## Seminars and Colloquia by Series

### Incorporating Invariance to Reduce the Complexity of Parametric Models

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 29, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
Alex CloningerUniversity of California, San Diego

Many scientific problems involve invariant structures, and learning functions that rely on a much lower dimensional set of features than the data itself.   Incorporating these invariances into a parametric model can significantly reduce the model complexity, and lead to a vast reduction in the number of labeled examples required to estimate the parameters.  We display this benefit in two settings.  The first setting concerns ReLU networks, and the size of networks and number of points required to learn certain functions and classification regions.  Here, we assume that the target function has built in invariances, namely that it only depends on the projection onto a very low dimensional, function defined manifold (with dimension possibly significantly smaller than even the intrinsic dimension of the data).  We use this manifold variant of a single or multi index model to establish network complexity and ERM rates that beat even the intrinsic dimension of the data.  We should note that a corollary of this result is developing intrinsic rates for a manifold plus noise data model without needing to assume the distribution of the noise decays exponentially, and we also discuss implications in two-sample testing and statistical distances.  The second setting for building invariances concerns linearized optimal transport (LOT), and using it to build supervised classifiers on distributions.  Here, we construct invariances to families of group actions (e.g., shifts and scalings of a fixed distribution), and show that LOT can learn a classifier on group orbits using a simple linear separator.   We demonstrate the benefit of this on MNIST by constructing robust classifiers with only a small number of labeled examples.  This talk covers joint work with Timo Klock, Xiuyuan Cheng, and Caroline Moosmueller.

### Topology of the Shift Locus via Big Mapping Class Groups by Yan Mary He

Series
Geometry Topology Seminar
Time
Monday, March 29, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Yan Mary HeUniversity of Toronto

The shift locus of (monic and centered) complex polynomials of degree d > 1 is the set of polynomials whose filled-in Julia set contains no critical points. Traversing a loop in the shift locus gives rise to a holomorphic motion of Cantor Julia sets, which can be extended to a homeomorphism of the plane minus a Cantor set up to isotopy. Therefore there is a well-defined monodromy representation from the fundamental group of the shift locus to the mapping class group of the plane minus a Cantor set. In this talk, I will discuss the image and the kernel of this map as well as a combinatorial model for the shift locus. This is joint work with J. Bavard, D. Calegari, S. Koch and A. Walker.

### Mathematics of Evolution: mutations, selection, and random environments

Series
Mathematical Biology Seminar
Time
Friday, March 26, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Natalia L. KomarovaUniversity of California, Irvine

Evolutionary dynamics permeates life and life-like systems. Mathematical methods can be used to study evolutionary processes, such as selection, mutation, and drift, and to make sense of many phenomena in life sciences. I will present two very general types of evolutionary patterns, loss-of-function and gain-of-function mutations, and discuss scenarios of population dynamics  -- including stochastic tunneling and calculating the rate of evolution. I will also talk about evolution in random environments.  The presence of temporal or spatial randomness significantly affects the competition dynamics in populations and gives rise to some counterintuitive observations. Applications include origins of cancer, passenger and driver mutations, and how aspirin might help prevent cancer.

### Branched Covers and Braided Embeddings

Series
Dissertation Defense
Time
Friday, March 26, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Sudipta KolayGeorgia Tech

We study braided embeddings, which is a natural generalization of closed braids in three dimensions. Braided embeddings give us an explicit way to construct lots of higher dimensional embeddings; and may turn out to be as instrumental in understanding higher dimensional embeddings as closed braids have been in understanding three and four dimensional topology. We will discuss two natural questions related to braided embeddings, the isotopy and lifting problem.

### Aldous-Broder theorem: extension to the non reversible case and new combinatorial proof

Series
Combinatorics Seminar
Time
Friday, March 26, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
Speaker
Jean-Francois MarckertUniversity of Bordeaux

Aldous-Broder algorithm is a famous algorithm used to sample a uniform spanning tree of any finite connected graph G, but it is more general: it states that given a reversible M Markov chain on G started at r, the tree rooted at r formed by the steps of successive first entrance in each node (different from the root) has a probability proportional to $\prod_{e=(e1,e2)∈Edges(t,r)}M_{e1,e2}$ , where the edges are directed toward r. As stated, it allows to sample many distributions on the set of spanning trees. In this paper we extend Aldous-Broder theorem by dropping the reversibility condition on M. We highlight that the general statement we prove is not the same as the original one (but it coincides in the reversible case with that of Aldous and Broder). We prove this extension in two ways: an adaptation of the classical argument, which is purely probabilistic, and a new proof based on combinatorial arguments. On the way we introduce a new combinatorial object that we call the golf sequences.

Based on joint work with Luis Fredes, see https://arxiv.org/abs/2102.08639

### Convergence over fractals for the periodic Schrödinger equation

Series
CDSNS Colloquium
Time
Friday, March 26, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Speaker
Daniel EceizabarrenaU Mass Amherst

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09 Meeting ID: 977 3221 5148 Passcode: 801074

In 1980, Lennart Carleson introduced the following problem for the free Schrödinger equation: when does the solution converge to the initial datum pointwise almost everywhere? Of course, the answer is immediate for regular functions like Schwartz functions. However, the question of what Sobolev regularity is necessary and sufficient for convergence turned out to be highly non-trivial. After the work of many people, it has been solved in 2019, following important advances in harmonic analysis. But interesting variations of the problem are still open. For instance, what happens with periodic solutions in the torus? And what if we refine the almost everywhere convergence to convergence with respect to fractal Hausdorff measures? Together with Renato Lucà (BCAM, Spain), we tackled these two questions. In the talk, I will present our results after explaining the basics of the problem.

### The steady-state degree and mixed volume of a chemical reaction network

Series
Student Algebraic Geometry Seminar
Time
Friday, March 26, 2021 - 09:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Cvetelina HillGeorgia Tech
Chemical reaction networks (CRNs), under the assumption of mass-action kinetics, are deterministic polynomial systems commonly used in systems biology. The steady-state of a CRN is the number of complex steady-states (solutions to the polynomial system), which is a measure of the algebraic complexity of solving the steady-state system. In general, the steady-state degree may be difficult to compute. Using three case studies of infinite families of networks, each generated by joining smaller networks to create larger ones, we give an upper bound to the steady-state degree of a CRN by utilizing the underlying polyhedral geometry associated with the corresponding system. In this talk I will give an overview of the necessary background for CRNs and the associated polyhedral geometry, and I will discuss the results on one of the case studies through examples.

### Equivariant completions for degenerations of toric varieties

Series
Algebra Seminar
Time
Wednesday, March 24, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Netanel FriedenbergGeorgia Tech

After reviewing classical results about existence of completions of varieties, I will talk about a class of degenerations of toric varieties which have a combinatorial classification - normal toric varieties over rank one valuation rings. I will then discuss recent results about the existence of equivariant completions of such degenerations. In particular, I will show a result from my thesis about the existence of normal equivariant completions of these degenerations.

### Dynamical sampling for burst-like forcing terms

Series
Analysis Seminar
Time
Wednesday, March 24, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Ilya KrishtalNorthern Illinois University

Dynamical sampling is a framework for studying the sampling and reconstruction problems for vectors that evolve under the action of a linear operator. In the first part of the talk I will review a few specific problems that have been part of the framework or motivated by it. In the second part of the talk I will concentrate on the problem of recovering a burst-like forcing term in an initial value problem for an abstract first order differential equation on a Hilbert space. We will see how the ideas of dynamical sampling lead to algorithms that allow one to stably and accurately approximate the burst-like portion of a forcing term as long as the background portion is sufficiently smooth.

### Speeds of hereditary properties and mutual algebricity

Series
Graph Theory Seminar
Time
Tuesday, March 23, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location