In the Spring 2026 semester the New York Group Theory Seminar will meet in a hybrid format, with most talks in-person and some talks online. The in-person talks will be on Fridays at 4:15pm eastern time, room 6417. The online Zoom talks will be on Fridays at 4:00pm U.S. eastern time.
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New York Group Theory Seminar: Friday, February 27, 2026, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Kunal Chawla (Princeton University)
Title: Non-realizability of the Poisson boundary
Abstract:
Given a countable group $G$ equipped with a probability measure $\mu$, one can define a random walk on $G$ as the Markov process whose increments are iid sampled by $\mu$. The large-scale properties of this random walk can exhibit a plethora of exotic behaviours, relating to the algebraic and geometric structure of $G$.
One way of capturing the large-scale behaviour is via the Poisson boundary of $(G,\mu)$, a canonical (abstract) measure space associated with the Markov chain. This object has been studied intensely over the decades. In particular, mathematicians have found concrete ‘realizations’ of this measure space as topological boundaries for the group (i.e. the Gromov boundary of a hyperbolic group). In 1983, Kaimanovich and Vershik asked whether the Poisson boundary can always be realized in this way.
I will describe the complete resolution of this problem in the negative. This is joint work with Joshua Frisch.
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New York Group Theory Seminar: Friday, March 6, 2026, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Eilidh McKemmie (Kean University)
Title: Monodromy groups of covers of genus 1 Riemann surfaces
Abstract:
Consider a cover of the Riemann sphere by a compact connected Riemann surface. The monodromy group of the cover is an important invariant describing how badly the cover degenerates. It is natural to ask which groups can appear in such a context. We will survey some applications of this question and discuss how to genus of the surface influences the answer. We will end by providing an answer for groups of Aschbacher-Scott type B in genus at most 1.
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New York Group Theory Seminar: Friday, March 13, 2026, 4:15pm, room 6417, CUNY Graduate Center
Rescheduled for Friday, May 15
Speaker: Doron Puder (Tel Aviv University and IAS Princeton)
Title: Aldous-type spectral gaps in Unitary groups
Abstract:
Around 1992, Aldous made the following bold conjecture. Let A be any set of transpositions in the symmetric group Sym(N). Then the spectral gap of the Cayley graph Cay(Sym(N),A) is identical to that of a relatively tiny N-vertex graph defined by A. So even though the spectrum of the Cayley graph contains N! eigenvalues, the largest non-trivial one always comes from a tiny pool of N of them. This conjecture was proven nearly 20 years later by Caputo, Liggett and Richthammer (JAMS, 2010).
In a joint work with Gil Alon, driven by the conviction that such a stunning phenomenon cannot possibly be isolated, we found a probable parallel of this phenomenon in the unitary group U(N). We have a concrete conjecture supported by simulations, and we prove it in several non-trivial special cases. As it turns out, the corresponding spectrum in the case of U(N) contains the one in Sym(N). Moreover, the critical part of the spectrum in U(N) coincides with the spectrum of an interesting discrete process.
I will explain the original conjecture of Aldous, the new conjecture and our findings.
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New York Group Theory Seminar: Friday, March 27, 2026, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Brett Berger (Stevens Institute of Technology)
Title: Interpreting free groups in their subgroup lattices
Abstract: For F a free group of (potentially infinite) rank at least 2 and L(F) its lattice of subgroups, we show that F is absolutely interpretable in L(F). This extends the original result of Sadovskii (1941) that free groups are strongly determined by their subgroup lattice (every automorphism of L(F) is induced by an automorphism of F). A lattice-theoretic description of this result was given by Yakovlev (1974). From this, we obtain a regular interpretation of F in L(F) which we then show is absolutizable. We also discuss related results such as the undecidability of the first-order theory of L(F) and that L(F) is rich. This is joint work with Alexei Miasnikov.
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New York Group Theory Seminar: Friday, April 10, 2026, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Yankun Wang (Stevens Institute of Technology)
Title: One-variable equations over the lamplighter group
Abstract:
We study one-variable equations over the lamplighter group $\MZ_2 \wr \MZ$. While the decidability of arbitrary equations over $L_2$ remains open, we prove that the Diophantine problem for single equations in one variable is decidable. Our approach reduces the problem to a divisibility question for families of parametric Laurent polynomials over $\MZ_2$, whose coefficients depend linearly on an integer parameter. We develop an automaton-theoretic framework to analyze divisibility of such polynomials, exploiting eventual periodicity phenomena arising from polynomial division over finite fields. This yields an explicit decision procedure, which is super-exponential in the worst case. On the other hand, we show that for a generic class of equations, solvability can be decided in nearly quadratic time. These results establish a sharp contrast between worst-case and typical computational behavior and provide new tools for the study of equations over wreath products.
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New York Group Theory Seminar: April 17, 2026, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Michael Chapman (IAS Princeton)
Title: Group Stability
Abstract:
In the 1940's, Ulam asked the following question, known today as Ulam's Group Stability Problem: Are almost homomorphisms close to actual ones? The answer to this question depends on the domain and target groups, as well as the appropriate notion of being an almost homomorphism and proximity between functions. In this talk I will present a specific viewpoint on this problem, which is related both to group approximation properties (soficity, hyperlinearity) as well as to theoretical computer science.
The talk will be introductory and will assume no previous background except for standard group theory notions. Some of the results presented are from joint (and ongoing) projects with Oren Becker, Irit Dinur, Alex Lubotzky and Doron Puder.
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New York Group Theory Seminar: Friday, April 24, 2026, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Lev Shneerson (Hunter College)
Title:
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New York Group Theory Seminar: Friday, May 1, 2026, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Jonathan Chaika (University of Utah and IAS Princeton)
Title:
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New York Group Theory Seminar: Friday, May 8, 2026, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Harald Andres Helfgott (Institut de Mathématiques de Jussieu)
Title:
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New York Group Theory Seminar: Friday, May 15, 2026, 4:15pm, room 6417, CUNY Graduate Center
Speaker: Doron Puder (Tel Aviv University and IAS Princeton)
Title: Aldous-type spectral gaps in Unitary groups
Abstract:
Around 1992, Aldous made the following bold conjecture. Let A be any set of transpositions in the symmetric group Sym(N). Then the spectral gap of the Cayley graph Cay(Sym(N),A) is identical to that of a relatively tiny N-vertex graph defined by A. So even though the spectrum of the Cayley graph contains N! eigenvalues, the largest non-trivial one always comes from a tiny pool of N of them. This conjecture was proven nearly 20 years later by Caputo, Liggett and Richthammer (JAMS, 2010).
In a joint work with Gil Alon, driven by the conviction that such a stunning phenomenon cannot possibly be isolated, we found a probable parallel of this phenomenon in the unitary group U(N). We have a concrete conjecture supported by simulations, and we prove it in several non-trivial special cases. As it turns out, the corresponding spectrum in the case of U(N) contains the one in Sym(N). Moreover, the critical part of the spectrum in U(N) coincides with the spectrum of an interesting discrete process.
I will explain the original conjecture of Aldous, the new conjecture and our findings.
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