Seminars archive

Most high-dimensional nonconvex optimization problems cannot be solved to optimality. It has been observed, however, that deep neural networks have a benign geometry that permits standard optimization methods to find acceptable solutions. However, solution times can be exorbitant. In addition, not all minimizers of the neural network loss functions are equally desirable, as some lead to prediction systems with better generalization properties than others. In this talk we discuss classical and new optimization methods in the light of these observations, and conclude with some open questions. BIO: Jorge Nocedal is the Walter P. Murphy Professor in the Department of Industrial Engineering and Management Sciences at Northwestern University. His research is in optimization, both deterministic and stochastic, and with emphasis on very large-scale problems. His current work is driven by applications in machine learning. He is a SIAM Fellow, was awarded the 2012 George B. Dantzig Prize, and the 2017 Von Neumann Theory Prize for contributions to theory and algorithms of optimization.

The Gauss Circle Problem asks how many lattice points are contained in a circle centered at the origin or radius R. A simple geometric argument establishes that this count is approximated by the area $\pi R^2$, with an error bounded by the perimeter $O(R)$. \\ ``Arithmetic statistics" is about arithmetic objects -- number fields, ideal class groups, and so on. Bhargava and many others have recently proved spectacular theorems by parametrizing such objects in terms of lattice points, and then using geometry to counting the lattice points. \\ Meanwhile, harmonic analysts have long known that you can do better than an error of $O(R)$ in Gauss's circle problem. I will describe a program to import such improvements into arithmetic statistics, and give an overview of the number theoretic results we hope to obtain. \\ This is ongoing joint work with Theresa Anderson and Takashi Taniguchi.

Erd\H{o}s, Faudree, Rousseau and Schelp observed the following fact for every fixed integer $k \geq 2$: Every graph on $n \geq k-1$ vertices with at least $(k-1)(n-k+2)+(k-2)(k-3)/2$ edges contains a subgraph with minimum degree at least k. However, there are examples in which the whole graph is the only such subgraph. Erdos et al. conjectured that having just one more edge implies the existence of a subgraph on at most $(1-\epsilon_k)n$ vertices with minimum degree at least $k$, where $\epsilon_k>0$ depends only on $k$. In this talk, we will sketch a proof of this conjecture. The proof relies on ideas from a paper of Mousset, Noever and $\check{S}kori\acute{c}$. We will discuss these ideas and how they can be extended to give a proof of the full conjecture.

The problem of bounding the total number of cycles in a graph is more than a century old. In 1897, Ahrens proved bounds on the number of cycles using the cyclomatic number of the graph and since then many results have appeared on the maximum number of cycles in graphs with different restrictions. <br> <br> In this talk I will consider a problem of maximizing the number of cycles for three classes of graphs: graphs with given number of edges (and unrestricted number of vertices), graphs with a given average degree, and graphs without a clique of a specific size. For the first two classes I will show that the maximum number of cycles in a graph has bounds exponential in the number of edges of the graph. I will also present exponentially tight bounds for the maximum number of cycles in a multigraph with a fixed number of vertices and edges.

For any fundamental discriminant $D > 0$, Hirzebruch and Zagier constructed a modular form of weight two whose Fourier coefficients are corrections of the Hurwitz class number sums $\sum_{r^2 \equiv 4n \, (D)} H((4n - r^2) / D)$. In this talk, we will discuss how one can reinterpret their result and remove the condition that $D$ is fundamental by working instead with vector-valued modular forms for Weil representations.

The gambler's fallacy is the erroneous belief that if (for example) a coin comes up heads often, then in the next toss it is more likely to be tails. In recent work with Robert Lemke Oliver, we found that funnily the primes exhibit a kind of gambler's fallacy: for example, consecutive primes do not like to have the same last digit. I'll show some of the data on this, and explain what we think is going on.

Probability theory rests on seemingly firm axioms, yet simple questions continue to confound philosophers and intrigue the public. We'll examine some of these questions and try to determine whether they uncover real problems with the foundations of probability theory, or just challenges to our flawed human intuition.

Rapid development of next generation sequencing technologies produces a plethora of large-scale epigenome profiling data. Given the quantity of available epigenome datasets, obtaining a clear and comprehensive picture of the underlying regulatory network remains a challenge. The multitude of cell type heterogeneity and temporal changes in the epigenome make it impossible to assay all epigenome events for each type of cell. Computational model shows its advantages in capturing intrinsic correlations among epigenetic features and adaptively predicting epigenome marks in a dynamic scenario. Current progress in machine learning provides opportunities to uncover higher level patterns of epigenome interactions and integrating regulatory signals from different resources. My works aim to utilize public data resources to characterize, predict and understand the epigenome-wide regulatory relationship. The first part of my work is a novel computational model to predict in vivo transcription factor (TF) binding using base-pair resolution methylation data. The model combines cell-type specific methylation patterns and static genomic features, and accurately predicts binding sites of a variety of TFs among diverse cell types. The second part of my work is a computational framework to integrate sequence, gene expression and epigenome data for genome wide TF binding prediction. This extended supervised framework integrates motif features, context-specific gene expression and chromatin accessibility profiles across multiple cell types and scale up the TF prediction task beyond the limits of candidate sites with limited known motifs. The third part of my work is a novel computational strategy for functional annotation of non-coding genomic regions. It takes advantage of the newly emerged, genome-wide and tissue-specific expression quantitative trait loci (eQTL) information to help annotate a set of genomic intervals in terms of transcription regulation. This method builds a bridge connecting genomic intervals with biological pathways and pre-defined biological-meaningful gene sets. Tissue specificity analysis provides additional evidence of the distinct roles of different tissues in the disease mechanisms

SQL (Structural Query Language) and Relational Algebra are two important languages to manipulate relational database. SQL is an international standard language used to express queries on data stored in a database. Relational Algebra is a Mathematical language with operations on sets. SQL queries are first translated to an equivalent expression in Relational Algebra in query processing. The thesis explores the translation from SQL to Relational Algebra to gain a deeper understanding in database systems. The thesis begins with an introduction to the problem (including motivation to working on the translation), the related background knowledge to handle the translation, and follows with the project design. It then discusses the evaluation of the result, reflects on my learning experience from the project, and makes suggestion about further improvement.