Vadim Alekseev, Stability for groups and quotients modulo Kazhdan subgroups
We provide several instances in which interesting approximation properties (e.g. RFD) and stability properties related to soficity (e.g. variants of P-stability) are inherited by quotients with respect to finitely generated normal subgroups or, more strongly, normal subgroups with Kazhdan's property (T). When these observations are combined with variations of the Rips construction due to Wise and Belegradek--Osin, this provides the existence of non-RFD hyperbolic groups, hyperbolic groups which are not flexibly P-stable, and other interesting phenomena related to soficity of groups and actions. This is joint work with Andreas Thom.
Jananan Arulseelan, Connections to Continuous Model Theory and Computable Structures
The computable model theory of operator algebras is a topic that arose from the ideas in MIP*=RE and has since developed quickly into an active area of its own. I will describe some of the short history of this subject, explain its main concepts, and discuss various applications to operator algebras. Finally, I will discuss some open problems, and advertise a new potential approach to the Kirchberg Embedding Problem.
Ionuţ Chifan, Rigidity results for operator algebras associated with infinite direct sum groups
In this talk we study rigidity phenomena for the group von Neumann algebra L(G) and the reduced C*-algebra Cr*(G) when G=⊕n Gn is an infinite direct sum of groups.
Using new techniques developed at the interface of geometric group theory and the theory of von Neumann algebras, we construct the first examples of ICC groups Gn for which the direct sum G=⊕n Gn exhibits a new rigidity phenomenon that we term McDuff W*-superrigidity. Specifically, for such groups G, an arbitrary group H satisfies L(H)=L(G) if and only if H=G x A for some ICC amenable group A. The groups Gn arise as property (T) wreath-like product extensions whose natural 2-cocycles satisfy a new boundedness condition.
In contrast to the von Neumann algebra case, at the C*-algebra level we obtain a stronger result. When the groups Gn are certain amalgamated free products of the property (T) wreath-like products described above, we prove that the infinite direct sum G=⊕n Gn is in fact C*-superrigid; that is, if H is any group such that Cr*(G)=Cr*(H), then necessarily H=G.
This talk is based on two recent joint works with J. F. Ariza Mejia, A. Fernandez Quero, D. Osin, and B. Sun.
Yotam Dikstein, On Cocycle Expansion, Stability and (non-)Sofic Groups
Cocycle expansion is a notion of stability; a cocycle expander is a space where `almost-cocycles' are always close to (actual) cocycles. We will define this notion formally and see how it is a high dimensional generalization to graph expansion. Then we will depict its connections to:
1. Group theory (conjectured existence of non-sofic groups),
2. Coding theory (locally testable codes),
3. Computational complexity (probabilistically checkable proofs), and
4. Algebraic topology (stability of covering maps).
Time permitting we will describe current methods of analyzing cocycle expansion and their limitations.
Changying Ding, Relative solidity for biexact groups in measure equivalence
I will present joint work with Daniel Drimbe establishing a relative solidity property for products of nonamenable biexact groups in the measure equivalence setting. As applications, we obtain a unique product decomposition result for such groups, strengthening work of Sako and extending Drimbe--Hoff--Ioana beyond the weakly amenable setting, as well as rigidity results for wreath products and infinite direct sums of biexact groups.
Francesco Fournier-Facio, The Local Lifting Property, Property FD and stability
Very flexible stability is a weakening of stability of approximate representations, that can be used in the same way as strict stability in the search for non-approximable groups. It is much more common than strict stability, for instance Eckhardt--Shulman proved it for all MAP amenable groups. In fact, the only things needed for this argument are Kirchberg's Local Lifting Property, and residual finite-dimensionality, for the group C^*-algebra. I will recall the argument and these two properties, and then talk about joint work with Rufus Willet where we establish the LP (stronger than LLP) and the Lubotzky-Shalom property FD (stronger than RFD) for 3-manifold groups, limit groups, and many one-relator groups and right-angled Artin groups.
Honghao Fu, Nonlocal Games in the High-Noise Regime: Optimal Quantum Values and Rigidity
Motivated by the limitations of near-term quantum devices, we study nonlocal games in the high-noise regime, where the two players may share arbitrarily many copies of a noisy entangled state. In this regime, existing rigidity theorems are unable to certify any nontrivial quantum structure. We first characterize the maximal quantum winning probabilities of the CHSH game, the Magic Square game, and their 2-out-of-n variants of these two games as explicit functions of the noise rate. These characterizations enable the construction of device-independent protocols for estimating the underlying noise level. Building on these results, we prove noise-robust rigidity theorems showing that these games certify one, two, and n pairs of anticommuting Pauli observables, respectively. To our knowledge, these are the first rigidity results of Pauli measurements that remain sound in the high-noise regime, which has applications in Measurement-Device-Independent (MDI) cryptography and studying the computational power of Multi-prover Interactive Proof System with entanglement and a vanishing completeness-soundness gap (MIP^*_0). Our proofs rely on Sum-of-Squares decompositions and Pauli analysis techniques.
David Gao, Sofic actions on sets and graphs
In this talk, we introduce a notion of sofic actions of groups on sets and graphs. While in works of Elek and Lippner and of Paunescu, notions of soficity in the context of pmp actions have already been developed, no satisfactory notions of sofic actions on discrete objects have heretofore been defined. This new notion of sofic actions allows us to strictly generalize the work of Hayes and Sale by showing a large class of generalized wreath product and graph wreath product groups are sofic. Interesting examples of sofic actions include transitive actions of sofic groups on graphs with amenable stabilizers, arbitrary actions of amenable groups and free groups on graphs, and arbitrary actions of LERF groups on sets. A majority of this talk is based on joint work with Srivatsav Kunnawalkam Elayavalli and Gregory Patchell.
Marius Junge, The connection between Connes embedding problem and Quantum Information Theory
In this talk we review the history of how Connes' embedding problem become entangled with Quantum Information Theory thanks to the work of Tsirelson and Kirchberg.
Junqiao Lin, MIPco = coRE
Recently, Ji et al. gave a negative answer to the Connes embedding problem by showing that MIP* = RE where the complexity class MIP* corresponds to an interactive proof system where the provers are given access to the tensor product model of entanglement. A natural follow-up question is whether the same uncomputability results also hold for the complexity class MIPco, a variant of MIP* where the provers are given access to the commuting operator model of entanglement instead. We show that this conjecture is true, and the complexity class MIPco is equal to coRE, the complexity class that is complete with respect to the non-halting problem.
In this talk, I will introduce tracially embeddable strategies, a class of commuting operator strategies that can be defined using tracial von Neumann algebras, which can be used to approximate the set of commuting operator strategies. I will describe how these techniques can be used to lift proofs from MIP*=RE to the commuting-operator case. If time permits, I will also sketch the idea for the original proof showing MIP\subseteq NEXP and explain how that fits into the answer reduction portion of the MIP*=RE proof.
Clark Lyons, The Aldous-Lyons Conjecture and Neighborhood Statistics Decision Problems
In joint work with Grigory Terlov and Zoltán Vidnyánszky, we gave a negative answer to a question of Lovász about effective regularity in bounded degree graphs. Our result relied on recent work of Bowen, Chapman, Lubotzky, and Vidick establishing undecidability of the values of subgroup tests, which they used to refute the Aldous-Lyons conjecture. In this talk I will present a converse result, showing that the undecidability of the values subgroup tests is in fact equivalent to the nonexistence of a computable sparse regularity bound. This motivates an alternative approach to refuting the Aldous-Lyons conjecture through a direct robust encoding of Turing machines into the neighborhood statistics of graphs. I will present some ideas related to this approach.
Aareyan Manzoor, The Connes embedding problem and the Aldous-Lyons problem
The Connes embedding problem was solved using non-local game and the landmark complexity result MIP*=RE. The idea is to consider each tracial von Neumann algebra as strategies on non-local game. I will explain how this can be used to find a measured equivalence relation whose von Neumann algebra is not Connes embeddable. I will then explain how this gives a disproof of the Aldous-Lyons conjecture.
Andrew Marks, The recursive compression method for proving undecidability results
We describe a general recursive compression lemma that abstracts the technique used to prove undecidability by Fitzsimons, Ji, Vidick, and Yuen (and subsequently refined by Ji, Natarajan, Vidick, Wright, and Yuen, and then further by Lin). We show that the halting problem is polynomial-time reducible to an r.e. language if and only if there is a recursive compression for that language. We explain how the idea of recursive compression was also used independently in mathematics in 2010 by Durand, Romashchenko, and Chen to give a new proof that the Wang tiling problem is undecidable. We give several other examples of recursive compression and explain how certain proofs in the literature can be viewed through this lens. We also describe some generalizations of the lemma that can prove hardness at arbitrarily high levels of the arithmetical hierarchy.
This is joint work with Seyed Sajjad Nezhadi and Henry Yuen.
Oriol Solé-Pi, Soficity and structural graph theory
We know that unimodularity is not a sufficient condition for a random rooted graph to be sofic. What kind of additional conditions can we impose in order to guarantee soficity? I will briefly overview what is known about this problem, and then I will discuss a new result in this direction: For any finite graph H, every unimodular random graph which is one-ended and does not have H as a minor must be sofic. The proof of this statement relies on some previously known results which connect the graph treewidth parameter to treeability.
Damian Osajda, Fit systolic groups, exactly
We prove that a class of systolic complexes (that is, complexes with simplicial non-positive curvature) satisfies Yu's property A, a coarse geometric property implying, for example, coarse embeddability into a Hilbert space. It follows that groups acting properly on such complexes are exact, or equivalently, boundary amenable. As a consequence, groups from a class containing all large-type Artin groups, as well as all finitely presented graphical C(3)-T(6) and classical C(6) small cancellation groups, are exact. Our proof uses the Špakula-Wright combinatorial criterion for Property A. This is joint work with Martín Blufstein, Victor Chepoi, and Huaitao Gui.
Koichi Oyakawa, Hyperfiniteness of the boundary action of virtually special groups
A Borel equivalence relation on a Polish space is called hyperfinite if it can be approximated by equivalence relations with finite classes. This notion has long been studied in descriptive set theory to measure complexity of Borel equivalence relations. Recently, a lot of research has been done on hyperfiniteness of the orbit equivalence relation on the Gromov boundary induced by various group actions on hyperbolic spaces. In this talk, I will explain my attempt to explore this connection of Borel complexity and geometric group theory for another intensively studied geometric object, which is CAT(0) cube complexes. More precisely, we prove that for any countable group acting virtually specially on a CAT(0) cube complex, the orbit equivalence relation induced by its action on the Roller boundary is hyperfinite.
Connor Paul-Paddock, A computable mapping from Turing machines to boolean constraint system nonlocal games using *-algebraic techniques
I will discuss a recent work (arXiv:2510.04943) in which we establish a computable mapping from Turing machines to boolean constraint system (BCS) nonlocal games where the commuting operator value of the game is strictly greater than 1/2 if and only if the Turing machine halts. As a corollary, it is undecidable to determine whether the commuting operator value of a nonlocal game is strictly greater than 1/2. Our results are entirely algebraic and distinct from those in MIP*=RE (and MIP^co=coRE), rather relying on the coRE-hardness of the positivity problem in tensor-products of *-algebras recently established by Mehta-Slofstra-Zhao. Furthermore, our result implies that there is a BCS nonlocal game for which the value of the Navascues-Pironio-Acin (NPA) semi-definite programming hierarchy does not attain the commuting operator value at any finite level.
Liviu Paunescu, Permutation stability of amalgamated products over finite groups
We provide some examples of permutation stable groups, obtained as amalgamated products over finite groups. We discuss some challenges that appear when the amalgamation is done over infinite groups. Joint work with Goulnara Arzhantseva.
Travis Russell, Projections in Operator Systems and Applications
A number of problems of great interest in quantum information theory can be described entirely in terms of projection operators satisfying some linear relations and states on the linear span of those operators. Consequently, these problems ought to be expressible as problems about the abstract operator systems generated by certain projection operators. In this talk, we describe how several problems, including Tsirelson's conjecture and Zauner's conjecture, can be formulated entirely in terms of the abstract data of certain operator systems. We will show how these operator systems can be "constructed" from the abstract data of finite-dimensional operator systems through an inductive limit process.
Bin Sun, L2-Betti numbers of Dehn filling
I will talk about joint work with Nansen Petrosyan about L2-Betti numbers of groups. Although L2-Betti numbers in general don't interact well with taking quotients of groups, we prove that group-theoretic Dehn filling, a certain quotienting process, preserves L2-Betti numbers. A key step in our proof is the Lück approximation of L2-Betti numbers for sofic groups.
Greg Terlov, Ineffectiveness of the regularity lemma for bounded degree graphs
In this talk, I will present a negative answer to a question of Lovász about effective approximation of bounded degree graphs. Specifically, that for every Δ ≥ 3, there is no computable bound, depending only on (ε, r), on the size of a graph required to approximate a graph of maximum degree at most Δ up to an ε error in r-neighborhood statistics. Our result follows from the recent celebrated work of Bowen, Chapman, Lubotzky, and Vidick, which refutes the Aldous--Lyons conjecture. This talk is based on a joint work with C. Lyons and Z. Vidnyánszky.
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