Past

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).

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.

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 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.

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.  
 

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.
 

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.

Dynamical interface motions are important flow patterns and fundamental free boundary problems in fluid mechanics, and have attracted huge attention in the mathematical community. Such waves for purely inviscid fluids are subject to various instabilities such as Kelvin-Helmholtz and Rayleigh-Taylor instabilities unless other stabilizing effects such as surface tension, Taylor-sign conditions or dissipations are imposed. However, in the presence of magnetic fields, it has been known that tangential magnetic fields may have stabilizing effects for free surface waves such as plasma-vacuum or plasma-plasma interfaces (at least locally in time), yet whether transversal magnetic fields (which occurs often for interfacial waves for astrophysical plasmas) can stabilize typical free interfacial waves remain to be some open problems. In this talk, I will show the stabilizing effects of the transversal magnetic fields for some interfacial waves for both compressible and incompressible multi-dimensional magnetohydrodynamics (MHD).

First, I will present the local (in time) well-posedness in Sobolev space of multi- dimensional compressible MHD contact discontinuities, which are the most typical interfacial waves for astrophysical plasma and prototypical fundamental waves for systems of hyperbolic conservations. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversally, which leads to a two-phase free boundary problem that may have nonlinear Rayleigh- Taylor instability and whose front symbols have no ellipticity. We overcome such difficulties by exploiting full the transversality of the magnetic fields and designing a nonlinear approximate problem, which yields the local well-posed without loss of derivatives and without any other conditions such as Rayleigh-Taylor sign conditions or surface tension. Second, I will discuss some results on the global well-posedness of free interface problems for the incompressible inviscid resistive MHD with transversal magnetic fields. Both plasma-vacuum and plasma-plasma interfaces are studied. The global in time well-posedness of both interface problems in a horizontally periodic slab impressed by a uniform non-horizontal magnetic field near an equilibrium are established, which reveals the strong stabilizing effect of the transversal field as the global well- posedness of the free boundary incompressible Euler equations (without the irrotational assumptions) around an equilibrium is unknown. This talk is based on joint work with Professor Yanjin Wang. 

Apart from the binomial coefficients which are ubiquitous in many counting problems, the Catalan and Fibonacci sequences seem to appear almost as frequently. There are also well-known interpretations of the Catalan numbers as lattice paths, or as the number of ways of connecting 2n points on a circle via non-intersecting lines. We start by obtaining some identities for sums involving the Catalan sequence. In addition, we use the beautiful binomial transform which allows us to obtain several pretty identities involving Fibonacci numbers, Catalan numbers, and trigonometric sums.

Speaker: James Taylor
Title: D-Modules and p-adic Representations

Abstract: The representation theory of finite groups is a beautiful and well-understood subject. However, when one considers more complicated groups things become more interesting, and to classify their representations is often a much harder problem. In this talk, I will introduce the classical theory, the particular groups I am interested in, and explain how one might hope to understand their representations through the use of D-modules - the algebraic incarnation of differential equations.

Speaker: Anthony Webster
Title: An Introduction to Epidemiology and Causal Inference

Abstract: This talk will introduce epidemiology and causal inference from the perspective of a statistician and former theoretical physicist. Despite their studies being underpinned by deep and often complex mathematics, epidemiologists are generally more concerned by seemingly mundane information about the relationships between potential risk factors and disease. Because of this, I will argue that a good epidemiologist with minimal statistical knowledge, will often do better than a highly trained statistician. I will also argue that causal assumptions are a necessary part of epidemiology, should be made more explicitly, and allow a much wider range of causal inferences to be explored. In the process, I will introduce ideas from epidemiology and causal inference such as Mendelian Randomisation and the "do calculus", methodological approaches that will increasingly underpin data-driven population research.  

Training deep learning models involves searching for a good model over the space of possible architectures and their parameters. Discovering models that exhibit robust generalization to unseen data and tasks is of paramount for accurate and reliable machine learning. Generalization, a hallmark of model efficacy, is conventionally gauged by a model's performance on data beyond its training set. Yet, the reliance on vast training datasets raises a pivotal question: how can deep learning models transcend the notorious hurdle of 'memorization' to generalize effectively? Is it feasible to assess and guarantee the generalization prowess of deep neural networks in advance of empirical testing, and notably, without any recourse to test data? This inquiry is not merely theoretical; it underpins the practical utility of deep learning across myriad applications. In this talk, I will show that scrutinizing the training dynamics of neural networks through the lens of topology, specifically using 'persistent-homology dimension', leads to novel bounds on the generalization gap and can help demystifying the inner workings of neural networks. Our work bridges deep learning with the abstract realms of topology and learning theory, while relating to information theory through compression.