# Events for 11/13/2019 from all calendars

## Inverse Problems and Machine Learning

**Time:** 12:00PM - 1:00PM

**Location:** BLOC 628

**Speaker:** Christoper Snyder, UT Austin

**Title:** *Combinatorial Complexity of Deep Networks: Think Weight Configurations, not Perturbations!*

**Abstract:** Did you know that (ReLU) Deep Neural Networks (DNNs) trained on linearly separable data are linear classifiers? While it is widely appreciated that some data assumptions are necessary to explain generalization in deep learning, we observe that very strong data assumptions induce regularity in gradient descent trained DNNs that is entirely combinatorial in nature. That is, strong constraints exist between the binary neuron states and binary output, which simplify the description of the classification map. We present a hierarchical decomposition of the DNN discrete classification map into logical (AND/OR) combinations of intermediate (True/False) classifiers of the input. Those classifiers that can not be further decomposed, called atoms, are (interpretable) linear classifiers. Taken together, we obtain a logical circuit with linear classifier inputs that computes the same label as the DNN. This circuit does not structurally resemble the network architecture, and it may require many fewer parameters, depending on the configuration of weights. In these cases, we obtain simultaneously an interpretation and generalization bound (for the original DNN), connecting two fronts which have historically been investigated separately. We study DNNs in simple, controlled settings, where we obtain superior generalization bounds despite using only combinatorial information (e.g. no margin information). On the MNIST dataset. We show that the learned, internal, logical computations correspond to semantically meaningful (unlabeled) categories that allow DNN descriptions in plain English. We improve the generalization of an already trained network by interpreting, diagnosing, and replacing components \textit{within} the logical circuit that is the DNN.

## Student Working Seminar in Groups and Dynamics

**Time:** 1:00PM - 2:00PM

**Location:** BLOC 628

**Speaker:** Diego Martínez

**Title:** *Quasidiagonality and relations to group theory*

**Abstract:** Given a C*-algebra A we say that it is quasidiagonal if there is a sequence of finite-dimensional almost representations that are almost isometric. Even though this seems to be a very C*-algebraic definition, it has proven useful in many other areas, such as geometric group thery, index theory and numerical analysis. In this introductory talk we'll discuss how quasidiagonality is related to those contexts, and the role quasidiagonality has had in the theory of operator algebras.

## Number Theory Seminar

**Time:** 1:45PM - 2:45PM

**Location:** BLOC 220

**Speaker:** Mike Fried, University of California, Irvine

**Title:** *Spaces of sphere covers and Riemann's two types of Θ functions*

**Abstract:** Riemann surface covers *X* → **P**^{1}_{z} of the sphere, uniformized by a complex variable *z*, arise by giving the branch points and generators *g*_{1} …*g*_{r} of a finite group *G* where the *g*_{i}s have product-one.

By taking any one such cover, and *dragging it by its branch points* you create a space of such covers.

**A Fundamental Problem:** For a given *G* and the conjugacy classes of the *g _{i}*s, describe the connected components of the space.

This talk will explain the following case/result: Spaces of

*r*-branch point 3-cycle covers, of degree

*n*, or their Galois closures

of degree

*n*!/2, have

*one*(resp.

*two*) component(s) if

*r*=

*n*-1 (resp.

*r*≥

*n*).

Each space is determined by the type of natural θ functions they support. This improves a Fried-Serre formula on when sphere covers with odd-order branching lift to

*unramified Spin covers*of the sphere. We will use the case

*n*=4, to

*see*these Θs and differentiate between their even and odd versions. Riemann used both for different purposes.

This is a special case of a general result about components of spaces of sphere covers. Hyperelliptic jacobians then appear as one case of a general problem entwining

*The Torsion Conjecture*and

*the Regular Inverse Galois problem.*A recent series of Ellenberg-Venkatesh-Westerland used these results, but only got to the hyperelliptic jacobian case.

**URL:** *Link*

## Groups and Dynamics Seminar

**Time:** 2:00PM - 3:00PM

**Location:** BLOC 628

**Speaker:** Tatiana Nagnibeda, University of Geneva

**Title:** *Various types of spectra and spectral measures on Schreier and Cayley graphs.*

**Abstract:** We will be interested in the Laplacian on graphs associated with finitely generated groups: Cayley graphs and more generally Schreier graphs corresponding to some natural group actions. The spectrum of such an operator is a compact subset of the closed interval [-1,1], but not much more can be said about it in general. We will discuss various techniques that allow to construct examples with different types of spectra: connected, union of two intervals, totally disconnected…, and how this depends on the choice of the generating set in the group. Types of spectral measures that can arise in these examples will also be discussed.

## Noncommutative Geometry Seminar

**Time:** 2:00PM - 3:00PM

**Location:** BLOC 628

**Speaker:** Tatiana Nagnibeda, University of Geneva

**Title:** *Various types of spectra and spectral measures on Schreier and Cayley graphs*

**Abstract:** We will be interested in the Laplacian on graphs associated with finitely generated groups: Cayley graphs and more generally Schreier graphs corresponding to some natural group actions. The spectrum of such an operator is a compact subset of the closed interval [-1,1], but not much more can be said about it in general. We will discuss various techniques that allow to construct examples with different types of spectra: connected, union of two intervals, totally disconnected…, and how this depends on the choice of the generating set in the group. Types of spectral measures that can arise in these examples will also be discussed.

## Groups and Dynamics Seminar

**Time:** 3:00PM - 4:00PM

**Location:** BLOC 220

**Speaker:** Paul Schupp, UIUC

**Title:** *Closures of Turing Degrees*

**Abstract:** This talk is on aspect of my general project with Carl Jockusch on “the coarsification of computability theory”, that is, bringing the asymptotic-generic point of view of geometric group theory into the theory of computability. Classically, computability theory studies Turing degrees, that is, equivalence classes of subsets of N which are computationally equivalent. Coarse computability studies how closely arbitrary subsets of N can be approximated by computable sets. The idea of coarse computabilty leads to a natural definition of the closure of a Turing degree in the space S of coarse similarity classes of subsets of N with the Besicovich metric. It turns out that S is an interesting space. We will discuss interactions of the topology of S and properties of Turing degrees.

## Noncommutative Geometry Seminar

**Time:** 3:00PM - 4:00PM

**Location:** BLOC 220

**Speaker:** Paul Schupp, University of Illinois at Urbana Champaign

**Title:** *Closures of Turing Degrees*

**Abstract:** This talk is on aspect of my general project with Carl Jockusch on “the coarsification of computability theory”, that is, bringing the asymptotic-generic point of view of geometric group theory into the theory of computability. Classically, computability theory studies Turing degrees, that is, equivalence classes of subsets of N which are computationally equivalent. Coarse computability studies how closely arbitrary subsets of N can be approximated by computable sets. The idea of coarse computabilty leads to a natural definition of the closure of a Turing degree in the space S of coarse similarity classes of subsets of N with the Besicovich metric. It turns out that S is an interesting space. We will discuss interactions of the topology of S and properties of Turing degrees.

## Graduate Student Organization Seminar

**Time:** 4:00PM - 5:00PM

**Location:** BLOC 628

**Speaker:** Diego Martinez

**Title:** *Coarse Geometry and Inverse Semigroups*

**Abstract:** In this talk we will study two seemingly disconnected notions: coarse geometry and inverse semigroups. Geometry often studies certain objects (such as sets or manifolds) equipped with a distance function. For instance, one classical problem would be to classify every compact manifold up to diffeomorphism. Coarse geometry shifts the point of view, and defines two sets to be coarse equivalent if they look the same from far away. In this way, for instance, a point and a sphere are indistinguishable from each other. Coarse geometry then studies properties that remain invariant under this weak equivalence relation, that is, properties of the space that only appear at infinity. On the other hand, an inverse semigroup is a natural generalization of the notion of group, and is closely related to the idea of groupoid. Starting with one of these objects we will introduce how to construct a metric space, in the same fashion as the Cayley graph construction in the context of groups. We will then study its coarse structure, in particular its property A and its amenability. Time permitting, we will also relate these properties to analogue properties in some operator algebras.

## Groups and Dynamics Seminar

**Time:** 4:00PM - 5:00PM

**Location:** BLOC 220

**Speaker:** Tullio Ceccherini-Silberstein

**Title:** *Hecke algebras of multiplicity-free induced representations*

**Abstract:** Given a finite group G and a subgroup K, one says that (G,K) is a Gelfand pair provided the associated permutation representation (\lambda, L(G/K)) is multiplicity-free (that is, decomposes into pairwise non-equivalent irreducible subrepresentations). This condition is equivalent to the algebra End_G(L(G/K)) of interwining operators being commutative. Observe that \lambda is nothing but the induced representation Ind_K^G \iota_K of the trival representation \iota_K of K. In [CS-S-T] we consider triples (G,K,\theta), where \theta is, more generally, an irreducible K-representation and introduce a Hecke-type algebra H(G,K,\theta) - analogous to End_G(L(G/K)) - and show that that Ind_K^G\theta is multiplicity-free if and only if H (G,K,\theta) is commutative. We apply our results in the context of the representation theory of GL_2(q), the general linear group of a field with q elements. [CS-S-T] Harmonic analysis and spherical functions for multiplicity-free induced representations of finite groups. Springer (to appear) arXiv: 1811.09526.

## Noncommutative Geometry Seminar

**Time:** 4:00PM - 5:00PM

**Location:** BLOC 220

**Speaker:** Tullio Ceccherini-Silberstein, University of Sannio

**Title:** *Hecke algebras of multiplicity-free induced representations*

**Abstract:** Given a finite group G and a subgroup K, one says that (G,K) is a Gelfand pair provided the associated permutation representation (\lambda, L(G/K)) is multiplicity-free (that is, decomposes into pairwise non-equivalent irreducible subrepresentations). This condition is equivalent to the algebra End_G(L(G/K)) of interwining operators being commutative. Observe that \lambda is nothing but the induced representation Ind_K^G \iota_K of the trival representation \iota_K of K. In [CS-S-T] we consider triples (G,K,\theta), where \theta is, more generally, an irreducible K-representation and introduce a Hecke-type algebra H(G,K,\theta) - analogous to End_G(L(G/K)) - and show that that Ind_K^G\theta is multiplicity-free if and only if H (G,K,\theta) is commutative. We apply our results in the context of the representation theory of GL_2(q), the general linear group of a field with q elements. [CS-S-T] Harmonic analysis and spherical functions for multiplicity-free induced representations of finite groups. Springer (to appear) arXiv: 1811.09526.