Past

If $n$ is congruent to 0 or 4 modulo 6, there are infinitely many primes of the form $x^2 + ny^2$ with both $x$ and $y$ prime. (Joint work with Mehtaab Sawhney, Columbia)

The backward error analysis is an important part of the perturbation theory and it is particularly useful for the study of the reliability of the numerical methods. We focus on the backward error for nonlinear eigenvalue problems. In this talk, the matrix-valued function is given as a linear combination of scalar functions multiplying matrix coefficients, and the perturbation is done on the coefficients. We provide theoretical results about the backward error of a set of approximate eigenpairs. Indeed, small backward errors for separate eigenpairs do not imply small backward errors for a set of approximate eigenpairs. In this talk, we provide inexpensive upper bounds, and a way to accurately compute the backward error by means of direct computations or through Riemannian optimization. We also discuss how the backward error can be determined when the matrix coefficients of the matrix-valued function have particular structures (such as symmetry, sparsity, or low-rank), and the perturbations are required to preserve them. For special cases (such as for symmetric coefficients), explicit and inexpensive formulas to compute the perturbed matrix coefficients are also given. This is a joint work with Leonardo Robol (University of Pisa).

Please attend if you would like to give a talk in the Logic Advanced Class this term.

How much can the order type - the list of element orders (with multiplicities)—reveal about the structure of a finite group G? Can it tell us whether G is abelian, nilpotent? Can it always determine whether G is solvable? 

This last question was posed in 1987 by John G. Thompson and I answered it negatively this year. The search for a counterexample was quite a puzzle hunt! It involved turning the problem into linear algebra and solving an integer matrix equation Ax=b. This would be easy if not for the fact that the size of A was 100,000 by 10,000…

I will explain the recent techniques developed with co-authors to obtain fine estimates about the large values of the Riemann zeta functions on the critical line. An emphasis will be put on the ideas originating from statistical mechanics and large deviations that may be of general interest for a stochastic analysis audience. No number theory knowledge will be assumed!

I will present joint work with Alessandro Fazzari in which we prove precise conditional estimates for the third (non-absolute) moment of the logarithm of the Riemann zeta function, beyond the Selberg central limit theorem, both for the real and imaginary part. These estimates match predictions made in work of Keating and Snaith. We require the Riemann Hypothesis, a conjecture for the triple correlation of Riemann zeros and another ``twisted'' pair correlation conjecture which captures the interaction of a prime power with Montgomery's pair correlation function. This conjecture can be proved on a certain subrange unconditionally, and on a larger range under the assumption of a variant of the Hardy-Littlewood conjecture with good uniformity.

In this talk, based on a joint work with Ilijas Farah, I will present an application of an old continuous selection theorem due to Michael to the study of II1 factors. More precisely, I'll show that if two strongly continuous paths (or loops) of projections (p_t), (q_t), for t in [0,1], in a II1 factor are such that every p_t is subequivalent to q_t, then the subequivalence can be realized by a strongly continuous path (or loop) of partial isometries. I will then use an extension of this result to solve affirmatively the so-called trace problem for factorial W*-bundles whose base space is 1-dimensional.

I will describe some recent developments in random walks on Gromov-hyperbolic spaces. I will focus in particular on the notions of Schottky sets and pivoting technique introduced respectively by Boulanger-Mathieu-S-Sisto and Gouëzel and mention some consequences. The talk will be introductory; I will not assume specialized knowledge in probability theory.

The problem of finding an explicit description of the centre of the restricted universal enveloping algebra of sl2 for a general prime characteristic p is still open. We use a computational approach to find a basis for the centre for small p. Building on this, we used a special central element t to construct a complete set of (p+1)/2 orthogonal primitive idempotents e_i, which decompose Z into one 1-dimensional and (p-1)/2 3-dimensional subspaces e_i Z. These allow us to compute e_i N as subspaces of the e_i Z, where N is the largest nilpotent ideal of Z. Looking forward, the results perhaps suggest N is a free k[T] / (T^{(p-1)/2}-1)-module of rank 2.

We show that an $n$-vertex $k$-uniform hypergraph, where all $(k-1)$-subsets that are supported by an edge are in fact supported by at least $n/2+o(n)$ edges, contains a spanning $(k-1)$-dimensional sphere. This generalises Dirac's theorem, and confirms a conjecture of Georgakopoulos, Haslegrave, Montgomery, and Narayanan. Unlike typical results in the area, our proof does not rely on the absorption method or the regularity lemma. Instead, we use a recently introduced framework that is based on covering the vertex set of the host hypergraph with a family of complete blow-ups.

This is joint work with Freddie Illingworth, Richard Lang, Olaf Parczyk, and Amedeo Sgueglia.

A well-known open problem for representations of symmetric groups is the Saxl conjecture. In this talk, we put Saxl's conjecture into a Lie-theoretical framework and present a natural generalisation to Weyl groups. After giving necessary preliminaries on spin representations and the Springer correspondence, we present our progress on the generalised conjecture. Next, we reveal connections to tensor product decomposition problems in symmetric groups and provide an alternative description of Lusztig’s cuspidal families. Finally, we propose a further generalisation to all finite Coxeter groups.

Come along for free Pizza and to hear about the Mathematrix events this term. 

Three-dimensional QFTs with 8 supercharges (N=4 supersymmetry) are a rich playground rife with connections to mathematics. For example, they admit two topological twists and furnish a three-dimensional analogue of the famous mirror symmetry of two-dimensional N=(2,2) QFTs, creatively called 3d mirror symmetry, that exchanges these twists. Recently, there has been increased interest in so-called rank 0 theories that typically do not admit Lagrangian descriptions with manifest N=4 supersymmetry, but their topological twists are expected to realize finite, semisimple TQFTs which are amenable to familiar descriptions in terms of, e.g., modular tensor categories and/or rational vertex operator algebras. In this talk, based off of joint work (arXiv:2406.00138) with Thomas Creutzig and Heeyeon Kim, I will introduce two families of rank 0 theories exchanged by 3d mirror symmetry and various mathematical conjectures stemming from our analysis thereof.

When studying interacting particle systems, two distinct categories emerge: indistinguishable systems, where particle identity does not influence system dynamics, and non-exchangeable systems, where particle identity plays a significant role. One way to conceptualize these second systems is to see them as particle systems on weighted graphs. In this talk, we focus on the latter category. Recent developments in graph theory have raised renewed interest in understanding largepopulation limits in these systems. Two main approaches have emerged: graph limits and mean-field limits. While mean-field limits were traditionally introduced for indistinguishable particles, they have been extended to the case of non-exchangeable particles recently. In this presentation, we introduce several models, mainly from the field of opinion dynamics, for which rigorous convergence results as N tends to infinity have been obtained. We also clarify the connection between the graph limit approach and the mean-field limit one. The works discussed draw from several papers, some co-authored with Nastassia Pouradier Duteil and David Poyato.

We give a gentle introduction to the theory of self-similar sets and self-similar measures. Connections of this topic to Diophantine approximation on Lie groups as well as to additive combinatorics will be exposed. In particular, we will discuss recent progress on Bernoulli convolutions. If time permits, we mention recent joint work with Samuel Kittle on absolutely continuous self-similar measures. 
 

We develop a model based on mean-field games of competitive firms producing similar goods according to a standard AK model with a depreciation rate of capital generating pollution as a byproduct. Our analysis focuses on the widely-used cap-and-trade pollution regulation. Under this regulation, firms have the flexibility to respond by implementing pollution abatement, reducing output, and participating in emission trading, while a regulator dynamically allocates emission allowances to each firm. The resulting mean-field game is of linear quadratic type and equivalent to a mean-field type control problem, i.e., it is a potential game. We find explicit solutions to this problem through the solutions to differential equations of Riccati type. Further, we investigate the carbon emission equilibrium price that satisfies the market clearing condition and find a specific form of FBSDE of McKean-Vlasov type with common noise. The solution to this equation provides an approximate equilibrium price. Additionally, we demonstrate that the degree of competition is vital in determining the economic consequences of pollution regulation.

This is based on joint work with Gianmarco Del Sarto and Marta Leocata. 

https://arxiv.org/pdf/2407.12754

In his final paper in 1954, Alan Turing wrote `No systematic method is yet known by which one can tell whether two knots are the same.' Within the next 20 years, Wolfgang Haken and Geoffrey Hemion had discovered such a method. However, the computational complexity of this problem remains unknown. In my talk, I will give a survey on this area, that draws on the work of many low-dimensional topologists and geometers. Unfortunately, the current upper bounds on the computational complexity of the knot equivalence problem remain quite poor. However, there are some recent results indicating that, perhaps, knots are more tractable than they first seem. Specifically, I will explain a theorem that provides, for each knot type K, a polynomial p_K with the property that any two diagrams of K with n_1 and n_2 crossings differ by at most p_K(n_1) + p_K(n_2) Reidemeister moves.

Bryant’s Laplacian flow is an analogue of Ricci flow that seeks to flow an arbitrary initial closed $G_2$-structure on a 7-manifold toward a torsion-free one, to obtain a Ricci-flat metric with holonomy $G_2$. This talk will give an overview of joint work with Mark Haskins and Rowan Juneman about complete self-similar solutions on the anti-self-dual bundles of ${\mathbb CP}^2$ and $S^4$, with cohomogeneity one actions by SU(3) and Sp(2) respectively. We exhibit examples of all three classes of soliton (steady, expander and shrinker) that are asymptotically conical. In the steady case these form a 1-parameter family, with a complete soliton with exponential volume growth at the boundary of the family. All complete Sp(2)-invariant expanders are asymptotically conical, but in the SU(3)-invariant case there appears to be a boundary of complete expanders with doubly exponential volume growth.

Can we efficiently find a local minimum of a nonconvex continuous optimization problem? 

We give a rather complete answer to this question for optimization problems defined by polynomial data. In the unconstrained case, the answer remains positive for polynomials of degree up to three: We show that while the seemingly easier task of finding a critical point of a cubic polynomial is NP-hard, the complexity of finding a local minimum of a cubic polynomial is equivalent to the complexity of semidefinite programming. In the constrained case, we prove that unless P=NP, there cannot be a polynomial-​time algorithm that finds a point within Euclidean distance $c^n$ (for any constant $c\geq 0$) of a local minimum of an $n$-​variate quadratic polynomial over a polytope. 
This result (with $c=0$) answers a question of Pardalos and Vavasis that appeared on a list of seven open problems in complexity theory for numerical optimization in 1992.

Based on joint work with Jeffrey Zhang (Yale).

Biography

One recent(ish) development in classical black hole thermodynamics is the inclusion of vacuum energy (cosmological constant) in the form of thermodynamic pressure. New thermodynamic phase transitions emerge in this extended phase space, beyond the usual Hawking—Page transition. This allows us to understand black holes from the viewpoint of chemistry in terms of concepts such as Van Der Waals fluids, reentrant phase transitions and triple points. I will review these developments and discuss the dictionary between the bulk laws and those of the dual CFT.
 

In this informal talk I will review some theoretical and practical aspects related to the finite element approximation of eigenvalue problems arising from PDEs.
The review will cover elliptic eigenvalue problems and eigenvalue problems in mixed form, with particular emphasis on the Maxwell eigenvalue problem.
Other topics can be discussed depending on the interests of the audience, including adaptive schemes, approximation of parametric problems, reduced order models.
 

In the first half of the talk, I will briefly survey the theory of matroids with coefficients, which was introduced by Andreas Dress and Walter Wenzel in the 1980s and refined by the speaker and Nathan Bowler in 2016. This theory provides a unification of vector subspaces, matroids, valuated matroids, and oriented matroids. Then, in the second half, I will outline an intriguing connection between Lorentzian polynomials, as defined by Petter Brändén and June Huh, and matroids with coefficients.  The second part of the talk represents joint work with June Huh, Mario Kummer, and Oliver Lorscheid.

We define a version of algebraic K-theory for bornological algebras, using the recently developed continuous K-theory by Efimov. In the commutative setting, we prove that this invariant satisfies descent for various topologies that arise in analytic geometry, generalising the results of Thomason-Trobaugh for schemes. Finally, we prove a version of the Grothendieck-Riemann-Roch Theorem for analytic spaces. Joint work with Jack Kelly and Federico Bambozzi. 

We discuss the trace problem and stable rank for tracialy complete C*-algebras

In the category of operator systems, identification comes via complete order isomorphisms, and so an operator system can be identified with a C*-algebra without itself being an algebra. So, when is an operator system a C*-algebra? This question has floated around the community for some time. From Choi and Effros, we know that injectivity is sufficient, but certainly not necessary outside of the finite-dimensional setting. In this talk, I will give a characterization in the separable nuclear setting coming from C*-encoding systems. This comes from joint work with Galke, van Lujik, and Stottmeister.

The "curse of dimensionality" refers to the host of difficulties that occur when we attempt to extend our intuition about what happens in low dimensions (i.e. when there are only a few features or variables)  to very high dimensions (when there are hundreds or thousands of features, such as in genomics or imaging).  With very high-dimensional data, there is often an intuition that although the data is nominally very high dimensional, it is typically concentrated around a much lower dimensional, although non-linear set. There are many approaches to identifying and representing these subsets.  We will discuss topological approaches, which represent non-linear sets with graphs and simplicial complexes, and permit the "measuring of the shape of the data" as a tool for identifying useful lower dimensional representations.

Since Gromov's celebrated polynomial growth theorem, the understanding of nilpotent groups has become a cornerstone of geometric group theory. An interesting aspect is the conjectural quasiisometry classification of nilpotent groups. One important quasiisometry invariant that plays a significant role in the pursuit of classifying these groups is the Dehn function, which quantifies the solvability of the world problem of a finitely presented group. Notably, Gersten, Holt, and Riley's work established that the Dehn function of a nilpotent group of class $c$ is bounded above by $n^{c+1}$.  

In this talk, I will explain recent results that allow us to compute Dehn functions for extensive families of nilpotent groups arising as central products. Consequently, we obtain a large collection of pairs of nilpotent groups with bilipschitz equivalent asymptotic cones but with different Dehn functions.

This talk is based on joint work with Claudio Llosa Isenrich and Gabriel Pallier.

Since Gromov's celebrated polynomial growth theorem, the understanding of nilpotent groups has become a cornerstone of geometric group theory. An interesting aspect is the conjectural quasiisometry classification of nilpotent groups. One important quasiisometry invariant that plays a significant role in the pursuit of classifying these groups is the Dehn function, which quantifies the solvability of the world problem of a finitely presented group. Notably, Gersten, Holt, and Riley's work established that the Dehn function of a nilpotent group of class $c$ is bounded above by $n^{c+1}$.  

In this talk, I will explain recent results that allow us to compute Dehn functions for extensive families of nilpotent groups arising as central products. Consequently, we obtain a large collection of pairs of nilpotent groups with bilipschitz equivalent asymptotic cones but with different Dehn functions.

This talk is based on joint work with Claudio Llosa Isenrich and Gabriel Pallier.

The Cuntz semigroup of a C*-algebra is an ordered monoid consisting of equivalence classes of positive elements in the stabilization of the algebra.  It can be thought of as a generalization of the Murray-von Neumann semigroup, and records substantial information about the structure of the algebra.  Here we examine the set of positive elements having a fixed equivalence class in the Cuntz semigroup of a simple, separable, exact and Z-stable C*-algebra and show that this set is path connected when the class is non-compact, i.e., does not correspond to the class of a projection in the C*-algebra.  This generalizes a known result from the setting of real rank zero C*-algebras.

The classification of simple, unital, nuclear UCT C*-algebras with finite nuclear dimension can be achieved using an invariant derived from K-theory and tracial information. In this talk, I will present a classification theorem for certain classes of C*-algebras that rely solely on tracial deformations.  

 Structural properties of C*-Algebras such as Stable Rank One, Real Rank Zero, and radius of comparison have played an important role in classification.  Crossed product C*-Algebras are useful examples to study because knowledge of the base Algebra can be leveraged to determine properties of the crossed product.  In this talk we will discuss the permanence of various structural properties when taking crossed products of several types.  Crossed products considered will include the usual C* crossed product by a group action along with generalizations such as crossed products by a partial automorphism.  

This talk is based on joint work with Julian Buck and N. Christopher Phillips and on joint work with Maria Stella Adamo, Marzieh Forough, Magdalena Georgescu, Ja A Jeong, Karen Strung, and Maria Grazia Viola.

In recent work, Aganagic proposed a categorification of quantum link invariants based on a category of A-branes. The theory is a generalization of Heegaard–Floer theory from gl(1|1) to arbitrary Lie algebras. It turns out that this theory is solvable explicitly and can be used to compute homological link invariants associated to any minuscule representation of a simple Lie algebra. This invariant coincides with Khovanov–Rozansky homology for type A and gives a new invariant for other types. In this talk, I will introduce the relevant category of A-branes, explain the explicit algorithm used to compute the link invariants, and give a sketch of the proof of invariance. This talk is based on 2305.13480 with Mina Aganagic and Miroslav Rapcak and work in progress with Mina Aganagic and Ivan Danilenko.

The Zappa–Szép (ZS) product of two groupoids is a generalization of the semi-direct product: instead of encoding one groupoid action by homomorphisms, the ZS product groupoid encodes two (non-homomorphic, but “compatible”) actions of the groupoids on each other. I will show how to construct the ZS product of two twists over such groupoids and give an example using Weyl twists from Cartan pairs arising from Kumjian--Renault theory.

 Based on joint work with Boyu Li, New Mexico State University

The set $M_n(\mathbb{R})$ of all $n \times n$ matrices over the real numbers is an example of an algebraic structure called a $C^*$-algebra. The subalgebra $D$ of diagonal matrices has special properties and is called a \emph{Cartan subalgebra} of $M_n(\mathbb{R})$. Given an arbitrary $C^*$-algebra, it can be very hard (but also very rewarding) to find a Cartan subalgebra, if one exists at all. However, if the $C^*$-algebra is generated by a cocycle $c$ and a group (or groupoid) $G$, then it is natural to look within $G$ for a subgroup (or subgroupoid) $S$ that may give rise to a Cartan subalgebra. In this talk, we identify sufficient conditions on $S$ and $c$ so that the subalgebra generated by $(S,c)$ is indeed a Cartan subalgebra of the $C^*$-algebra generated by $(G,c)$. We then apply our theorem to $C^*$-algebras generated by $k$-graphs, which are directed graphs in higher dimensions. This is joint work with J. Briones Torres, A. Duwenig, L. Gallagher, E. Gillaspy, S. Reznikoff, H. Vu, and S. Wright.

A self-similar k-graph is a pair consisting of a (discrete countable) group and a k-graph, such that the group acts on the k-graph self-similarly. For such a pair, one can associate it with a universal C*-algebra, called the self-similar k-graph C*-algebra. This class of C*-algebras embraces many important and interesting C*-algebras,  such as the higher rank graph C*-algebras of Kumjian-Pask, the Katsura algebras,  the Nekrashevych algebras constructed from self-similar groups, and the Exel-Pardo algebra. 

In this talk, we will survey some results on self-similar k-graph C*-algebras. 

Dimensions of mapping class groups of orientable and non-orientable surfaces

1:30pm

Luis Jorge Sánchez Saldaña (UNAM)

Mapping class groups have been studied extensively for several decades. Still in these days these groups keep being studied from several point of views. In this talk I will talk about several notions of dimension that have been computed (and some that are not yet known) for mapping class groups of both orientable and non-orientable manifolds. Among the dimensions that I will mention are the virtual cohomological dimension, the proper geometric dimension, the virtually cyclic dimension and the virtually abelian dimension. Some of the results presented are in collaboration with several colleagues: Trujillo-Negrete, Hidber, León Álvarez and Jimaénez Rolland.

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Diffeomorphisms of reducible 3-manifolds

2:45pm

Rachael Boyd (Glasgow)

I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space B Diff(M), for M a compact, connected, reducible 3-manifold. We prove that when M is orientable and has non-empty boundary, B Diff(M rel ∂M) has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where M is irreducible by Hatcher and McCullough.

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Nonunique ergodicity in strata of geodesic laminations and the boundary of Outer space

4:00pm

Mladen Bestvina (Utah)

It follows from the work of Gabai and Lenzhen-Masur that the maximal number of projectively distinct ergodic transverse measures on a filling geodesic lamination on a hyperbolic surface is equal to the number of curves in a pants decomposition. In a joint work with Jon Chaika and Sebastian Hensel, we answer the analogous question when the lamination is restricted to have specified polygons as complementary components. If there is enough time, I will also talk about the joint work with Elizabeth Field and Sanghoon Kwak where we consider the question of the maximal number of projectively distinct ergodic length functions on a given arational tree on the boundary of Culler-Vogtmann's Outer space of a free group.
 

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

In this talk, I will describe some of the ways in which mathematical modelling contributed to the Covid-19 pandemic response in New Zealand. New Zealand adopted an elimination strategy at the beginning of the pandemic and used a combination of public health measures and border restrictions to keep incidence of Covid-19 low until high vaccination rates were achieved. The low or zero prevalence for first 18 months of the pandemic called for a different set of modelling tools compared to high-prevalence settings. It also generated some unique data that can give valuable insights into epidemiological characteristics and dynamics. As well as describing some of the modelling approaches used, I will reflect on the value modelling can add to decision making and some of the challenges and opportunities in working with stakeholders in government and public health.        

The goal of Post-Quantum Cryptography (PQC) is to design cryptosystems which are secure against classical and quantum adversaries. A topic of fundamental research for decades, the status of PQC drastically changed with the NIST PQC standardization process. Recently there have been AI attacks on some of the proposed systems to PQC. In this talk, we will give an overview of the progress of quantum computing and how it will affect the security landscape. 

Group-based cryptography is a relatively new family in post-quantum cryptography, with high potential. I will give a general survey of the status of post-quantum group-based cryptography and present some recent results.

In the second part of my talk, I speak about Post-quantum hash functions using special linear groups with implication to post-quantum blockchain technologies.

The goal of Post-Quantum Cryptography (PQC) is to design cryptosystems which are secure against classical and quantum adversaries. A topic of fundamental research for decades, the status of PQC drastically changed with the NIST PQC standardization process. Recently there have been AI attacks on some of the proposed systems to PQC. In this talk, we will give an overview of the progress of quantum computing and how it will affect the security landscape. 

Group-based cryptography is a relatively new family in post-quantum cryptography, with high potential. I will give a general survey of the status of post-quantum group-based cryptography and present some recent results.

In the second part of my talk, I speak about Post-quantum hash functions using special linear groups with implication to post-quantum blockchain technologies.