Seminars & What’s On Calendar

Abstract: There are several topological invariants that one may associate to a finitely generated group G – the characteristic varieties, the resonance varieties, and the Bieri-Neumann-Strebel invariants – which keep track of various finiteness properties of certain subgroups of G.  These invariants are interconnected in ways that makes them both more computable and more informative.  I will describe one such connection, made possible by tropical geometry, and I will provide examples and applications pertaining to complex geometry and low-dimensional topology. 

Abstract: The full N-body problem addresses the dynamics of N rigid bodies under mutual gravitational interactions. This physical system has powered the fabric of science and especially mathematics for centuries, having a decisive role in developing geometric mechanics, qualitative theory of dynamical systems, or KAM theory. In this talk, we briefly survey this problem and focus on analyzing special solutions called relative equilibria.

After determining the hamiltonian equations of motion, our approach identifies and uses the existence of translational and rotational symmetries of the N-body problem. In particular, we provide very compact equations characterizing relative equilibria solutions, which become linear by fixing the values of the invariants associated with the action of the symmetry group.

In the existing literature, relative equilibria have been classified into Lagrangian and non-Lagrangian, respectively, corresponding to whether the center of mass of all bodies is in the same plane. Our analysis determines what kind of configurations allow for each type of equilibrium and provides necessary conditions for non-Lagrangian equilibria.

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Abstract: Given a string Coxeter system (W,S), we construct highly regular quotients of the 1-skeleton of its universal polytope P, which form an infinite family of expander graphs when (W,S) is indefinite and P has finite vertex links. The regularity of the graphs in this family depends on the Coxeter diagram of (W,S). The expansion stems from superapproximation applied to (W,S). This construction is also extended to cover Wythoffian polytopes. As a direct application, we obtain several notable families of expander graphs with high levels of regularity, answering in particular a question posed by Chapman, Linial and Peled positively.

This talk is based on joint work with Marston Conder, Alexander Lubotzky and Francois Thilmany.

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Abstract: The classical Trudinger-Moser inequality is a borderline case of Sobolev inequalities and plays an important role in geometric analysis and PDEs in general. Aubin in 1979 showed that the best constant in the Trudinger-Moser inequality can be improved by reducing to one half if the functions are restricted to the complement of a three dimensional subspace of the Sobolev space $H^1$, while Onofri in 1982 discovered an elegant optimal form of Trudinger-Moser inequality on sphere. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without mass center constraints.

One such inequality, for example, incorporates the mass center deviation (from the origin) into the optimal inequality of Aubin on the sphere, which is for functions with mass centered at the origin. The main ingredient leading to the above inequalities is a novel geometric inequality: Sphere Covering Inequality.

Efforts have also been made to show similar inequalities in higher dimensions. Among the preliminary results, we have improved Beckner’s inequality for axially symmetric functions when the dimension $n=4, 6, 8$. Many questions remain open.

The talk is based on collaborations with Amir Moradifam, Sun-Yung Alice Chang, Yeyao Hu and Weihong Xie.

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Abstract: Extreme value theory deals with large values and rare events. These large values tend to cluster in case of temporal dependence. This clustering behaviour is widely observed in practice.
I will start with a mild introduction to extreme value theory, discussing probabilistic and statistical issues. This part will be accessible to a broader audience.

Then, I will talk about a more specific problem of statistical theory for cluster functionals and rare events. Two types of estimators are of a primary importance: disjoint and sliding blocks estimators. It has been conjectured that sliding blocks estimators are “better” (to be made precise in the talk). We proved in a recent series of papers that this is not the case and in fact both disjoint and sliding blocks estimators are asymptotically equivalent. This part will be aimed at probabilistic and statisticians.

I will conclude with recent directions in extreme value theory, such as extremes in high dimension, extremes of graphs and networks.

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Abstract: Discrete two-generator subgroups of PSL(2,R) have been extensively studied by investigating their action by Möbius transformations on the hyperbolic plane. Due to work of Gilman, Rosenberger, Purzitsky and many others, there is a complete classification of such groups by isomorphism type, and an algorithm to decide whether or not a two-generator subgroup of PSL(2,R) is discrete.

Here we completely classify discrete two-generator subgroups of PSL(2,Q_p) over the p-adic numbers Q_p by studying their action by isometries on the corresponding Bruhat-Tits tree. We give an algorithm to decide whether or not a two-generator subgroup of PSL(2,Q_p) is discrete, and discuss how this can be used to decide whether or not a two-generator subgroup of SL(2,Q_p) is dense. This is joint work with Jeroen Schillewaert.

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Abstract: Deep learning (the training of deep neural nets) is a very simple idea. Yet it has led to many striking applications throughout science and industry over the last decade. It has also become a major tool for applied mathematicians. In pure mathematics the impact has so-far been modest. I will discuss a few instances where it has proved useful, and led to interesting results in pure mathematics. I will also reflect on my experience as a pure mathematician interacting with deep learning.

Finally, I will discuss what can be learned from the successful examples that I understand, and try to guess an answer to the question in the title. (Deep learning also raises interesting mathematical questions, but this talk won’t be about this.)

Abstract: In this talk I’ll provide a brief overview of some of my research interests: understanding the stability of feature selection, robust modelling methods and multi-class modelling.

– Is the model you’ve selected uniquely “best”? [Spoiler: probably not]

– How can we develop methods that are resistant to data corruption?

– How can we take advantage of known structures in our data? For example, discover which groups in our data respond similarly to various inputs.

There are great funding opportunities associated with applied research. I’ve established a long-term association with Meat and Livestock Australia, particularly around predicting beef and sheep eating quality, which has spawned many projects. I’ll share some of my experiences of working with industry partners, interdisciplinary collaborators, and PhD students.

Abstract: Category theory takes a bird’s eye view of mathematics, allowing mathematicians to spot new patterns and interconnections. Their abstract nature has proved incredibly useful in mathematics, and its applications have reached areas like neuroscience, chemistry, electrical circuits and computer science. In particular, triangulated categories play a central role in every branch of mathematics that uses homological algebra: representation theory, algebraic geometry and stable homotopy theory. Given a metric on a triangulated category, there is a reasonable notion of Fourier series, which we can use to ‘approximate’ objects. This powerful technical tool, while relatively new, has already been used to powerful effect.

In this talk, I will try to convince you that you should use the tool of approximation in triangulated categories. Failing that, I will at least try to convince you that categories are everywhere, and their language is incredibly useful.

Abstract: A singular space often comes equipped with (1) a resolution, given by a morphism from a smooth space, and (2) a smoothing, namely a deformation with smooth generic fibre. I will discuss work in progress on how these may be related homologically, starting with the threefold ordinary double point as a key example.

Biography: Will Donovan is currently an associate professor at Yau MSC, Tsinghua University, Beijing. He is also a member of the adjunct faculty at BIMSA, Yanqi Lake, Huairou, Beijing and a visiting associate scientist at Kavli IPMU, University of Tokyo. He received his PhD in Mathematics in 2011 from Imperial College London. His interests are algebraic geometry, noncommutative geometry, representation theory, string theory and symplectic geometry.
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Seminar notes [PDF]

Abstract: I will explain what a Hodge module is starting from Poincare.

Biography: Kari Vilonen is a professor of mathematics at the University of Melbourne. His research is in geometric aspects of representation theory and the Langlands program. He has also worked on foundational questions on perverse sheaves and D modules including the microlocal point of view.

Abstract: In various application areas it is crucial to make predictions or decisions based on sequentially incoming observations and previous existing knowledge on the system of interest. The prior knowledge is often given in the form of evolution equations (e.g., ODEs derived via first principles or fitted based on previously collected data), from here on referred to as model. Despite the available observation and prior model information, accurate predictions of the „true“ reference dynamics can be very difficult.
Common reasons that make this problem so challenging are: ( i ) the underlying system is extremely complex (e.g., highly nonlinear) and chaotic (i.e., crucially dependent on the initial conditions), (ii) the associate state and/or parameter space is very high dimensional (e.g., worst case 10^8) (iii) Observations are noisy, partial in space and discrete in time.
In practice these obstacles are combated with a series of approximations (the most important ones being based on assuming Gaussian densities and using Monte Carlo type estimations) and numerical tools that work surprisingly well in some settings. Yet the mathematical understanding of the signal tracking ability of a lot of these methods is still lacking. Additionally, solutions of some of the more complicated problems that require simultaneous state and parameter estimation (including control parameters that can be understood as decisions/actions performed) can still not be approximated in a computationally feasible fashion. Here we will try to address the first layer of these issues step by step and discuss the next advances that need to be made in these many layered problems. More specifically a stability and accuracy analysis of a family of the most popular sequential data assimilation methods typically used in practice is presented. Then we will discuss how techniques from the world of machine learning can aid to overcome some of the computational challenges.
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Abstract: In this mini-course, I will talk about a celebrated theorem of Yum-Tong Siu asserting that given a smooth projective family f:X->Y of complex manifolds over an irreducible base and given any positive integer m, the dimension of the space of pluricanonical forms H^0(X_y, mK_{X_y}) is independent of Y. After recasting the result in its historical context, I will mention the Ohsawa-Takegoshi extension theorem which plays a central role of the proof. Finally, I will sketch the main steps following Mihai Paun’s streamlined proof of the theorem.

Abstract: Moduli spaces are geometric spaces which parametrize equivalence classes of algebraic varieties. I will discuss the moduli space of algebraic curves equivalently Riemann surfaces) of genus g, and use this example to motivate some interesting questions in higher dimensions.

Biography: Kenneth Ascher is an assistant professor in the department of mathematics at the University of California Irvine. His research area is algebraic and arithmetic geometry, with specific focuses on moduli spaces of higher dimensional varieties and applications to questions in arithmetic. He received his PhD in 2017 from Brown University under the direction of Dan Abramovich , and was a postdoctoral fellow at the Massachusetts Institute of Technology and Princeton University.
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Abstract: Virtual knots were introduced by Louis Kauffman in 1999 as a completion of classical knot theory in which planarity is no longer required. Virtual knots have been studied using a variety of approaches, including algebra, combinatorics, and geometric methods. They also have strong connections to quantum topology and finite type invariants. This talk will survey some fascinating results that have been established and present also open problems and directions for future research.

Biography: Dr Hans U. Boden is a professor of mathematics at McMaster University in Canada. He is visiting the University of Sydney and SMRI from May 17 to June 11. His research interests are on the geometry and topology of manifolds, especially gauge theory and low-dimensional topology. In recent years, his work has focused on developing geometric methods to understand knotting and linking in 3-dimensional manifolds. While in Sydney, he will be working closely with Dr Zsuzsi Dancso on a collaborative project related to the Tait conjectures in knot theory.

Abstract: Clustering is an important task in almost every area of knowledge: medicine and epidemiology, genomics, environmental science, economics, visual sciences, among others.

Methodologies to perform inference on the number of clusters have often been proved to be inconsistent and introducing a dependence structure among the clusters implies additional difficulties in the estimation process. In a Bayesian setting, clustering in the situation where the number of clusters is unknown is often performed by using Dirichlet process priors or finite mixture models. However, the posterior distributions on the number of groups have been recently proved to be inconsistent.

This seminar aims at reviewing the Bayesian approaches available to perform via mixture models and give some new insights.
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Abstract: Manifolds, that is, spaces that locally look like Euclidean space, occur in many settings and fields of research within and outside mathematics. In dimension two, manifolds are called surfaces.
Geometric topology is the study of manifolds. But unlike in most other settings where manifolds occur, we are not primarily interested in their shapes, but in their properties that remain unchanged under continuous deformations.
I will explain how manifolds can be most conveniently represented; go over some methods to study these representations; and demonstrate how these methods can give rise to deterministic algorithms solving fundamental problems in the field.
The flavour of this research highly depends on an integer: the dimension of the manifolds under investigation. In this talk the flavour will be mostly three.

Abstract: Many interesting invariants in geometry satisfy certain glueing or factorisation conditions, that are often useful when doing calculations. Topological quantum field theories (TQFTs) emerged in the 1980s as an organising structure for invariants that are governed by bordisms. In 2 dimensions, bordisms are surfaces with boundaries and the TQFT has a simple algebraic description. By remembering the diffeomorphisms of the surfaces, one is lead to the notion of a cohomological field theory.

This talk will give an overview of these ideas and be aimed at a broad audience.
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Abstract: Young diagrams and standard tableaux on them parameterize irreducible representations of the symmetric group and their bases, respectively. There is a similar story for the double affine Hecke algebra (DAHA) taking periodic tableaux, or for the rational Cherednik algebra (a.k.a. rational DAHA) with appropriate modifications. This construction of the basis makes use of an alternate presentation of the rational DAHA and the basis diagonalizes the action of its Dunkl-Opdam subalgebra. We make use of the combinatorics to construct explicit maps between standard modules parameterized by hooks, thus recovering the BGG resolution of the simple module parameterized by the trivial hook.

We can also describe this simple module using the geometry of parabolic Hilbert schemes of points on plane curve singularities. The “tableau” basis that diagonalizes the Dunkl-Opdam subalgebra is the basis of equivariant homology that comes from torus fixed points.

This is joint work with Eugene Gorsky and José Simental.

Bio: Monica Vazirani is a professor at UC Davis. She received her PhD from UC Berkeley, after which she had an NSF postdoc she spent at UC San Diego and UC Berkeley, as well as postdoctoral positions at MSRI and Caltech. Dr. Vazirani’s research interests center on the representation theory of algebras related to the symmetric group and how to express algebraic phenomena via the combinatorics of partitions, tableaux, crystal graphs and parking functions.
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Abstract: Suppose we want to solve a given task (say, a decision problem) and have a randomised algorithm for it which is correct; but only with some non-trivial probability, for instance .51. We would like to “amplify” this probability of success to an arbitrarily small amount, as close to 1 as possible: how to do this? And, more importantly, how to do this using as little extra randomness as possible?

I will first discuss why one would want to do this, then how to achieve it naively, and — quite surprisingly — how we can do much better than this naive approach using expander graphs.
Bio: Clément Canonne is a Lecturer in the School of Computer Science of the University of Sydney; he obtained his Ph.D. in 2017 from Columbia University, before joining Stanford as a Motwani Postdoctoral Fellow, then IBM Research as a Goldstine Postdoctoral Fellow. His main areas of research are distribution testing (and, broadly speaking, property testing) and learning theory; focusing, in particular, on understanding the computational aspects of learning and statistical inference subject to various resource or information constraints.
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Abstract: Canards are solutions of singularly perturbed ODEs that organise the dynamics in phase and parameter space. In this talk, we explore two aspects of canard theory: their applications in the life sciences and their ability to generate new phenomena.
More specifically, we will use canard theory to analyse a canonical model of the electrical activity in a heart muscle cell. We demonstrate that pathological heart rhythms, called early afterdepolarisations, are canard-induced phenomena. We use this knowledge to explain the rich set of model behaviours, some of which have also been observed in experiments. Then, we explore a new class of canard-induced patterns in reaction-diffusion PDEs which exhibit coexisting domains of mutually synchronised oscillators and complementary domains of decoherent (asynchronous) oscillators.
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Abstract: We prove that for each finite index subgroup H of the mapping class group of a closed hyperbolic surface, and for each real number r>1 there does not exist a faithful C^r-action (in Hölder’s sense) of H on a circle. For this, we partially determine the optimal regularity of faithful actions by right-angled Artin groups on a circle. (Joint with Thomas Koberda and Cristobal Rivas)
Bio: Sang-hyun Kim works at Korea Institute for Advanced Study as Professor in the School of Mathematics since 2019. Before this, he worked at Seoul National University, KAIST, Tufts University, the University of Texas at Austin and MSRI. He received Ph.D in 2007 at Yale University under the supervision of Andrew Casson. His research interests focus on the interplay between geometric group theory and low–dimensional topology, particularly motivated by right-angled Artin groups and manifold diffeomorphism groups. He was selected as the Scientist of the Month by the Korean Ministry of Science and ICT in 2020.
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Abstract: In recent years, the field of optimal transport has attracted the attention of many high-profile mathematicians with a wide range of applications.  In this talk we will discuss some of its recent applications in financial mathematics, particularly on the problems of model calibration, robust finance and portfolio optimisation.  Classical topological duality results are extended to probabilistic settings, connecting stochastic control problems with non-linear partial differential equations and providing interesting practical interpretations in finance.  We will also look at how numerical methods, including machine learning algorithms, can be implemented to solve these problems.
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Abstract: In spite of their enormous success as black box function approximators in many fields such as computer vision, natural language processing and automated decision making, Deep Neural Networks often fall short of providing interpretable models of data. In applications where aiding human understanding is the main goal, describing regularities in data with compact formuli promises improved interpretability and better generalization. In this talk I will introduce the resulting problem of Symbolic Regression and its generalization to Program Induction, highlight some learning methods from the literature and discuss challenges and limitations of searching for algorithmic descriptions of data.
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Abstract: In this talk, we introduce some deep learning based approaches for modelling sequence to sequence relationships that are gaining popularity in many applied fields, such as time-series analysis, natural language processing, and data-driven science and engineering. We will also discuss some interesting mathematical issues underlying these methodologies, including approximation theory and optimization dynamics.

Deep Learning demo on GitHub
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Abstract: Pure model-based approaches are today often insufficient for solving complex inverse problems in medical imaging. At the same time, methods based on artificial intelligence, in particular, deep neural networks, are extremely successful, often quickly leading to state-of-the-art algorithms. However, pure deep learning approaches often neglect known and valuable information from the modeling world and suffer from a lack of interpretability.

In this talk, we will develop a conceptual approach towards inverse problems in imaging sciences by combining the model-based method of sparse regularization by shearlets with the data-driven method of deep learning. Our solvers pay particular attention to the singularity structures of the data. Focussing then on the inverse problem of (limited-angle) computed tomography, we will show that our algorithms significantly outperform previous methodologies, including methods entirely based on deep learning. Finally, we will also touch upon the issue of how to interpret the results of such algorithms, and present a novel, state-of-the-art explainability method based on information theory.
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Abstract: Studying the set of exact solutions of a system of polynomial equations largely depends on a single iterative algorithm, known as Buchberger’s algorithm. Optimized versions of this algorithm are crucial for many computer algebra systems (e.g. Mathematica, Maple, Sage). After discussing the problem and what makes it challenging, I will discuss a new approach to Buchberger’s algorithm that uses reinforcement learning agents to perform S-pair selection, a key step in the algorithm. In certain domains, the trained model outperforms state-of-the-art selection heuristics in total number of polynomial additions performed, which provides a proof-of-concept that recent developments in machine learning have the potential to improve performance of algorithms in symbolic computation.
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Abstract: In 2016 AlphaGo defeated the world champion go player Lee Sedol in a historic 5 game match. In this lecture we will discuss the research behind this system and the innovations that ultimately lead to AlphaZero, which can learn to play multiple board games, including Go, from scratch without human knowledge.
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Abstract: We’ll start by looking at the structure of classification proofs for finite semigroups and how to program these in Pytorch. (That could be the subject of the tutorial.) A proof by cuts generates a proof tree—think of solving Sudoku. Its size depends on the choice of cut locations at each stage. This leads to the question of how to choose the cuts in an optimal way. We’ll discuss the Value-Policy approach to RL for this, and discuss some of the difficulties notably in sampling. Then we’ll look at another approach, somewhat more heuristic, that aims to provide a faster learning process with the goal of obtaining an overall gain in time when the training plus the proof are counted together.
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Abstract: Semi-supervised learning (SSL) is the problem of extending label information from a small subset of a data set to the entire set. In low-label regimes the geometry of the unlabelled set is a crucial aspect that should be leveraged in order to obtain algorithms that outperform standard supervised learning. In this talk I will introduce graphical SSL algorithms that rely on manifold regularization in order to incorporate this geometric information. I will discuss interesting connections to linear algebra and matrix perturbations, kernel methods, and theory of elliptic partial differential equations.

Abstract: In this talk we will leverage a reinforcement learning method, specifically the cross-entropy method, to search for counterexamples to several conjectures in graph theory and combinatorics. We will present a very simplistic setup, in which only minimal changes need to be made (namely the reward function used for RL) in order to successfully attack a wide variety of problems. As a result we will resolve several open problems, and find more elegant counterexamples to previously disproved ones.
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Geometric Deep Learning II (Georg Gottwald): YouTube video
Saliency + Combinatorial Invariance (Geordie Williamson): YouTube video

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Regularisation (Georg Gottwald): YouTube video
Recurrent Neural Nets (Joel Gibson): YouTube video

Description: Borrowing from statistical mechanics, dynamical systems and numerical analysis to better understand deep learning.
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Description: Universal approximation theorem and convolutional neural networks.
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Description: Classic problems in machine learning, kernel methods, deep neural networks, supervised learning, and basic examples.
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Abstract: The talk will be on the algebraic structure present in both parts of the title. This algebraic story is most pronounced for E2-algebras in the category of 2-vector spaces (also known as braided fusion categories). Condensation gives rise to an equivalence relation on such E2-algebras (Witt equivalence) with sets of equivalence classes exhibiting nice properties, e.g. being abelian groups (Witt groups). The Witt group of braided fusion categories is a countably generated abelian group with the torsion part annihilated by 32.
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Abstract: This is joint work in progress with Shuji Saito. Many cohomology theories of interest (l-adic cohomology, de Rham cohomology, motivic cohomology, K-theory…) have long exact sequences associated to blowups. Such a property can be neatly encoded in a Grothendieck topology such as the cdh-topology or the h-topology. These topologies appeared in Voevodsky’s proof of the Bloch-Kato conjecture, and more recently in Beilinson’s simple proof of Fontaine’s CdR conjecture, and in Bhatt and Scholze’s work on projectivity of the affine Grassmanian.

A feature of these topologies which often turns out to be a bug is that the associated sheaves cannot see nilpotents. In this talk I will discuss a modification which can see nilpotents, and which still has long exact sequences for many blowups.

Bio: Shane Kelly is an Associate Professor at Tokyo Institute of Technology. His research area is algebraic K-theory and motivic homotopy theory, and more recently he is interested in applications to representation theory. His graduate studies were mostly based in Paris; in 2012 he received a PhD jointly from Université Sorbonne Paris Nord and The Australian National University under the joint supervision of Cisinski and Neeman, respectively.
Seminar notes (PDF)

Abstract: On fundamental groups, the discriminant

∏_i≠k(z_i – z_k) ∈ ^×

of a finite collection of points of the plane defines the abelianization homomorphism
sending a braid to its number of overcrossings less undercrossings or writhe.
In terms of diffeomorphisms of the punctured plane, it defnes a kind of
`invertible topological quantum field theory’ associated to the Burau representation,
and in the classical physics of point particles the real part of
its logarithmic derivative is the potential energy of a collection of Coulomb
charges, while its imaginary part is essentially the Nambu-Goto area of a
loop of string in the three-sphere.
Its higher homotopy theory defines a very interesting a double-loop map

× Ω²S³ → 𝒫ic(S⁰)

to the category of lines over the stable homotopy ring-spectrum, related
to Hopkins and Mahowald’s exotic (E2) multiplication on classical integral
homology, perhaps related to the `anyons’ of nonclassical physics.

Bio: Jack Johnson Morava, of Czech and Appalachian descent, studied under Eldon Dyer and Sir Michael Atiyah, graduating with a PhD from Rice University in 1968, followed by an Academy of Sciences postdoc in Moscow with Yuri Manin and Sergei Novikov. He joined the Johns Hopkins faculty in 1979 where he was involved in the Japan-US mathematical institute, and from roughly 2003 to 2010 he worked half-time on the DARPA FunBio initiative. He retired in 2017 to live with his anthropological linguist wife in Charlottesville, Virginia and get some work done.
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Abstract: In this lecture I will explain a relationship between incidence theorems in elementary
geometry and the theory of integrable systems, both classical and quantum. We will
study geometric consistency relations between angles of 3-dimensional (3D) circular
quadrilateral lattices — lattices whose faces are planar quadrilaterals inscribable
into a circle. We show that these relations generate canonical transformations of a
remarkable “ultra-local” Poisson bracket algebra defined on discrete 2D surfaces
consisting of circular quadrilaterals. Quantization of this structure allowed us to
obtain new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter
equation) as well as reproduce all those that were previously known. These solutions
generate an infinite number of non-trivial solutions of the Yang-Baxter equation and
also define integrable 3D models of statistical mechanics and quantum field theory. The
latter can be thought of as describing quantum fluctuations of lattice geometry.
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Abstract: Under the geometric Satake equivalence, slices in the affine Grassmannian give a geometric incarnation of dominant weight spaces in representations of reductive groups. These affine Grassmannian slices are quantized by algebras known as truncated shifted Yangians. From this perspective, we expect to categorify these weight spaces using category O for these truncated shifted Yangians.

The slices in the affine Grassmannian and truncated shifted Yangians can also be defined as special cases of the Coulomb branch construction of Braverman-Finkelberg-Nakajima. From this perspective, we find many insights. First, we can generalize affine Grassmannian slices to the case of non-dominant weights and arbitrary symmetric Kac-Moody Lie algebras. Second, we establish a link with modules for KLRW algebras. Finally, we defined a categorical g-action on the categories O, using Hamiltonian reduction.

Bio: Joel Kamnitzer is a Professor of Mathematics at the University of Toronto. His research concerns complex reductive groups and their representations, focusing on canonical bases, categorification, and geometric constructions. His 2005 Ph.D. thesis from UC Berkeley focused on the study of Mirkovic-Vilonen cycles in Affine Grassmannians. He received the 2011 Andre Aisenstadt Prize, a 2012 Sloan Research Fellowship, a 2018 E.W.R. Steacie Memorial Fellowship, a 2018 Poincare Chair, and the 2021 Jeffrey-Williams Prize.
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Abstract: Group theory is littered with undecidable problems. A classic example is the word problem: there are groups for which there exists no algorithm that can decide if a product of generators represents the trivial element or not. Many problems (the word problem included) are at least semidecidable, meaning that there is a correct algorithm guaranteed to terminate if the answer is “yes”, but with no guarantee on how long one has to wait. I will discuss strategies to try and tackle various semidecidable problems computationally with the key example being the discovery of a counterexample to the Kaplansky unit conjecture.
Bio: Giles Gardam is a research associate at the University of Münster working in geometric group theory. He studied mathematics and computer science at the University of Sydney, receiving his Bachelor’s degree in 2012, and completed his doctorate at Oxford in 2017. He was then a postdoc at the Technion before starting at Münster in 2019.
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Abstract: The cohomology ring H^*(BG) (with coefficients in a field k of characteristic p)
is a very special graded commutative ring, but this comes out much more clearly if one uses
the cochains C^*(BG), which can be viewed as a commutative ring up to homotopy. For
example C^*(BG) is always Gorenstein (whilst this is not quite true for H^*(BG)).

This leads one to study C^*(BG) as if it was a commutative local Noetherian ring, though of
course one has to use homotopy invariant methods. The ring C^*(BG) is regular
if and only if G is p-nilpoent and so it is natural to look for ways of deciding where C^*(BG) lies
on a the spectrum between regular and Gorenstein rings. For a commutative Noetherian ring,
one considers the singularity category D_{sg}(R) (the quotient of finite complexes of finitely
generated modules by finitely generated projectives). This is trivial if and only if R is regular,
so is the appropriate tool. The talk will describe how to define this for C^*(BG), show it has
good basic properties and describe the singularity category in the simplest case it is not
trivial (when G has a cyclic Sylow p-subgroup).
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Course information: Professor Vladimir Bazhanov will give a short course of lectures on Yang-Baxter maps.
The topic lies on the intersection of the theory of quantum groups and discrete integrable equations.
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Abstract: A Coxeter system is a presentation of a group by generators and a specific form of relations, namely the braid relations and the reflection relations. The Coxeter presentation leads to, among others, a similar presentation of the (Iwahori-)Hecke algebras and the Kazhdan-Lusztig theory, which provides combinatorial answers to major problems in Lie theoretic representation theory and geometry. Extending such applications to the `singular land’ requires the singular version of the Hecke algebra. Underlying combinatorics of the singular Hecke algebra/category comes from the parabolic double cosets and is the first step in understanding the singular Hecke category. In this talk, I will present a Coxeter theory of the parabolic double cosets developed in a joint work with Ben Elias. In particular, I will explain a generalization of the reduced expressions and describe the braid and non-braid relations.
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Abstract: The totally asymmetric simple exclusion process (TASEP) was introduced around 1970 as a model for translation in protein synthesis and traffic flow.
It has interesting physical properties (e.g. boundary-induced phase transitions) and also beautiful mathematical properties. The inhomogeneous TASEP is a Markov chain of weighted particles hopping on a ring, in which the probability that two particles interchange depends on the weight of those particles. If each particle has a distinct weight, then we can think of this as a Markov chain on permutations. In many cases, the steady state probabilities can be expressed in terms of Schubert polynomials. Based on joint work with Donghyun Kim.
Speaker bio: Lauren Williams is the Robinson professor of mathematics at Harvard and the Seaver Professor at the Harvard Radcliffe Institute. Her research is in algebraic combinatorics. Williams received her BA in mathematics from Harvard College in 2000, and her PhD from MIT in 2005. Subsequently, she was a postdoc at UC Berkeley and Harvard, then a faculty member at UC Berkeley from 2009 to 2018, before returning to Harvard in 2018. She is the recipient of a Sloan Research Fellowship, an NSF CAREER award, the AWM-Microsoft research prize, and is an Honorary member of the London Mathematical Society.
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Abstract: Let G(ℂ) be a complex reductive group and W be its Weyl group. In 1966, Tits introduced a certain subgroup of G(ℂ), which is an extension of W by an elementary abelian 2-group. This group is called the Tits group and provides a nice lifting of W.
In this talk, I will discuss a generalization of the notion of the Tits group T to a reductive p-adic group G. Such T, if exists, gives a nice lifting of the Iwahori-Weyl group of G. I will show that the Tits group exists when the reductive group splits over an unramified extension of the p-adic field and will provide an example in the ramified case that such a Tits group does not exist. Finally, as an application, we will provide a nice presentation of the Hecke algebra of the p-adic group G with ln-level structure.
This talk is based on the recent joint work with Ganapathy.
Speaker bio: Xuhua He is the Choh-Ming Professor of Mathematics at the Chinese University of Hong Kong. He works in pure mathematics. His research interests include Arithmetic geometry, Algebraic groups, and representation theory. He received his Bachelor’s degree in mathematics from Peking University in 2001 and a Ph.D. degree from MIT in 2005 under the supervision of George Lusztig. He worked as a member at the Institute for Advanced Study during 2005-2006 and Simons Instructor at Stony Brook University during 2006-2008. He worked at the Hong Kong University of Science and Technology during 2008-2014 as an assistant Professor and associated Professor, and then moved to the University of Maryland during 2014-2019 as a Full Professor of Mathematics before joining CUHK in 2019. He received the Morningside Gold Medal of Mathematics in 2013, the Xplorer Prize in 2020 and is an invited sectional speaker of the International Congress of Mathematicians in 2018.
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Abstract: This is a joint work with Samuel Jeralds. Let 𝔤 be an affine Kac-Moody Lie algebra and let λ, µ be two dominant integral weights for 𝔤. We prove that under some mild restriction, for any positive root β, V(λ) ⊗ V(µ) contains V(λ + µ – β) as a component, where V(λ) denotes the integrable highest weight (irreducible) 𝔤-module with highest weight λ. This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V(λ) ⊗ V(µ). Then, we prove the corresponding geometric results including the higher cohomology vanishing on the 𝒢-Schubert varieties in the product partial flag variety 𝒢/𝒫 × 𝒢/𝒫 with coefficients in certain sheaves coming from the ideal sheaves of 𝒢-sub Schubert varieties. This allows us to prove the surjectivity of the Gaussian map.
Seminar notes: PDF document
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Abstract: Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4 – the symplectically primitive but complex imprimitive groups – and 10 exceptional groups up to dimension 10, for which it is still open. Recently, we have proven that for all but possibly 39 cases in the remaining infinite series there is no symplectic resolution. We have thereby reduced the classification problem to finitely many open cases. We do not expect any of the remaining cases to admit a symplectic resolution. This is joint work with Gwyn Bellamy and Johannes Schmitt.

Abstract: We describe a K-theoretic version of the equivariant constructible derived category. We state (with evidence!) a ‘geometric Satake’ conjecture relating its value on the affine Grassmannian to representations of the Langlands dual group.
About the speaker: Gus Lonergan is the Chief Mathematician at A Priori Investment Management LLC. He was previously a L.E. Dickson Instructor in the mathematics department at the University of Chicago. He is interested in representation theory.
Lonergan completed his PhD at MIT under Roman Bezrukavnikov; the thesis was an attempt to understand mod p phenomena in algebraic topology in the context of geometric representation theory. He attended Cambridge University for his undergraduate and master’s degree. He plays a little music on the side.
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Abstract: We introduce the notion of a symmetric quiver as provided by Derksen and Weyman in 2002 and discuss symmetric degenerations in this context (which correspond to orbit closure relations in the symmetric representation variety). After motivating our particular interest in the latter by presenting connections to group actions in algebraic Lie Theory, we explain our main questions: are symmetric degenerations induced by “usual” degenerations in the representation variety of the underlying quiver? We look at (counter)examples and recent results. This is joint work with Giovanni Cerulli Irelli.
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Abstract: In the 1920s, building on Fermat’s Last Theorem, Mordell conjectured that the set of rational points of any smooth projective curve of genus at least two, over any number field, is finite. In the 1960s, Shafarevich turned this into a purely algebro-geometric conjecture involving families of smooth projective curves. Parshin, Arakelov and Faltings settled this conjecture by showing that the base spaces of such families are in some sense hyperbolic, as long as there is some variation in the algebraic structure of the fibers. Inspired by recent advances in birational geometry, Kebekus and Kovacs conjectured that these hyperbolicity type properties should hold for a vast class of projective families, with fibers of arbitrary dimension. In this talk I will discuss this conjecture and my solution to it. I will also talk about more recent progress in this area, based on a joint work with Kovacs (University of Washington).
Seminar slides: PDF document

Abstract: In the first part of the talk, I will review Grobner degenerations of matrix Schubert varieties, following Knutson et al and others. I will interpret this in terms of quantum integrable systems and discuss how this construction is not entirely satisfactory and needs to be generalized. Then I will provide such a generalization; we’ll work with the “lower-upper scheme”; one component of which is closely related to the commuting variety. We’ll discuss applications to the latter.

Abstract: A group is said to have polynomial representation growth if the sequence enumerating the isomorphism classes of finite dimensional irreducible representations according to their dimension is polynomially bounded. The representation zeta function of such group is the associated Dirichlet generating series. In this talk I will focus on representation zeta functions of arithmetic groups and their properties. I will explain the ideas behind a proof of a variant of the Larsen-Lubotzky conjecture on the representation growth of arithmetic lattices in high rank semisimple Lie groups (joint with Nir Avni, Benjamin Klopsch and Christopher Voll) and analogous results for arithmetic groups of type A_2 in positive characteristic (joint with Amritanshu Prasad and Pooja Singla).
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Abstract: This talk will be in two parts.
I will outline joint work with Daryl Cooper concerning the space of holonomies of properly convex real projective structures on manifolds whose fundamental group satisfies a few natural properties. This generalises previous work by Benoist for closed manifolds. A key example, computed with Joan Porti, is used to illustrate the main results.
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Abstract: This talk will be in two parts.
The first part will be introductory, and will address the question:
Given an ordinary differential equation (ODE) with certain physical/geometric properties (for example a preserved measure, first and/or second integrals), how can one preserve these properties under discretization?
The second part of the talk will cover some more recent work, and address the question:
How can one deduce hard to find properties of an ODE from its discretization?
Bio: Reinout Quispel was an undergraduate at the University of Utrecht (the Netherlands) for nine years, before obtaining a PhD on the discretization of soliton theory from Leiden University in 1983. He moved to Australia for a three-year position in 1986 and is still there 35 years later. His main areas of expertise are in integrable systems and in the geometric numerical integration of differential equations. He was awarded the Onsager Professorship and Medal by the Norwegian University of Science and Technology (NTNU) in 2013.
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Abstract: In the first half of the talk we shall introduce the notion of Lie superalgebras, and then give a quick outline of the classification of finite-dimensional complex simple Lie superalgebras.
In the second part of the talk we shall discuss the representation theory of these Lie superalgebras and explain the irreducible character problem in the BGG category. Our main focus will be on our computation of the irreducible characters for two of the exceptional Lie superalgebras. This part is based on recent joint works with C.-W. Chen, L. Li, and W. Wang.
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Abstract: Welded tangles are knotted surfaces in R^4. Bar-Natan and Dancso described a class of welded tangles which have “foamed vertices” where one allows surfaces to merge and split. The resulting welded tangled foams carry an algebraic structure, similar to the planar algebras of Jones, called a circuit algebra. In joint work with Dancso and Halacheva we provide a one-to-one correspondence between circuit algebras and a form of rigid tensor category called “wheeled props.” This is a higher dimensional version of the well-known algebraic classification of planar algebras as certain pivotal categories.
This classification allows us to connect these “welded tangled foams” to the Kashiwara-Vergne conjecture in Lie theory. In work in progress, we show that the group of homotopy automorphisms of the (rational completion of) the wheeled prop of welded foams is isomorphic to the group of symmetries KV, which acts on the solutions to the Kashiwara-Vergne conjecture. Moreover, we explain how this approach illuminates the close relationship between the group KV and the pro-unipotent Grothendieck–Teichmueller group.
Bio: Marcy Robertson obtained her PhD in Algebraic Topology from the University of Illinois at Chicago in 2010. From there she worked in Canada, France and her native United States before settling down in Australia 2015. She is now a Senior Lecturer of Pure Mathematics at the University of Melbourne.
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Abstract: The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe a stationary moving wave patterns consisting of two plane solitary waves moving at an angle to each other. The results obtained within the approximate asymptotic theory is validated by comparison with the exact two-soliton solution of the Kadomtsev-Petviashvili equation.
The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin-Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.
Bio: Yury Stepanyants graduated in 1973 with the HD of MSc Diploma from the Gorky State University (Russia) and started to work as the Engineer with the Research Radiophysical Institute in Gorky. He proceeded his career with the Institute of Applied Physics of the Russian Academy of Sciences (Nizhny Novgorod) from 1977 to 1997. In 1983 Yury obtained a PhD in Physical Oceanography, and in 1992 he obtained a degree of Doctor of Sciences in Geophysics. After immigration in Australia in 1998, Yury worked for 12 years as the Senior Research Scientist with the Australian Nuclear Science and Technology Organisation in Sydney. Since July 2009 he holds a position of Full Professor at the University of Southern Queensland in Toowoomba, Australia. Yury has published more than 100 journal papers, three books, several review papers and has obtained three patents.
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Piggott Abstract: A program of research, started in the 1980s, seeks to classify the groups that can be presented by various classes of length-reducing rewriting systems. We discuss the resolution of one part of the program (joint work with Andy Eisenberg (Temple University), and recent related work with Murray Elder (UTS).
Elder Abstract: The growth function of a finitely generated group is a powerful and well-studied invariant. Gromov’s celebrated theorem states that a group has a polynomial growth function if and only if the group is ‘virtually nilpotent’. Of interest is a variant called the ‘geodesic growth function’ which counts the number of minimal-length words in a group with respect to some finite generating set. I will explain progress made towards an analogue of Gromov’s theorem in this case.
I will start by defining all of the terms used in this abstract (finitely generated group; growth function; virtual property of a group; nilpotent) and then give some details of the recent progress made. The talk is based on the papers arxiv.org/abs/1009.5051, arxiv.org/abs/1908.07294 and arxiv.org/abs/2007.06834 by myself, Alex Bishop, Martin Brisdon, José Burillo and Zoran Šunić.
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Abstract: Traditional Indigenous marriage rules have been studied extensively since the mid 1800s. Despite this, they have historically been cast aside as having very little utility. Here, I will walk through some of the interesting mathematics of the Gamilaraay system and show that, instead, they are in fact a very clever construction.
Indeed, the Gamilaraay system dynamically trades off kin avoidance to minimise incidence of recessive diseases against pairwise cooperation, as understood formally through Hamilton’s rule.
Bio:Jared Field completed his undergraduate studies at the University of Sydney in Mathematics and French literature, before reading for a DPhil in Mathematical Biology at Balliol College, Oxford. He is now a McKenzie Fellow in the School of Mathematics and Statistics at the University of Melbourne, with broad interests at the intersection of mathematics, evolution and ecology.

Abstract: The theory of complex representations of p-adic groups can feel very technical and unwelcoming, but its central role in the conjectural local Langlands correspondence has pushed us to pursue its understanding.
In this talk, I will aim to share the spirit of, and open questions in, the representation theory of G, through the lens of restricting these representations to maximal compact open subgroups.
Our point of departure: the Bruhat-Tits building of G, a 50-year-old combinatorial and geometric object that continues to reveal secrets about the structure and representation theory of G today.
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Abstract: In 1983, Gerd Faltings proved the Mordell conjecture stating that curves of
genus at least two have only finitely many rational points. This can be understood as
the statement that most polynomial equations (in a precise sense)

f(x,y)=0

of degree at least 4 have at most finitely many solutions. However, the effective
version of this problem, that of constructing an algorithm for listing all rational
solutions, is still unresolved. To get a sense of the difficulty, recall how long it
took to prove that there are no solutions to

x^n+y^n=1

other than the obvious ones. In this talk, I will survey some of the recent progress on
an approach to this problem that proceeds by encoding rational solutions into arithmetic
principal bundles and studying their moduli in a manner reminiscent of geometric gauge
theory.
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Abstract: Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs.
Each k-graph can be described in terms of a coloured graph, called its skeleton, and some factorisation rules that describe how 2-coloured paths pair up into commuting squares. C*-algebras of k-graphs generalise Cuntz-Krieger algebras, and have been the subject of sustained interest essentially because questions about crossed products of C*-algebras by higher-rank free abelian groups are hard, and k-graph algebras constitute a comparably tractable class of examples that could point the way to general theorems.
A particularly obstinate question in this vein is that of determining the K-theory of a k-graph algebra, or even just whether the K-theory depends on the factorisation rules, or only on the skeleton. I’ll outline some joint work with James Fletcher and Elizabeth Gillaspy that uses a homotopy argument to establish a surprising link between this question and the question of connectedness (or otherwise) of the space of solutions to a Yang-Baxter-like equation. I won’t assume any background about C*-algebras, k-graphs, or the Yang-Baxter equations, and all are welcome—and people who might know about connectedness (or otherwise) of the spaces of solutions to Yang-Baxter-like equations are especially welcome!

Abstract: A group G acting faithfully by homeomorphisms of the Cantor set is called piecewise full if any homeomorphism assembled piecewise from elements of G is itself an element of G.
They first appeared in the work of Giordano, Putnam and Skau in the context of Cantor minimal systems. Recently they have received significant attention as a source of new examples of finitely generated infinite simple groups. I will present a number of results about these groups obtained in joint work with Alejandra Garrido and Colin Reid.

Abstract: The aim of these two talks is to give an overview of the geometric aka stacky approach to various cohomology theories for schemes: de Rham, Hodge, crystalline and prismatic (due to Simpson and later Drinfel’d). The basic observation is that interesting cohomology theories for schemes can be realised as the (humble) coherent cohomology an associated stack. Interesting aspects of the cohomology theories e.g. comparison theorems, theories of coefficients, perfectness etc can then be naturally expressed and proven in terms of the geometry of the associated stacks.
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Abstract: We explain an equivalence of categories between a module category of quiver Hecke algebras associated with the Kronecker quiver and a category of equivariant perverse coherent sheaves on the nilpotent cone of type A. This provides a link between two different monoidal categorifications of the open quantum unipotent cell of affine type A₁, one given by Kang–Kashiwara–Kim–Oh–Park in terms of quiver Hecke algebras, the other given by Cautis–Williams in terms of equivariant perverse coherent sheaves on affine Grassmannians. The first part of the talk will be devoted to introduction to quiver Hecke algebras and categorification of quantum cluster algebras. The main result will be explained in the second part. This is a joint work with Michela Varagnolo and Eric Vasserot.
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Abstract: Autoequivalences of triangulated categories are an interesting and understudied class of groups. In large part due to the development of the theory of Bridgeland stability conditions, there are suggestive parallels between these groups and mapping class groups of surfaces. The goal of this talk will be to explain how some of the geometric group theory which appears in the study of mapping class groups also arises in the study of triangulated autoequivalences.

Abstract: There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio: Trace(ρ(g)) / dim(ρ), for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.
Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank.
Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms”, which is (since the 60s) the main organization principle, and is based on the (huge collection) of “Large” representations.
This talk will discuss the notion of rank for the group GLₙ over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.
This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried with Steve Goldstein (Madison) and John Cannon (Sydney).

Abstract: This is joint work with Justin Hilburn. We will explain a theorem showing that D-modules on the Tate vector space of Laurent series are equivalent to ind-coherent sheaves on the space of rank 1 de Rham local systems on the punctured disc equipped with a flat section. Time permitting, we will also describe an application of this result in the global setting. Our results may be understood as a geometric refinement of Tate’s ideas in the setting of harmonic analysis. They also may be understood as a proof of a strong form of the 3d mirror symmetry conjectures in a special case.
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Abstract: In this talk I will state a conjecture which predicts that the derived category of smooth representations of a p-adic split reductive group admits a fully faithful embedding into the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We will make the conjecture precise in the case of the principal block of GLn and relate it to the construction of a family of representations on the stack of L-parameters that interpolates a modified version of the local Langlands correspondence. The existence of this family is suggested by the work of Helm and Emerton–Helm. I will explain why the derived tensor product with this “Emerton–Helm family” should realize the expected embedding of derived categories and discuss some explicit examples.
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Abstract: This talk is based on joint work with Benson and Etingof. We say that a symmetric tensor category is incompressible if there is no symmetric tensor functor from this category to a smaller tensor category. Our main result is a construction of new examples of incompressible tensor categories in positive characteristic.
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Abstract: I will discuss some of the structures suggested by the physics of supersymmetric gauge theory of relevance to the geometric Langlands program. The discussion will include defects of various dimensions in field theory, with an emphasis on the role of boundary conditions and how they lead to the suggestion of a duality between hamiltonian actions of Langlands dual groups.
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Abstract: These lectures will cover some basic results about derived categories of coherent sheaves (e.g. the structure of the derived category of a curve, Fourier-Mukai transforms and how to construct them, tilting bundles, auto-equivalence groups, perhaps spaces of stability conditions). I will assume that the audience is vaguely familiar with the definition of derived and triangulated categories, but I will spend a fair amount of time in the first few lectures trying to give some intuitive feel for these general constructions, and explaining how one makes calculations in practice. I will also need to assume some familiarity with basic algebraic geometry and sheaf theory.

Abstract: Statistical science has a 200-year history of advances in theory and application. Data science is a relatively newly defined area of enquiry developing from the explosion in the ubiquitous collection of data. The interplay between these fields, and their interactions with science, are a topic of lively discussion among statisticians. This talk will overview some of the current research in statistical science that is motivated by new developments in data science.
Bio: Nancy Reid is University Professor and Canada Research Chair in Statistical Theory and Applications at the University of Toronto. Her research interests include statistical theory, likelihood inference, design of studies, and statistical science in public policy. Her main research contributions have been to the field of theoretical statistics. Professor Reid is a Fellow of the Royal Society, the Royal Society of Canada, the American Association for the Advancement of Science, and a Foreign Associate of the National Academy of Sciences. In 2014 she was appointed Officer of the Order of Canada.