on the heavy-tailed large deviations and local stability framework developed in the first part. This leads to the heavy-tailed counterparts of the classical Freidlin-Wentzell and Eyring-Kramers theories. Moreover, we reveal a fascinating phenomenon in deep learning: by injecting and then truncating heavy-tailed noises during the training phase, SGD can almost completely avoid sharp minima and hence achieve better generalization performance for the test data.  

This talk is based on the joint work with Mihail Bazhba, Jose Blanchet, Bohan Chen, Sewoong Oh, Zhe Su, Xingyu Wang, and Bert Zwart.
 

While the typical behaviors of stochastic systems are often deceptively oblivious to the tail distributions of the underlying uncertainties, the ways rare events arise are vastly different depending on whether the underlying tail distributions are light-tailed or heavy-tailed. Roughly speaking, in light-tailed settings, a system-wide rare event arises because everything goes wrong a little bit as if the entire system has conspired up to provoke the rare event (conspiracy principle), whereas, in heavy-tailed settings, a system-wide rare event arises because a small number of components fail catastrophically (catastrophe principle). In the first part of this talk, I will introduce the recent developments in the theory of large deviations for heavy-tailed stochastic processes at the sample path level and rigorously characterize the catastrophe principle for such processes. 

The empirical success of deep learning is often attributed to the mysterious ability of stochastic gradient descents (SGDs) to avoid sharp local minima in the loss landscape, as sharp minima are believed to lead to poor generalization. To unravel this mystery and potentially further enhance such capability of SGDs, it is imperative to go beyond the traditional local convergence analysis and obtain a comprehensive understanding of SGDs' global dynamics within complex non-convex loss landscapes. In the second part of this talk, I will characterize the global dynamics of SGDs building on the heavy-tailed large deviations and local stability framework developed in the first part. This leads to the heavy-tailed counterparts of the classical Freidlin-Wentzell and Eyring-Kramers theories. Moreover, we reveal a fascinating phenomenon in deep learning: by injecting and then truncating heavy-tailed noises during the training phase, SGD can almost completely avoid sharp minima and hence achieve better generalization performance for the test data.  

This talk is based on the joint work with Mihail Bazhba, Jose Blanchet, Bohan Chen, Sewoong Oh, Zhe Su, Xingyu Wang, and Bert Zwart.

While the typical behaviors of stochastic systems are often deceptively oblivious to the tail distributions of the underlying uncertainties, the ways rare events arise are vastly different depending on whether the underlying tail distributions are light-tailed or heavy-tailed. Roughly speaking, in light-tailed settings, a system-wide rare event arises because everything goes wrong a little bit as if the entire system has conspired up to provoke the rare event (conspiracy principle), whereas, in heavy-tailed settings, a system-wide rare event arises because a small number of components fail catastrophically (catastrophe principle). In the first part of this talk, I will introduce the recent developments in the theory of large deviations for heavy-tailed stochastic processes at the sample path level and rigorously characterize the catastrophe principle for such processes. 
The empirical success of deep learning is often attributed to the mysterious ability of stochastic gradient descents (SGDs) to avoid sharp local minima in the loss landscape, as sharp minima are believed to lead to poor generalization. To unravel this mystery and potentially further enhance such capability of SGDs, it is imperative to go beyond the traditional local convergence analysis and obtain a comprehensive understanding of SGDs' global dynamics within complex non-convex loss landscapes. In the second part of this talk, I will characterize the global dynamics of SGDs building on the heavy-tailed large deviations and local stability framework developed in the first part. This leads to the heavy-tailed counterparts of the classical Freidlin-Wentzell and Eyring-Kramers theories. Moreover, we reveal a fascinating phenomenon in deep learning: by injecting and then truncating heavy-tailed noises during the training phase, SGD can almost completely avoid sharp minima and hence achieve better generalization performance for the test data.  

This talk is based on the joint work with Mihail Bazhba, Jose Blanchet, Bohan Chen, Sewoong Oh, Zhe Su, Xingyu Wang, and Bert Zwart.

Bio:

Chang-Han Rhee is an Assistant Professor in Industrial Engineering and Management Sciences at Northwestern University. Before joining Northwestern University, he was a postdoctoral researcher at Centrum Wiskunde & Informatica and Georgia Tech. He received his Ph.D. from Stanford University. His research interests include applied probability, stochastic simulation, experimental design, and the theoretical foundation of machine learning. His research has been recognized with the 2016 INFORMS Simulation Society Outstanding Publication Award, the 2012 Winter Simulation Conference Best Student Paper Award, the 2023 INFORMS George Nicholson Student Paper Competition (2^nd place), and the 2013 INFORMS George Nicholson Student Paper Competition (finalist). Since 2022, his research has been supported by the NSF CAREER Award.  
 

Scientific knowledge, written in the form of differential equations, plays a vital role in various deep learning fields. In this talk, I will present a graph neural network (GNN) design based on reaction-diffusion equations, which addresses the notorious oversmoothing problem of GNNs. Since the self-attention of Transformers can also be viewed as a special case of graph processing, I will present how we can enhance Transformers in a similar way. I will also introduce a spatiotemporal forecasting model based on neural controlled differential equations (NCDEs). NCDEs were designed to process irregular time series in a continuous manner and for spatiotemporal processing, it needs to be combined with a spatial processing module, i.e., GNN. I will show how this can be done. 

The compatibility of local and global Langlands correspondences is a central problem in algebraic number theory. A possible approach to resolving it relies on the existence of global Galois representations with prescribed local monodromy.  I will provide a partial solution by relating the question to its topological analogue. Both the topological and arithmetic version can be solved using the same family of projective hypersurfaces, which was first studied by Dwork.

In this short course, we survey the celebrated weak and strong compactness theorems proved by Karen Uhlenbeck in 1982. These results are fundamental to the gauge theory and have found numerous applications to geometry, topology, and theoretical physics. The proof is based on the ingenious idea of putting connections into ``Uhlenbeck--Coulomb gauge'', which enables the use of standard elliptic and/or nonlinear PDE techniques, as well as involved local-to-global patching arguments. We aim at giving detailed explanation of the proof, and we shall also discuss the relation between Uhlenbeck's compactness and the classical geometric problem of isometric immersions of submanifolds into Euclidean spaces.

The feedback loop between simulations and observations is the driving force behind almost all discoveries in astronomy. However, as technological innovations allow us to create ever more complex simulations and make ever more detailed observations, it becomes increasingly difficult to combine the two: since we cannot do controlled experiments, we need to simulate whatever we can observe. This requires efficient simulation pipelines, including (general-relativistic-)(magneto-)hydrodynamics, particle physics, chemistry, and radiation transport. In this talk, we explore the challenges associated with these modelling efforts and discuss how adopting data-driven surrogate modelling and proper control over model uncertainties, promises to unlock a gold mine of future discoveries. For instance, the application to stellar wind simulations can teach us about the origin of chemistry in our Universe and the building blocks for life, while supernova simulations can reveal exotic states of matter and elucidate the formation black holes.

The notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying solidity of von Neumann algebras. We introduce the notion of biexactness for von Neumann algebras, which allows us to place many previous solidity results in a more systematic context, and naturally leads to extensions of these results. We will also discuss examples of solid factors that are not biexact. This is a joint work with Jesse Peterson.

 In this short course, we will discuss the Euler equations and applications in gas dynamics and geometry. First, the basic theory of Euler equations and mixed-type problems will be reviewed. Then we will present the results on the transonic flows past obstacles, transonic flows in the fluid dynamic formulation of isometric embeddings, and the transonic flows in nozzles. We will discuss global solutions and stability obtained through various techniques and approaches. The short course consists of three parts and is accessible to PhD students and young researchers.