### Fall 2019 Mathematics Colloquium

###### Time and Location:

Minard 212 at 3:00 PM (refreshments at 2:30 PM in the Math Conference Room (Minard 404))

Special Colloquia or Tri-College Colloquia venues and times may vary, please consult the individual listing.

#### Tuesday, September 10 Yulia Gel (UT-Dallas)

What Do Network Local Geometry Tell Us about Functionality of Complex Networks?

Abstract: The past decade has seen an increasing interest in the application of statistical tools developed in the interdisciplinary field of network analysis to improve our understanding of complex systems and critical infrastructures, e.g., power grids, financial systems, brain networks, and ecosystems. However, most approaches in network studies still focus on global topological characteristics, and the role of local geometry in network functionality, along with its associated statistical properties, still remain largely under-investigated. In turn, the emerging tools of topological data analysis, and in particular, persistent homology, associated with the systematic study of progressively finer simplical complexes, enable to unveil some critical characteristics behind organization of complex networks and interactions of their components at multi-scale levels, which are otherwise largely inaccessible with conventional analytical approaches. In this talk we discuss the utility of integrating persistent homology, network motifs and statistical data depth concepts for more systematic, data-driven and geometrically enhanced inference for complex networks in a broad range of real-world scenarios, from power grid robustness to anomaly detection in cryptocurrencies.

#### Tuesday, September 17 Azer Akhmedov (NDSU)

Lattices of diffeomorphism groups of compact one-manifolds.

Abstract: The study of discrete subgroups of SL(2,R) has been started in the late 19th century in the works of Klein, Poincare and Fuchs. In the 20th century, a very rich and powerful theory of lattices of Lie groups has been established starting with the works of Selberg, Borel, Mostow, Margulis, Zimmer and others. A lattice of a Lie group is a discrete subgroup of finite covolume. For example, Z^n is a lattice of R^n, and SL(2,Z) (despite being non-cocompact) is a lattice of SL(2,R). A lattice tends to capture the algebraic and geometric content of the ambient Lie group. Starting with the works of Thurston, Koppel, et al. it has been observed that the diffeomorphism group Diff^r(M^1) of higher regularity r > 1 of a compact one-manifold M^1 has a much more tame behavior (compared with the homeomorphism group) and indeed does resemble Lie groups in many regards. Since then researchers have studied finitely generated subgroups of it mostly guided by the study of subgroups of Lie groups. The study of discrete subgroups of Diff^r(M) though has been initiated only recently (by myself and others) , and already many parallels with the Lie theory has been observed. We introduce the notion of a lattice of Diff^r(M) and push the theory (parallels) further.

This talk is aimed at a general audience. I will briefly review some basic and relevant notions and facts from the theory of lattices Lie groups. Then I will state main results of mine emphasizing the analogies.

#### Tuesday, September 24 Kiseop Lee (Purdue)

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Systemic Risk in Market Microstructure of Crude Oil and Gasoline Futures Prices: A Hawkes Flocking Model Approach

Abstract: We propose the Hawkes flocking model that assesses systemic risk in high-frequency processes at the two perspectives – endogeneity and interactivity. We examine the futures markets of WTI crude oil and gasoline for the past decade, and perform a comparative analysis with conditional value-at-risk as a benchmark measure. In terms of high-frequency structure, we derive the empirical findings. The endoge- nous systemic risk in WTI was significantly higher than that in gasoline, and the level at which gasoline affects WTI was constantly higher than in the opposite case. Moreover, although the relative influence’s degree was asymmetric, its difference has gradually reduced.

#### Tuesday, October 15 Deborshee Sen (Duke)

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Monte Carlo algorithms on distributed architectures

Abstract: The collection of vast amounts of data and cheap access to multiple computer cores are important aspects of the modern scientific era. This leads to the necessity of developing distributed algorithms to tackle such problems. Markov chain Monte Carlo (MCMC) algorithms are a very popular class of algorithms which are used in Bayesian statistics and elsewhere. However, these lose attractive theoretical guarantees when implemented on parallel architectures. Sequential Monte Carlo (SMC) algorithms are an alternate class which preserves theoretical guarantees on distributed architectures. I will review some aspects of scaling up MCMC algorithms to big data and introduce distributed SMC algorithms. I will also talk about the optimal way of distributing computing resources across a network in this context.

#### Tuesday, October 22 Tri-College Colloquium

Damiano Fulghesu at NDSU

On the Intersection Ring of the Moduli Space of Irreducible Plane Curves

Abstract: Moduli spaces of curves are without a doubt a very hot topic in modern algebraic geometry. The study of the intersection theory of such moduli spaces started in the 80s and is still seeing significant developments, especially in connection with enumerative geometry. In this talk, we will tackle the special case of irreducible plane curves of a fixed degree. We will show some general results about the integral intersection ring of their moduli space and we will present a more detailed structure of the intersection ring of the moduli space of irreducible cubics and quartics. This work is a collaboration with Andrea Di Lorenzo and Angelo Vistoli.

#### Tuesday, November 12 Tri-College Colloquium

Torin Greenwood at MSUM

Examples in Analytic Combinatorics

TIME and ROOM : 3:00 in MacLean 274 (Refreshments starting at 2:30 in The Presidents Conference Room MacLean 268)

Abstract: Discovering exact formulae for the nth term in a sequence of numbers can be challenging. Even when it is possible, the formulae are often too complicated to be practical. Instead, a central goal in analytic combinatorics is to find simple approximate expressions for sequences. This process begins with the “symbolic method,” a codified set of rules for converting sequences into their corresponding generating functions. Once we are equipped with the generating function, singularity analysis can lead to asymptotics. In this talk, we will explore several examples of this machinery, including applications to molecular biology and RNA.