# Math Calendar

### Upcoming Events

## The Relationship of Supernilpotence to Nilpotence (4)

Andrew Moorhead, Vanderbilt University

Location: Stevenson 1312

Supernilpotence is a condition on an algebra that is definable with a higher arity commutator that generalizes the classical binary commutator for general algebras. Supernilpotent algebras have received attention lately because of theorems of the flavor ‘nice property true satisfied by finite nilpotent groups’ is satisfied by finite supernilpotent Mal’cev algebras of finite type. For example, it is now know that there is a polynomial time algorithm to solve the equation equation satisfiability problem for such algebras. The exact relationship between supernilpotence and nilpotence had been unclear. We will discuss how supernilpotence implies nilpotence for algebras with a Taylor term, but that in general the two notions are independent.

## Results on Boundary Actions, cont’d

Lauren C. Ruth, Vanderbilt University

Location: Stevenson 1308

Last time, we defined boundary actions and saw how the Gleason–Yamabe theorem (crucial in Montgomery Zippin’s solution to Hilbert’s 5th Problem) gives rise to the group splitting in Furman’s result. This time, we will take a closer look at the maximal G-boundary, also called the Furstenberg boundary. After showing existence and uniqueness, we will prove that G acts faithfully on its Furstenberg boundary if and only if its amenable radical is trivial, following notes of Ozawa. Time permitting, we will go through Haagerup’s proof of Breuillard–Kalantar–Kennedy–Ozawa’s result that the amenable radical of G is trivial if and only if G has the unique trace property. Sources:Haagerup, A new look at C* simplicity and the unique trace property of a group, arXiv. Ozawa, Lecture on the Furstenburg boundary and C*-simplicity, online lecture notes.

## This Title is False

Hayden Jananthan, Vanderbilt University

Location: Stevenson 1206

To some, Gödel’s name is synonymous with the disruption of mathematics in the early 1900s, due in part to his famous ‘Incompleteness Theorems’, which put bounds on the expressiveness of formal systems of arithmetic. Frequently misunderstood and misstated, we will formally describe what Gödel’s First Incompleteness Theorem states and give a sketch of its proof. Analyzing our proof closely, we will find an even more general statement that both deepens and clarifies our original statement of his theorem.

## Homotopy Theory, Fixed Point Theory and the Cyclotomic Trace

Jonathan Campbell, Vanderbilt University

Location: Stevenson 1308

Fixed point theory has been the motivation for many of the most celebrated results of 20th century mathematics: the Lefschetz fixed point theorem, the Atiyah-Singer index theorem, and the development of etale cohomology. In this talk I’ll describe work, joint with Kate Ponto, that relates classical fixed point theory to an important homotopy theoretic invariant called algebraic K-theory. The relationship seems to clarify both domains, and readily suggests generalizations that relate to dynamical zeta functions. The link turns on careful considerations of the bicategorical structure of THH. Prerequisites: An appetite for (or, lacking that, a tolerance of) category theory. I will try to define and motivate most objects.

## Neural Network Quantization with a Probabilistic Version of Gradient Descent

Jonathan Ashbrock, Vanderbilt University

Location: Stevenson 1310

## K-groups and Rings of Integers

Adebisi Agboola, UC Santa Barbara

Location: Stevenson 5211

Suppose that F is a number field and that G is a finite group. The inverse Galois problem asks whether or not there exists an extension of F whose Galois group is isomorphic to G. This question is known to have an affirmative answer in many cases, but is unsolved in general. I shall discuss a conjecture in relative algebraic K-theory (in essence, a conjectural Hasse or local-global principle applied to certain relative algebraic K-groups) that implies an affirmative answer to both the inverse Galois problem and to an analogous problem concerning the Galois module structure of rings of integers in tame extensions of F. The K-theoretic conjecture can be proved in many cases (subject to mild technical conditions) e.g. for groups of odd order, giving an analogue of a classical theorem of Shafarevich in this setting. While this approach does not, as yet, resolve any new cases of the inverse Galois problem, it does yield a quite substantial advance in our knowledge concerning the Galois module structure of rings of integers. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Dietmar Bisch)

## Annual Town Hall Meeting of Chair and DGS with Graduate Students

Location: Stevenson 1206

The annual town hall meeting is an informal gathering where Department Chair Mike Neamtu along with Director of Graduate Studies Denis Osin meet with graduate students to answer questions and discuss updates about the graduate program.

## Module Categories, Graph Planar Algebra Embeddings, and Extended Haagerup

Noah Snyder, Indiana University

Location: Stevenson 1432

A natural question that Pinhas Grossman and I have been studying is given a finite collection of finite index N-N bimodules what are all factors M containing N which can be built as a sum of these bimodules. This question is closely related to several “representation theoretic” questions about fusion categories, namely classifying module categories, finding Ocneanu’s “maximal atlas”, and finding Etingof-Nikshych-Ostrik’s “Brauer-Picard groupoid.” Unrelated to all of this, Vaughan Jones asked given a fixed subfactor planar algebra P, can you find all bipartite graphs \Gamma such that P embeds into the Graph Planar Algebra of \Gamma. Emily Peters gave partial evidence that for the Haagerup subfactor there were exactly three such graphs (the two principal graphs, and the broom). My main goal in this talk is to explain why these two questions are basically the same as each other. The key result is a GPA embedding theorem for module categories, which says that P embeds in the GPA(\Gamma) if and only if \Gamma is the fusion graph for some module category. In particular, I will show that it follows from my first paper with Pinhas that Emily’s three graphs are the only graphs with Haagerup GPA embeddings. We are also able to use this approach to answer all these questions for the Extended Haagerup subfactor, showing that there are two new fusion categories EH3 and EH4 which still appear to be exceptional. This is joint work with Grossman, Morrison, Penneys, and Peters as part of our AIM Square.

## Complemented Envelopes of Commutative Bimonoids (Part I)

Adam Prenosil, Vanderbilt University

Location: Stevenson 1312

A recurrent theme in (especially ordered) algebra is embedding algebraic structures in which certain elements are “missing” into richer structures: completions and densifications of ordered structures are examples of this phenomenon. In this talk we consider embeddings into complemented and into complete complemented structures. Classical examples of such extensions are the group of differences of a cancellative commutative monoid and the Boolean envelope of a distributive lattice.

We show how such examples fit into a common framework of complemented extensions of what we call bimonoids. It turns out that each commutative bimonoid embeds in a canonical doubly dense way into what we call its complemented Dedekind–MacNeille completion. If the bimonoid is already complemented (e.g. a Boolean algebra), this construction coincides with the ordinary Dedekind–MacNeille completion.

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Srivatsav Kunnawalkam Elayavalli, Vanderbilt University

Location: Stevenson 1206

## Relatively Hyperbolic Groups With Free Abelian Second Cohomology

Mike Mihalik, Vanderbilt University

Location: Stevenson 1310

H. Hopf conjectured (probably in the 1940’s) that if G is a finitely presented group then H2(G;ZG) is free abelian. While many classes of groups are known to satisfy this conjecture (including all word hyperbolic groups), the conjecture remains open today. For a group G we consider a condition on H∞1(G), the first homology of the end of G that is equivalent to H2(G;Z) being free. Suppose G is a 1-ended finitely presented group that is hyperbolic relative to P a finite collection of 1-ended finitely presented proper subgroups of G. Our main theorem states that if the boundary ∂(G,P) is locally connected and the second cohomology group H2(P,ZP) is free abelian for each P∈P, then H2(G,ZG) is free abelian. When G is 1-ended it is conjectured that ∂(G,P) is always locally connected. When G and each member of P is 1-ended and ∂(G,P) is locally connected, we prove that the “Cusped Space” for this pair has semistable fundamental group at ∞. This provides a starting point in our proof of the main theorem.

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Scott Atkinson, Vanderbilt University

Location: Stevenson 1432

## Complemented Envelopes of Commutative Bimonoids (Part II)

Adam Prenosil, Vanderbilt University

Location: Stevenson 1312

Having introduced the complemented Dedekind–MacNeille completion of a commutative bimonoid in the previous talk, we now look inside this completion for some tighter complemented envelopes. In particular, it is natural to ask if a commutative bimonoid has an envelope akin to the group of differences, where each element has the form a – b. We provide some sufficient conditions for the existence of such complemented envelopes and use them to obtain categorical equivalences between varieties of integral and involutive residuated structures, unifying and extending some existing equivalences.

## Harmonic Hecke Eigenlines and Mazur’s Problem

Ian Wagner, Emory University

Location: Stevenson 1310

We construct two families of harmonic Maass Hecke eigenforms. Using these families we construct p-adic harmonic Maass forms in the sense of Serre. The p-adic properties of these forms answer a question of Mazur about the existence of an “eigencurve-type” object in the world of harmonic Maass forms.

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James Farre, University of Utah

Location: Stevenson 1310

## The Combinatorics of RNA Branching

Christine Heitsch, Georgia Tech

Location: Stevenson 5211

Understanding the folding of RNA sequences into three-dimensional structures is one of the fundamental challenges in molecular biology. For example, the branching of an RNA secondary structure is an important molecular characteristic yet difficult to predict correctly, especially for sequences on the scale of viral genomes. However, results from enumerative, probabilistic, analytic, and geometric combinatorics yield insights into RNA structure formation, and suggest new directions in viral capsid assembly. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Mark Ellingham)

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Corey Jones, Ohio State University

Location: Stevenson 1432

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Alexander Olshanskiy, Vanderbilt University

Location: Stevenson 1308

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Ilya Kapovich, University of Illinois at Urbana-Champaign

Location: Stevenson 5211

Tea at 3:33 pm in Stevenson 1425. (Contact Person: Denis Osin)

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Josh Edge, Indiana University

Location: Stevenson 1432

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Paramita Das, Indian Statistical Institute

Location: Stevenson 1432

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Andrei Okounkov, Columbia University

Location: Stevenson 5211

Tea at 3:33 pm in Stevenson 1425. (Contact Person: Vaughan Jones)

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Shamindra Ghosh, Indian Statistical Institute

Location: Stevenson 1432

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Justin Lanier, Georgia Tech

Location: Stevenson 1310

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Susan Friedlander, University of Southern California

Location: Stevenson 5211

Tea at 3:33 pm in Stevenson 1425. (Contact Person: Giusy Mazzone)

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Scott Wilson, CUNY Queens College

Location: Stevenson 1310

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Yasu Kawahigashi, University of Tokyo

Location: Stevenson 1432

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Wenqing Hu, Missouri University of Science and Technology.

Location: Stevenson 1307

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Frank Thorne, University of South Carolina

Location: Stevenson 1310

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Willie Wongu, Michigan State University

Location: Stevenson Center 1307

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Gueo Grancharov, Florida International University

Location: Stevenson 1310

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Dennis Sullivan, Suny at Stony Brook

Location: Stevenson 5211

Tea at 3:33 pm in Stevenson 1425. (Contact Person: Marcelo Disconzi)