Past

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.