Past

Wreath-like products are a new class of groups, which are close relatives of the classical wreath products. Examples of wreath-like product groups arise from every non-elementary hyperbolic groups by taking suitable quotients. As a consequence, unlike classical wreath products, many wreath-like products have Kazhdan's property (T). 

I will present several rigidity results for von Neumann algebras of wreath-like product groups. We show that any group G in a natural family of wreath-like products with property (T) is W*-superrigid: the group von Neumann algebra L(G) remembers the isomorphism class of G. This provides the first examples of W*-superrigid groups with property (T). For a wider class wreath-like products with property (T), we show that any isomorphism of their group von Neumann algebras arises from an isomorphism of the groups. As an application, we prove that any countable group can be realized as the outer automorphism group of L(G), for an icc property (T) group G. These results are joint with Ionut Chifan, Denis Osin and Bin Sun.  

Time permitting, I will mention an additional application of wreath-like products obtained in joint work with Ionut Chifan and Daniel Drimbe, and showing that any separable II_1 factor is contained in one with property (T). This provides an operator algebraic counterpart of the group theoretic fact that every countable group is contained in one with property (T).

COURSE_PROPOSAL (12).pdf

The space of unordered tuples. The notion of differentiability and the theory of metric Sobolev in the context of multi-valued functions. Multivalued maximum principle and Holder regularity. Estimates on the Hausdorff dimension of the singular set of Dir-minimizing functions. If time permits: mass minimizing currents and their link with Dir-minimizers. 

To a group theorist ribbon groups look like knot groups, except  that we know everything about knot groups and next to nothing about ribbon groups.

I will talk about an old paper of mine with Peter Teichner where several questions on ribbon groups naturally arise.

"Colouring" a tournament means partitioning its vertex set into acylic subsets; and the "domination number" is the size of the smallest set of vertices with no common in-neighbour. In some ways these are like the corresponding concepts for graphs, but in some ways they are very different. We give a survey of some recent results and open questions on these topics.

Joint with Tung Nguyen and Alex Scott.

The Satake isomorphism is a fundamental result in p-adic groups, and the affine Grassmannian is the natural setting where this geometrizes to the Geometric Satake Correspondence. In fact, it suffices to work with affine Grassmannian slices, which retain all of the information.

Recently, Braverman, Finkelberg, and Nakajima showed that affine Grassmannian slices arise as Coulomb branches of certain quiver gauge theories. Remarkably, their construction works in Kac-Moody type as well. Their work opens the door to studying affine Grassmannians and Geometric Satake Correspondence for Kac-Moody groups. Unfortunately, it is difficult at present to do any explicit geometry with the Coulomb branch definition. For example, a basic feature is that affine Grassmannian slices embed into one another. However, this is not apparent from the Coulomb branch definition. In this talk, I will explain why these embeddings are necessarily subtle. Nonetheless, I will show a way to construct the embeddings using fundamental monopole operators.

This is joint work with Alex Weekes.

Supersymmetric localization allows one to reduce the computation of the partition function of a supersymmetric theory to a finite-dimensional integral, but the space over which one integrates is often singular. In this talk I will explain how one can use shifted symplectic geometry to get rigorous definitions of partition functions and state spaces in theories with extended supersymmetry. For instance, this gives a field-theoretic origin of DT invariants of CY4 manifolds. This is a report on joint work with Brian Williams.

Diffusion Limited Aggregation is a very simple mathematical model which describes a wide range of natural phenomena. Despite its simplicity, there is very little progress in understanding its large-scale structure. Since its introduction by Witten and Sander over 40 years ago, there was only one mathematical result. In 1987 Kesten obtained an upper bound on the growth rate. In this talk I will discuss DLA and some related models and the recent progress in understanding DLA. In particular, a new simpler proof of Kesten result which generalizes to other aggregation models.

Quasiconvexity is the fundamental existence condition for variational problems, yet it is poorly understood. Two outstanding problems remain: 

• 1) does rank-one convexity, a simple necessary condition, imply quasiconvexity in two dimensions? 

• 2) can one prove existence theorems for quasiconvex energies in the context of nonlinear Elasticity? 

In this talk we show that both problems have a positive answer in a special class of isotropic energies. Our proof combines complex analysis with the theory of gradient Young measures. On the way to the main result, we establish quasiconvexity inequalities for the Burkholder function which yield, in particular, many sharp higher integrability results. 
The talk is based on joint work with Kari Astala, Daniel Faraco, Aleksis Koski and Jan Kristensen.

I will construct the differential geometric, gauge-theoretic, and duality covariant model of classical four-dimensional Abelian gauge theory on an orientable four-manifold of arbitrary topology. I will do so by implementing the Dirac-Schwinger-Zwanziger (DSZ) integrality condition in classical Abelian gauge theories with general duality structure and interpreting the associated sheaf cohomology groups geometrically. As a result, I will obtain that four-dimensional Abelian gauge theories are theories of connections on Siegel bundles, namely principal bundles whose structure group is the generically non-abelian disconnected group of automorphisms of an integral affine symplectic torus. This differential-geometric model includes the electric and magnetic gauge potentials on an equal footing and describes the equations of motion through a first-order polarized self-duality condition for the curvature of a connection. This condition is reminiscent of the theory of four-dimensional Euclidean instantons, even though we consider a two-derivative theory in Lorentzian signature. Finally, I will elaborate on various applications of this differential-geometric model, including a mathematically rigorous description of electromagnetic duality in Abelian gauge theory and the reduction of the polarized self-duality condition to a Riemannian three-manifold, which gives as a result a new type of Bogomolny equation.

The affinity between statistical physics and machine learning has a long history. Theoretical physics often proceeds in terms of solvable synthetic models; I will describe the related line of work on solvable models of simple feed-forward neural networks. I will then discuss how this approach allows us to analyze uncertainty quantification in neural networks, a topic that gained urgency in the dawn of widely deployed artificial intelligence. I will conclude with what I perceive as important specific open questions in the field.

 

In this lecture, I will discuss the resolution of the Arthur-Barbasch-Vogan conjecture on the unitarity of special unipotent representations for any real form of a connected reductive complex Lie group, with contributions by several groups of authors (Barbasch-Ma-Sun-Zhu, Adams-Arancibia-Mezo, and Adams-Miller-van Leeuwen-Vogan). The main part of the lecture will be on the approach of the first group of authors for the case of real classical groups: counting by coherent families (combinatorial aspect), construction by theta lifting (analytic aspect), and distinguishing by invariants (algebraic-geometric aspect), resulting in a full classification, and with unitarity as a direct consequence of the construction.

Multi-parameter persistence is a main research topic in topological data analysis. Major questions involve the computation and the structural properties
of persistence modules. In this context, I will sketch two very recent results:

(1) We define a natural bifiltration called the localized union-of-balls bifiltration that contains filtrations studied in the context of local persistent homology as slices. This bifiltration is not k-critical for any finite k. Still, we show that a representation of it (involving algebraic curves of low degree) can be computed exactly and efficiently. This is joint work with Matthias Soels (TU Graz).

(2) Every persistence modules permits a unique decomposition into indecomposable summands. Intervals are the simplest type of summands, but more complicated indecomposables can appear, and usually do appear in examples. We prove that for homology-dimension 0 and density-Rips bifiltration, at least a quarter of the indecomposables are intervals in expectation for a rather general class of point samples. Moreover, these intervals can be ``peeled off'' the module efficiently. This is joint work with Angel Alonso (TU Graz).

Critical levels appear as singularities of waves propagating in shear flows. When magnetic field exists, critical levels are located where the phase velocity of the wave relative to the basic flow matches the velocity of Alfvén waves. Critical levels are known for locally strong wave amplitude in its vicinity, known as the critical layers. In this talk, I will demonstrate the situation where magnetic critical layers can contribute to the instability of the MHD flow.  We consider two different flow configurations. One is the shallow water flow, and the other is the 2D flow on a sphere. Asymptotic analysis has been used to explore deeper insights of the instability mechanism.

Data-driven modeling and deep learning present powerful tools that are opening up new paradigms and opportunities in the understanding, discovery, and design of soft and biological materials. I will describe our recent applications of deep representational learning to expose the sequence-function relationship within homologous protein families and to use these principles for the data-driven design and experimental testing of synthetic proteins with elevated function. I will then describe an approach based on latent space simulators to learn ultra-fast surrogate models of protein folding and biomolecular assembly by stacking three specialized deep learning networks to (i) encode a molecular system into a slow latent space, (ii) propagate dynamics in this latent space, and (iii) generatively decode a synthetic molecular trajectory.

Given a group G and a field k, we can "twist" the multiplication of the group algebra kG by a 2-cocycle, and the result is a twisted group algebra. Twisted group algebras arise as direct sums of blocks of group algebras, and so are of interest in representation and block theory. In this talk we will discuss some recently obtained results on the first Hochschild cohomology of twisted group algebras, in particular that these cohomology groups are nontrivial whenever G is a finite simple group.

The usual language of algebraic geometry is not appropriate for Arithmetical geometry: addition is singular at the real prime. We developed two languages that overcome this problem: one replace rings by the collection of “vectors” or by bi-operads and another based on “matrices” or props. These are the two languages of [Har17], but we omit the involutions which brings considerable simplifications. Once one understands the delicate commutativity condition one can proceed following Grothendieck footsteps exactly. The square matrices, when viewed up to conjugation, give us new commutative rings with Frobenius endomorphisms.

The closest thing we have to a general method for finding primes in sets is to use sieve methods to turn the problem into some other (hopefully easier) arithmetic questions about the set.

Unfortunately this process is still poorly understood - we don’t know ‘how much’ arithmetic information is sufficient to guarantee the existence of primes, and how much is not sufficient. Often arguments are rather ad-hoc.

I’ll talk about work-in-progress with Kevin Ford which shows that many of our common techniques are not optimal and can be refined, and in many cases these new refinements are provably optimal.

Stochastic portfolio theory is a framework to study large equity markets over long time horizons. In such settings investors are often confined to trading in an “open market” setup consisting of only assets with high capitalizations. In this work we relax previously studied notions of open markets and develop a tractable framework for them under mild structural conditions on the market.

Within this framework we also introduce a large parametric class of processes, which we call rank Jacobi processes. They produce a stable capital distribution curve consistent with empirical observations. Moreover, there are explicit expressions for the growth-optimal portfolio, and they are also shown to serve as worst-case models for a robust asymptotic growth problem under model ambiguity.

Time permitting, I will also present an extended class of models and illustrate calibration results to CRSP Equity Data.

This talk is based on joint work with Martin Larsson.

The pioneering work of Murray and von Neumann shows that any countable discrete group G gives rise in a canonical way to a group von Neumann algebra, denoted L(G). A main theme in operator algebras is to classify group von Neumann algebras, and hence, to understand how much information does L(G) remember of the underlying group G. In the amenable case, the classification problem is completed by the work of Connes from 1970s asserting that for all infinite conjugacy classes amenable groups, their von Neumann algebras are isomorphic.

In sharp contrast, in the non-amenable case, Popa's deformation rigidity/theory (2001) has led to the discovery of several instances when various properties of the group G are remembered by L(G). The goal of this talk is to survey some recent progress in this direction.

Lightweight compliant surfaces are commonly used as roofs (awnings), filtration systems or propulsive appendages, that operate in a fluid environment. Their flexibility allows for shape to change in fluid flows, to better endure harsh or fluctuating conditions, or enhance flight performance of insect wings for example. The way the structure deforms is however key to fulfill its function, prompting the need for control levers. In this talk, we will consider two ways to tailor the deformation of surfaces in a flow, making use of the properties of origami (folded sheet) and kirigami (sheet with a network of cuts). Previous literature showed that the substructure of folds or cuts allows for sophisticated shape morphing, and produces tunable mechanical properties. We will discuss how those original features impact the way the structure interacts with a flow, through combined experiments and theory. We will notably show that a sheet with a symmetric cutting pattern can produce an asymmetric deformation, and study the underlying fluid-structure couplings to further program shape morphing through the cuts arrangement. We will also show that extreme shape reconfiguration through origami folding can cap fluid drag.

We will give an introduction to hyperbolic cusps and their Dehn fillings. In particular, we will give a brief survey of quantitive results in the field. To motivate this work, we will sketch how these techniques are used for studying the classical question of characteristic slopes on knots.

The notion of amenable actions by discrete groups on C*-algebras has been introduced by Claire Amantharaman-Delaroche more than thirty years ago, and has become a well understood theory with many applications. So it is somewhat surprising that an established theory of amenable actions by general locally compact groups has been missed until 2020. We now present a theory which extends the discrete case and unifies several notions of approximation properties of actions which have been discussed in the literature. We also present far reaching results towards the weak containment problem which asks wether an action $\alpha:G\to \Aut(A)$ is amenable if and only if the maximal and reduced crossed products coincide.

In this lecture we report on joint work with Alcides Buss and Rufus Willett.

Given a group G, one can seek to understand (some of) its subgroups. Centralisers of elements are easy to define, but maybe not so easy to understand: even in such well studied groups as Out(Fn) they are not yet understood in general. I'll discuss recent work with Armando Martino where we extend what is known in Out(Fn), involving a (surprising?) connection to free-by-cyclic groups and their automorphisms as well as working with actions on trees. The strategies seem like they should apply in many more cases, and if time allows I'll discuss ongoing work (with Gilbert Levitt and Armando Martino) exploring these possibilities.

In this talk, we introduce Newton-MR variants for solving nonconvex optimization problems. Unlike the overwhelming majority of Newton-type methods, which rely on conjugate gradient method as the primary workhorse for their respective sub-problems, Newton-MR employs minimum residual (MINRES) method. With certain useful monotonicity properties of MINRES as well as its inherent ability to detect non-positive curvature directions as soon as they arise, we show that our algorithms come with desirable properties including the optimal first and second-order worst-case complexities. Numerical examples demonstrate the performance of our proposed algorithms.

We determine the minimal forbidden minors characterising the class of countable graphs that embed into some compact surface. We will also discuss Thomas’s conjecture that the Robertson—Seymour Graph Minor Theorem extends to countably infinite graphs. [https://arxiv.org/abs/2301.11042]

What do football passes and financial transactions have in common? Both are observable events in some real-world walk process that is happening over some network that is, however, not directly observable. In both cases, the basis for record-keeping is that these events move something tangible from one node to another. Here we explore process-driven approaches towards analyzing such data, with the goal of answering domain-specific research questions. First, we consider transaction data from a digital community currency recorded over 16 months. Because these are records of a real-world walk process, we know that the time-aggregated network is a flow network. Flow-based network analysis techniques let us concisely describe where and among whom this community currency was circulating. Second, we use a technique called trajectory extraction to transform football match event data into passing sequence data. This allows us to replicate classic results from sports science about possessions and uncover intriguing dynamics of play in five first-tier domestic leagues in Europe during the 2017-18 club season. Taken together, these two applied examples demonstrate the interpretability of process-driven approaches as opposed to, e.g., temporal network analysis, when the data are records of a real-world walk processes.

The plethysm product on symmetric functions corresponds to composition of polynomial representations of general linear groups. Decomposing a plethysm product into Schur functions, or equivalently, writing the corresponding composition of Schur functors as a direct sum of Schur functors, is one of the main open problems in algebraic combinatorics. I will give an introduction to these mathematical objects emphasising the beautiful interplay between representation theory and combinatorics. I will end with new results obtained in joint work with Rowena Paget (University of Kent) on stability on plethysm coefficients. No specialist background knowledge will be assumed.

We propose a nematic model for polyatomic gas, intending to study anisotropic phenomena. Such phenomena stem from the orientational degree of freedom associated with the rod-like molecules composing the gas. We adopt as a primer the Curitss-Boltzmann equation. The main difference with respect to Curtiss theory of hard convex body fluids is the fact that the model here presented takes into account the emergence of a nematic ordering. We will also derive from a kinetic point of view an energy functional similar to the Oseen-Frank energy. The application of the Noll-Coleman procedure to derive new expressions for the stress tensor and the couple-stress tensor will lead to a model capable of taking into account anisotropic effects caused by the emergence of a nematic ordering. In the near future, we hope to adopt finite-element discretisations together with multi-scale methods to simulate the integro-differential equation arising from our model.

Holography, which reveals a specific correspondence between gravitational and quantum theories, is an ongoing area of research that has known a lot of interest these past decades. The duality of holography has many applications: it provides an interpretation for black hole entropy in terms of microstates, it helps our understanding of solid state properties such as superconductivity and strongly coupled quantum systems, and it even offers insight into the mysterious realm of quantum gravity. 

In this talk, I will first introduce the concept of holography and some of its applications. I will then discuss some notions of string theory and geometry that are commonly used in holography. Finally, if time permits, I will present some of our latest results, where we match the energy of membranes in supergravity to properties of the dual quantum models.

In his pioneering work in 1860, Riemann proposed the Riemann problem and solved it for isentropic gas in terms of shocks and rarefaction waves. It eventually became the foundation of the theory of one-dimension conservation laws developed in the 20th century. We prove the non-nonlinear structural stability of the Riemann problem for multi-dimensional isentropic Euler equations in the regime of rarefaction waves. This is a joint work with Tian-Wen Luo.

In 1993, Erdős, Sárközy and Sós posed the question of whether there exists a set $S$ of positive integers that is both a Sidon set and an asymptotic basis of order $3$. This means that the sums of two elements of $S$ are all distinct, while the sums of three elements of $S$ cover all sufficiently large integers. In this talk, I will present a construction of such a set, building on ideas of Ruzsa and Cilleruelo. The proof uses a powerful number-theoretic result of Sawin, which is established using cutting-edge algebraic geometry techniques.

We describe an approach to Bialynicki-Birula theory for holomorphic $\mathbb{C}^*$ actions on complex analytic spaces and Morse-Bott theory for Hamiltonian functions for the induced circle actions. A key principle is that positivity of a suitably defined "virtual Morse-Bott index" at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum even when it is a singular point in the moduli space. Inspired by Hitchin’s 1987 study of the moduli space of Higgs monopoles over Riemann surfaces, we apply our method in the context of the moduli space of non-Abelian monopoles or, equivalently, stable holomorphic pairs over a closed, complex, Kaehler surface. We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and show that these indices are positive in a setting motivated by a conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type (including symplectic four-manifolds) obey the Bogomolov-Miyaoka-Yau inequality.

In this talk I will discuss a novel mechanism  that couples matter fields to three-dimensional de Sitter quantum gravity. This construction is based on the Chern-Simons formulation of three-dimensional Euclidean gravity, and it centers on a collection of Wilson loops winding around Euclidean de Sitter space. We coin this object a Wilson spool.  To construct the spool, we build novel representations of su(2). To evaluate the spool, we adapt and exploit several known exact results in Chern-Simons theory. Our proposal correctly reproduces the one-loop determinant of a free massive scalar field on S^3 as G_N->0. Moreover, allowing for quantum metric fluctuations, it can be systematically evaluated to any order in perturbation theory.   

Join us for our first Fridays@4 session of Trinity about different academic routes people take post-PhD, with a particular focus on fellowships and grants. We’ll hear from Jason Lotay about his experiences on both sides of the application process, as well as hear about the experiences of ECRs in the South Wing, North Wing, and Statistics. Towards the end of the hour we’ll have a Q+A session with the whole panel, where you can ask any questions you have around this topic!

One-dimensional persistent homology encodes geometric information of data by means of a barcode decomposition. Often, one needs to relate the persistence barcodes of two datasets which are intrinsically linked, e.g. consider a sample from a large point cloud. Such connections are encoded through persistence morphisms; as in linear algebra, a (one-dimensional) persistence morphism is fully understood by fixing a basis in the domain and codomain and computing the associated matrix. However, in the literature and existing software, the focus is often restricted to interval decompositions of images, kernels and cokernels. This is the case of the Bauer-Lesnick matching, which is computed using the intervals from the image. Unfortunately, this matching has substantial differences from the structure of the persistence morphism in very simple examples. In this talk I will present an induced block function that is well-behaved in such examples. This block function is computed using the associated matrix of a persistence morphism and is additive with respect to taking direct sums of persistence morphisms. This is joint work with M. Soriano-Trigueros and R. Gonzalez-Díaz from Universidad de Sevilla.

Abstract: Neurotransmission at chemical synapses relies on the calcium-induced fusion of synaptic vesicles with the presynaptic membrane. The distance of the vesicle to the calcium channels determines the fusion probability and consequently the postsynaptic signal. After a fusion event, both the release site and the vesicle undergo a recovery process before becoming available for reuse again. For all these process components, stochastic effects are widely recognized to play an important role. In this talk, I will present our recent efforts on how to describe and structurally understand neurotransmission dynamics using stochastic modeling approaches. Starting with a linear reaction scheme, a method to directly compute the exact first- and second-order moments of the filtered output signal is proposed. For a modification of the model including explicit recovery steps, the stochastic dynamics are compared to the mean-field approximation in terms of reaction rate equations. Finally, we reflect on spatial extensions of the model, as well as on their approximation by hybrid methods.

References:

- A. Ernst, C. Schütte, S. Sigrist, S. Winkelmann. Mathematical Biosciences, 343, 108760, 2022.

- A. Ernst, N. Unger, C. Schütte, A. Walter, S. Winkelmann. Under Review. https://arxiv.org/abs/2302.01635

In this talk, I give a statement of the “Galois to automorphic” direction of categorical geometric Langlands. I will describe the Galois and automorphic side, the Hecke action on both sides, and the definition of Hecke eigensheaves. On the way, I hope to give motivation for the various objects at play : the stack of $G^L$ local systems on the fixed curve $X$, the stack of $G$ bundles on $X$, $D$-modules, arc groups, loop groups, the affine Grassmannian, and geometric Satake.

Affine logic is the fragment of continuous logic in which the connectives are limited to affine functions. I will discuss the basics of this logic, first studied by Bagheri, and present the results of a recent joint work with I. Ben Yaacov and T. Tsankov in which we initiate the study of extreme types and extremal models in affine logic.

In particular, I will discuss an extremal decomposition result for models of simplicial affine theories, which generalizes the ergodic decomposition theorem.

Hoheisel used zero-density results to prove that for all x large enough there is a prime number in the interval $[x−x^{\theta}, x]$ with $θ < 1$. The connection between zero-density estimates and primes in short intervals was explicitly described in the work of Ingham in 1937. The approach of Ingham combined with the zero-density estimates of Huxley (1972) provides us with the distribution of primes in $[x−x^{\theta}, x]$ with $\theta > 7/12$. Further improvement upon the value of \theta was achieved by combining sieves with the weighted zero-density estimates in the works of Iwaniec and Jutila, Heath-Brown and Iwaniec, and Baker and Harman. The last work provides the best result achieved using zero-density estimates. We will discuss the main ideas of the paper by Baker and Harman and simplify some parts of it to show a more explicit connection between zero-density results and the sieved sums, which are used in the paper. This connection will provide a better understanding on which parts should be optimised for further improvements and on what the limits of the methods are. This project is still in progress.

Correlation operators are used in data assimilation algorithms
to weight the contribution of prior and observation information.
Efficient implementation of these operators is therefore crucial for
operational implementations. Diffusion-based correlation operators are popular in ocean data assimilation, but can require a large number of serial matrix-vector products. An all-at-once formulation removes this requirement, and offers the opportunity to exploit modern computer architectures. High quality preconditioners for the all-at-once approach are well-known, but impossible to apply in practice for the
high-dimensional problems that occur in oceanography. In this talk we
consider a nested preconditioning approach which retains many of the
beneficial properties of the ideal analytic preconditioner while
remaining affordable in terms of memory and computational resource.

My group is developing a roadmap to replace bulky electric motors in miniature robots requiring large mechanical work output.

First, I will describe the mechanics of coiled muscles made by twisting nylon fishing lines, and how these actuators use internal strain energy to achieve a “record breaking” performance. Then I will describe intriguing hierarchical super-, and hyper-coiled artificial muscles which exploit the interplay between nonlinear mechanics and material microstructure. Next, I will describe their use to actuate the dynamic snapping of insect-scale jumping robots. The combination of strong but slow muscles with a fast-snapping beam gives rise to dynamic buckling cascade phenomena leading to effective robotic jumping mechanisms.

These examples shed light on the future of automation propelled by new bioinspired materials, nonlinear mechanics, and unusual manufacturing processes.

This talk is concerned with the question of generating the homology of a covering space by lifts of simple closed curves (from topological viewpoint), and generating the first homology of a subgroup by powers of elements outside certain filtrations (from group-theoretic viewpoint). I will sketch Malestein's and Putman's construction of examples of branched covers where lifts of scc's span a proper subspace. I will discuss the relation of their proof to the Magnus embedding, and present recent results on similar embeddings of surface groups which facilitate extending their theorems to unbranched covers.

With conference season fast approaching, we will be meeting up to discuss our experiences of going to conferences and how best to prepare for them as a minority in Mathematics.

Orthogonal Intertwiners for Infinite Particle Systems On The Continuum:

Interacting particle systems are studied using powerful tools, including 
duality. Recently, dualities have been explored for inclusion processes, 
exclusion processes, and independent random walkers on discrete sets 
using univariate orthogonal polynomials. This talk generalizes these 
dualities to intertwiners for particle systems on more general spaces, 
including the continuum. Instead of univariate orthogonal polynomials, 
the talk dives into the theory of infinite-dimensional polynomials 
related to chaos decompositions and multiple stochastic integrals. The 
new framework is applied to consistent particle systems containing a 
finite or infinite number of particles, including sticky and correlated 
Brownian motions.

Spectral gap of the symmetric inclusion process:

In this talk, we consider the symmetric inclusion process on a general finite graph. Our main result establishes universal upper and lower bounds for the spectral gap of this interacting particle system in terms of the spectral gap of the random walk on the same graph. In the regime in which the gamma-like reversible measures of the particle system are log-concave, our bounds match, yielding a version for the symmetric inclusion process of the celebrated Aldous' spectral gap conjecture --- originally formulated for the interchange process and proved by Caputo, Liggett and Richthammer (JAMS 2010). Finally, by means of duality techniques, we draw analogous conclusions for an interacting diffusion-like unbounded conservative spin system known as Brownian energy process, which may be interpreted as a spatial version of the Wright-Fisher diffusion with mutation. Based on a joint work with Seonwoo Kim (SNU, South Korea).

The resolvents of finite volume restricted Hamiltonians, G^(⍵), have long been used to describe the localization of quantum systems. More recently, projected Green's functions (pGfs) -- finite volume restrictions of the resolvent -- have been applied to translation invariant free fermion systems, and the pGf zero eigenvalues have been shown to determine topological edge modes in free-fermion systems with bulk-edge correspondence. In this talk, I will connect the pGfs to the G^(⍵) appearing in the transfer matrices of quasi-periodic systems and discuss what pGF zeros can tell us about the solutions to transfer matrix equations. Using these methods, we re-examine the critical almost-Matthieu operator and notice new guarantees on analytic regions of its resolvent for Liouville irrationals.