Georgia Institute of Technology College of Sciences

In recent years, various notions of capacity and complexity have been proposed for characterizing the generalization properties of stochastic gradient descent (SGD) in deep learning. Some of the popular notions that correlate well with the performance on unseen data are (i) the flatness of the local minimum found by SGD, which is related to the eigenvalues of the Hessian, (ii) the ratio of the stepsize to the batch-size, which essentially controls the magnitude of the stochastic gradient noise, and (iii) the tail-index, which measures the heaviness of the tails of the network weights at convergence. In this paper, we argue that these three seemingly unrelated perspectives for generalization are deeply linked to each other. We claim that depending on the structure of the Hessian of the loss at the minimum, and the choices of the algorithm parameters, the distribution of the SGD iterates will converge to a heavy-tailed stationary distribution. We rigorously prove this claim in the setting of quadratic optimization: we show that even in a simple linear regression problem with independent and identically distributed data whose distribution has finite moments of all order, the iterates can be heavy-tailed with infinite variance. We further characterize the behavior of the tails with respect to algorithm parameters, the dimension, and the curvature. We then translate our results into insights about the behavior of SGD in deep learning. We support our theory with experiments conducted on synthetic data, fully connected, and convolutional neural networks. This is based on the joint work with Mert Gurbuzbalaban and Umut Simsekli.

Symmetric unions are an interesting class of knots. Although they have not been studied much for their own sake, they frequently appear in the literature. One such instance regards the question of whether there is a nontrivial knot with trivial Jones polynomial. In my talk, I will describe a class of symmetric unions, constructed by Tanaka, such that if any are amphichiral, they would have trivial Jones polynomial. Then I will show how such a knot not only answers the above question but also gives rise to a counterexample to the Cosmetic Surgery Conjecture. However, I will prove that such a knot is in fact trivial and hence cannot be used to answer any of these questions. Finally, I will discuss how the arguments that go into this proof can be generalized to study amphichiral symmetric unions.

I’ll describe various aspects of my joint work with Oliver Lorscheid on the foundation of a matroid.

In this talk, we will discuss the main results of our paper, Counting Colorings of Triangle-Free Graphs, in which we prove the Johansson-Molloy theorem for the upper bound on the chromatic number of a triangle free graph using a novel counting approach developed by Matthieu Rosenfeld, and also extend this result to obtain a lower bound on the number of proper q-colorings for a triangle free graph.  The talk will go over the history of the problem, an outline of our approach, and a high-level sketch of the main proofs. This is joint work with Anton Bernshteyn, Tyler Brazelton, and Akum Kang.

The term cancer covers a multitude of bodily diseases, broadly categorized by having cells which do not behave normally. Cancer cells can arise from any type of cell in the body; cancers can grow in or around any tissue or organ making the disease highly complex. My research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modelling. In this talk I shall present a 3D individual-based force-based model for tumour growth and development in which we simulate  the behavior of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels. Each agent is fully realised, for example, cells are described as viscoelastic sphere with radius and centre given within the off-lattice model. Interactions are primarily governed by mechanical forces between elements. However, as well as the mechanical interactions we also consider chemical interactions, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells, as well as intercellular aspects such as cell phenotypes. 

Neural codes are inspired by John O'Keefe's discovery of the place cell, a neuron in the mammalian brain which fires if and only if its owner is in a particular region of physical space. Mathematically, a neural code $C$ on n neurons is a collection of subsets of $\{1,...,n\}$, with the subsets called codewords. The implication is that $C$ encodes how the members of some collection $\{U_i\}_{i=1}^n$ of subsets of $\mathbb{R}^d$ intersect one another. 

The principal question driving the study of neural codes is that of convexity. Given just the codewords of $C$, can we determine if there is a collection of open convex subsets $ \{U_i\}_{i=1}^n$ of some $\mathbb{R}^d$ for which $C$ is the code? A convex code is a code for which there is such a realization of open convex sets. While the question of determining which codes are convex remains open, there has been significant progress as many large families of codes can now be ruled as convex or nonconvex. In this talk, I will give an overview of some of the results from this work. In particular, I will focus on a phenomenon called a local obstruction, which if found in a code forbids convexity.    

Given two knots K_1 and K_2, their 0-surgery manifolds S_0^3(K_1) and S_0^3(K_2) are homology cobordant rel meridian if they are homology cobordant preserving the homology class of the positively oriented meridian. It is known that if K_1 ∼ K_2, then S_0^3(K_1) and S_0^3(K_2) are homology cobordant rel meridian. The converse of this statement was first disproved by Cochran-Franklin-Hedden-Horn.  In this talk we will provide a new counterexample, the pair of knots 4_1 and M(4_1) where M is the Mazur satellite operator. S_0^3(4_1) and S_0^3(M(4_1)) are homology cobordant rel meridian, but 4_1 and M(4_1) are non-concordant and have concordance orders 2 and infinity, respectively. 

In this talk, we present an operator theoretic analogue of the F. and M. Riesz Theorem. We first recast the classical theorem in operator theoretic terms. We then establish an analogous result in the context of representations of the Cuntz algebra, highlighting notable differences from the classical setting. At the end, we will discuss some extensions of these ideas. This is joint work with R. Clouâtre and R. Martin.

Zoom Link:  

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

The problem of classifying collections of objects (graphs, manifolds, operators, etc.) up to some notion of equivalence (isomorphism, diffeomorphism, conjugacy, etc.) is central in every domain of mathematical activity. Invariant descriptive set-theory provides a formal framework for measuring the intrinsic complexity of such classification problems and for deciding, in each case, which types of invariants are “too simple” to be used for a complete classification. It also provides a very interesting link between topological dynamics and the meta-mathematics of classification. In this talk I will discuss various forms of classification which naturally occur in mathematical practice (concrete classification, classification by countable structures, classification by cohomological invariants, etc.) and I will provide criteria for showing when some classification problem cannot be solved using these forms of classification.

Nonnegative polynomials are of fundamental interest in the field of real algebraic geometry. We will discuss a model of nonnegative polynomials over an interesting class of algebraic varieties which have potential applications in optimization theory. In particular, we will discuss connections between this subject and algebraic topology and the geometry of simplicial complexes.

We consider a service system consisting of parallel single server queues of infinite capacity. Work of different classes arrives as correlated Gaussian processes with known drifts but unknown covariances, and it is deterministically routed to the different queues according to some routing matrix. In this setting we show that, under some conditions, the covariance matrix of the arrival processes can be directly recovered from the large deviations behavior of the queue lengths. Also, we show that in some cases this covariance matrix cannot be directly recovered this way, as there is an inherent loss of information produced by the dynamics of the queues. Finally, we show how this can be used to quickly learn an optimal routing matrix with respect to some utility function.

Living and active systems exhibit various emergent dynamics necessary for system regulation, growth, and motility. However, how robust dynamics arises from stochastic components remains unclear. Towards understanding this, I develop topological theories that support robust edge states, effectively reducing the system dynamics to a lower-dimensional subspace. In particular, I will introduce stochastic networks in molecular configuration space that enable different phenomena from a global clock, stochastic growth and shrinkage, to synchronization. These out-of-equilibrium systems further possess uniquely non-Hermitian features such as exceptional points and vorticity. More broadly, my work  provides a blueprint for the design and control of novel and robust function in correlated and active systems.

What are all the ways to draw the lines, when you're dividing up a state to get representation? If you can't find them all, can you choose a good sample? I'll discuss some surprisingly simple questions about graphs and geometry that can help us make advances in policy and civil rights.