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Title: Instanton Floer homology and Heegaard diagrams Speaker: Zhenkun Li Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: D. Savchuk

Instanton Floer homology was introduced by Floer in 1980s and has become a power invariants for three manifolds and knots since then. It has lead to many milestone results, such as the approval of Property \(P\) conjecture. Heegaard diagrams, on the other hand, is a combinatorial methods to describe 3-manifolds. In principle, Heegaard diagrams determine 3-manifolds and hence determine their instanton Floer homology as well. However, no explicit relations between these two objects were known before. In this talk, for a 3-manifold \(Y\), I will talk about how to extract some information about the instanton Floer homology of \(Y\) from the Heegaard diagrams of \(Y\). Additionally, I will explore some of the applications and future directions of this work. This is a joint work with Baldwin and Ye.

Title: Amenability of groups and its implications for random walks and asymptotic properties of group actions Speaker: Maksym Chaudkhari, University of Texas at Austin Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: D. Savchuk

Amenability is an analytic property of groups introduced by von Neumann in response to the famous Banach-Tarski paradox. We will review the applications of amenability in dynamical systems and in probability theory. Furthermore, we are going to discuss extensions of several classical results, including Kesten's theorem, from the case of discrete amenable groups to the general setting of amenable non-locally compact topological groups. The talk is based on a joint work with Kate Juschenko and Friedrich Martin Schneider.

Title: Polynomials Systems over Finite Fields and Error Correcting Codes Speaker: Sudhir Ghorpade Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: G. Micheli

It is elementary and well known that a nonzero polynomial in one variable of degree d with coefficients in a field \(F\) has at most \(d\) zeros in \(F\). It is meaningful to ask similar questions for systems of several polynomials in several variables of a fixed degree, provided the base field F is finite. These questions become particularly interesting and challenging when one restricts to polynomials that are homogeneous, and considers zeros (other than the origin) that are non-proportional to each other. More precisely, we consider the following question: Given a system of a fixed number of linearly independent homogeneous polynomial equations of a fixed degree with coefficients in a fixed finite field \(F\), what is the maximum number of common zeros they can have in the corresponding protective space over \(F\)?

The case of a single homogeneous polynomial (or in geometric terms, a projective hypersurface) corresponds to a classical inequality proved by Serre in 1989. For the general case, an elaborate conjecture was made by Tsfasman and Boguslavsky, which was open for almost two decades. Recently significant progress in this direction has been made, and it is shown that while the Tsfasman–Boguslavsky Conjecture is true in certain cases, it can be false in general. Some new conjectures have also been proposed. We will give a motivated outline of these developments. These conjectures are intimately related to questions in the theory of linear error correcting codes as well as the problem of counting points of sections of Veronese varieties by linear subvarieties of a fixed dimension. We shall also outline these connections.

This talk is mainly based on joint works with Mrinmoy Datta and with Peter Beelen and Mrinmoy Datta.

Title: Asymptotic topology: coarse equivalence, absolute extensors, asymptotic dimension Speaker: Mykhailo Zarichnyi, The Ivan Franko National University of Lviv Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: D. Savchuk

The asymptotic topology (or coarse geometry) deals with the large scale properties of metric spaces and more general objects, the so-called coarse spaces. One of the most important asymptotic invariants is the asymptotic dimension introduced by M. Gromov who used it in his investigations of finitely generated groups. Different notions of classical topology and metric geometry have their counterparts in the asymptotic topology. Connections between the classical and asymptotic cases can be established by means of the various coronas, i.e., the remainders of compactifications of metric spaces with respect to special algebras of functions (in particular, the Higson corona).

The talk is mostly devoted to properties of the asymptotic dimension, in particular, existence of universal spaces for this dimension. There will also be considered absolute extensors in asymptotic categories. Together with review of known results, new results will be presented.

Title: The fifty-fifty conjecture for elliptic curves Speaker: Dorian Goldfeld Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: G. Micheli

In 1979, I conjectured that 50% of elliptic curves over \(Q\) have only finitely many rational points while the other 50% have infinitely many rational points with rank \(=1\). Computational evidence obtained over the years (after calculating hundreds of thousands of elliptic curves) showed that about 20% of elliptic curves have infinitely many rational points and are of rank \(2\), i.e., there are two independent points each of which generates infinitely many other points on the curve. This data contradicted my conjecture and for many years people did not believe the conjecture. In this talk I will discuss recent progress on the fifty-fifty conjecture which strongly supports that the conjecture is true. Here is a quote from Peter Sarnak: “The data will eventually turn around, when elliptic curves with larger coefficients could be calculated in substantial numbers.”

Note: This lecture has been postcancelled.

Title: Semifields, and their relations to coding theory and cryptography Speaker: Lukas Kölsch Time: 3:30pm–4:30pm Place: CMC 130 Sponsor: H. Pendharkar

Finite semifields are finite dimensional non-associative algebras over a finite field. They have a rich history going back to the work of Dickson in the start of the 20th century, mostly due to their connections to finite geometry and difference sets. Recently, applications to coding theory and code-based cryptography have renewed interest in semifields.

In this talk, I am going to present new constructions and structural results on semifields (joint work with Faruk Gologlu) and their impacts in coding theory and cryptography.

Title: Framework for assessing and easing global COVID-19 travel restrictions Speaker: Thien-Minh Le, Department of Biostatistics, Harvard T.H. Chan School of Public Health Time: 2:30pm–3:30pm Place: CMC 130 or Teams Sponsor L. Lu

In the second half of February 2020, millions of people around the globe found themselves stranded outside of their home countries or unable to undertake essential business or personal travel. This was due to airline travel bans that governments implemented worldwide in an attempt to limit the spread of a new coronavirus. Despite their best efforts in using travel restrictions to protect public health, by April 15, 2020, at least 175 countries worldwide had reported their first confirmed positive COVID-19 case. Cases then started to surge all over the world and it became evident that the pandemic was not going to be short-lived. Although various types of travel restrictions were implemented, their effectiveness in balancing health outcomes and travel is not well understood. Here we develop a flexible network metapopulation model and propose two global traffic-regulation policies with the goal of simultaneously controlling the pandemic while avoiding excessive shutdowns and damage to the global economy. We then compare their effectiveness with existing policies. Based on simulation and theoretical insights we find that, under our proposed policy, international airline travel may resume up to 58% of the pre-pandemic level with pandemic control comparable to that of a complete shutdown of all airline travel. Our findings suggest that lifting travel restrictions during the pandemic is possible if done appropriately.

Title: Riemann-Hilbert Problems and Structured Determinants in Random Matrix Theory and Statistical Mechanics Speaker: Roozbeh Gharakhloo, University of Michigan Time: 3:45pm–4:45pm Place: CMC 109 Sponsor: H. Pendharkar

In the last three decades or so, the Riemann-Hilbert approach has established itself as a powerful tool for obtaining precise asymptotic information about problems arising in various disciplines. These applications range from the theory of integrable nonlinear PDEs and random growth models, to random matrix theory, statistical physics and analytic number theory. Structured determinants are quite prevalent as they characterize principal objects of interest in various fields of research, especially in random matrix theory and statistical mechanics. Aside from the (pure) Toeplitz and Hankel determinants, there has been a growing interest in recent years in studying other deformed structured determinants. Among those are Toeplitz\(+\)Hankel, bordered Toeplitz, bordered Hankel, and "\(pj+qk\)" or the so-called slant-determinants. An \(m\)-bordered structured determinant has a single structure for all its entries except for \(m\)-columns, in which the entries correspond to different generating functions, while "\(pj+qk\)"-determinants are generalizations of Toeplitz and Hankel determinants where the \(jk\)-th matrix element is the \((pj+qk)\)-th Fourier coefficient of a generating function, \(p,q\in\mathbb{Z}\). In my talk, I will describe some aspects of these structured determinants and their Riemann-Hilbert characterizations, mainly focusing on:

Title: Identifying Group-wise Mixtures in High-Dimensional Multi-Task Learning Model with an Application in Constructing Brain Subnetworks Speaker: Yisha Yao, Department of Biostatistics, Yale University Time: 2:30pm–3:30pm Place: CMC 13 or Teams Sponsor L. Lu

Many important applications require the identification of latent clusters, communities, or subnetwork structures. Existing methods for detecting subnetworks tend to depend on the assumption that the observable information manifests the latent subnetwork structures. In practice, this assumption may not hold. We propose and develop a method that does not require this assumption. The key idea is to incorporate group-specific mixtures into a high-dimensional multi-task learning model which enables us to infer the latent subnetworks by detecting the unknown mixtures among the groups of coefficients. We develop an efficient algorithm to fit our model and provide both theoretical guarantees and extensive simulation results to support the convergence of the algorithm and accuracy of the estimates. To demonstrate the practical utility of our method, we present an analysis of imaging genetic data and elaborate on how we integrate genetic information for constructing brain subnetworks. This is feasible because gene-brain interactions carry useful information for the brain subnetworks. Our brain subnetwork map complements the existing brain map that does not make use of genetic information.

Title: Long transient dynamics in stochastic systems with applications Speaker: Weiwei Qi Time: 3:45pm–4:45pm Place: NES 104 Sponsor: H. Pendharkar

Systems' essential dynamics over finite time are referred to as transient dynamics. They are thought to be more relevant to physically observed dynamical behaviors. In this talk, we are mainly concerned about long transient dynamics in stochastic systems. In particular, we consider two different models with small noise perturbation arising from population dynamics, where species only coexist over a long finite time period and go to extinction in the long run. To capture such transient persistent dynamics, we use quasi-stationary distributions (QSDs) and study their noise-vanishing asymptotic. Since QSD is closely related to the spectrum of the Fokker-Planck operator, our method is mainly based on PDE analysis. It is worthwhile to mention that the second-order coefficients of the Fokker-Planck operator are degenerate on the boundary for any fixed noise and vanish in the zero-noise limit, resulting in essential technical difficulties. In the end of the talk, I would list some topics for future work.

Note: This lecture has been postponed until further notice.

Title: Nonparametric estimation of regression functions, densities, density derivatives and quantiles Speaker: Olga Savchuk, University of South Florida Time: 2:30pm–3:30pm Place: CMC 120 or Teams Sponsor L. Lu

Nonparametric methods for estimating densities, regression functions, regression function derivatives and population quantiles require choosing a smoothing parameter that is frequently called the bandwidth. The bandwidth selection problems are discussed and some of the developed bandwidth selection methods are outlined. The problem of estimating the second derivative of a regression function is embedded in the framework of estimating the residual stress in SiC thin films used in semiconductor industry. The approaches for estimating the population quantiles are outlined and the problem of selecting the bandwidth for the kernel quantile estimator is discussed.

Title: Topological quantum numbers Speaker: Gamal Mograby Time: 3:45pm–4:45pm Place: CMC 109 Sponsor: H. Pendharkar

The first part of the talk will be concerned with the spectral analysis of a new class of fractal-type diamond graphs and provide a gap-labeling theorem in the sense of Bellissard. We will show that labeling the spectral gaps by the integrated density of states provides a set of topological quantum numbers that reflect the branching parameter of the graph construction and the decimation structure.

The second part of the talk is a gentle introduction to several research topics ranging from analysis on fractals through quantum walks on graphs to perfect quantum state transfer.

Title: Utilizing Healthcare Data for Precision Medicine Speaker: Xuan Wang, Department of Biomedical Informatics/Department of Biostatistics, Harvard University Time: 2:30pm–3:30pm Place: CMC 13 or Teams Sponsor L. Lu

The wide adoption of electronic health records (EHR) systems has led to the availability of large clinical datasets available for precision medicine research. EHR data is a valuable new source for deriving real-word, data-driven prediction models of disease risk and treatment response. However, they also come with analytical difficulties. Precise information on clinical outcomes is usually not readily available and requires labor intensive manual chart review. Integrating and generalizing information across healthcare systems is also challenging due to heterogeneity and privacy. In this talk, Iï¿½ll discuss analytical approaches for mining EHR data with a focus on risk prediction and transportability. These methods will be illustrated using EHR data from multiple healthcare centers. Yet, randomized clinical trial remains the gold standard to evaluate the new treatments but time consuming. I am also going to talk about how to evaluate the effectiveness of an early/easily measured surrogate for studying the treatment effect on the primary outcome.

Title: A generalized likelihood ratio test and its applications in genetics and non-linear regression Speaker: Abhijit Mandal, Department of Mathematical Sciences, University of Texas at El Paso Time: 2:30pm–3:30pm Place: CMC 130 or Teams Sponsor L. Lu

The single locus association analysis is a common approach to detect variants in genome-wide association studies (GWASs). But it often fails to detect variants with small effect sizes and cannot capture the joint effects of these variants. We have proposed a novel method for the multilocus association analysis that provides a powerful test by jointly modeling the variants within a gene. In this talk, we present a flexible nonparametric model for the association between these variants and the related covariates and the trait to avoid power loss due to model misspecification. The proposed test is computationally efficient to apply at a GWAS level. We analyzed a dataset from the Atherosclerosis Risk in Communities (ARIC) study to detect genes associated with pulmonary function in Caucasians. We also applied our method in testing additive components in non-linear regression using the generalized additive model.

Title: The Yang-Baxter equation and quantum invariants of knots and \(3\)-manifolds Speaker: Emanuele Zappala, Yale University Time: 3:00pm–4:00pm Place: CMC 130 Sponsor M. Elhamdadi

In this colloquium talk I will give an overview of several topics in low-dimensional topology and quantum algebra. The talk will revolve around the Yang-Baxter equation, which I will present from a knot theoretic perspective (pertaining to operator invariants of knots/links and representations of the braid group), and a statistical mechanics perspective (related to scattering theory and \(S\)-matrices). I will also present some constructions of \(3\)-manifold invariants (due to Turaev-Viro) closely related to knot theory, and that have found profound applications in quantum gravity. Lastly, I will consider some recent joint work with Mohamed Elhamdadi concerning new correspondences between \(n\)-Lie algebras and Yang-Baxter operators, and I will describe certain results for their second cohomology groups.

This talk will lack most of the technical details, but will hopefully give a useful overview of several topics in low-dimensional and quantum topology that are of interest in theoretical physics as well.

Title: Success with less: erasure recovery and error correction using fewer bits Speaker: Gretchen Matthews Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: G. Micheli

Decoding algorithms for error-correcting codes typically take as input all symbols of a received word and attempt to determine the original codeword. In fractional decoding, only an \(\alpha\)-proportion of symbols are used where \(\alpha<1\).

Linear exact repair recovers erasures in codes over field extensions while limiting the amount of subsymbols shared over the network. In this talk, we will discuss such strategies for getting by with fewer bits than are typically used, focusing on evaluation codes and codes from curves.

Title: Intrinsic geometry, topology and complexity of 3-manifolds Speaker: Anastasiia Tsvietkova, Rutgers University Time: 3:00pm–4:00pm Place: NES 104 Sponsor: N. Jonoska

Every 3-manifold can be canonically decomposed into pieces, and each piece has a certain geometric structure (Geometrization Theorem). Thus, on a global scale, one can match topological information for the manifold with the respective geometry. However, on a local scale, i.e. intrinsically, the interplay between geometric, topological, and combinatorial properties of a 3-manifold often is not well understood. In this talk, we will focus on one topic in this framework: how the number of surfaces embedded in a 3-manifold is related to the hyperbolic volume and combinatorial "complexity" (number of crossings or number of tetrahedra) of a 3-manifold.

While results of this nature are of interest in low-dimensional geometry and topology, the obtained insight can help to establish connections with other areas of mathematics. Among these areas are quantum topology, differential geometry, representation theory, number theory and computational complexity theory. If the time allows, we will briefly discuss some of these connections.

Title: Borsuk–Ulam and the coindex of embedding spaces Speaker: Michael Harrison, Institute for Advanced Study Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: N. Jonoska

The celebrated Borsuk–Ulam Theorem states that for any continuous map \(f\) from the \(n\)-dimensional sphere to \(R^n\), there exists \(x\) in \(S^n\) such that \(f(x)=f(-x)\). Equivalently, any continuous odd map \(S^n\) to \(R^n\) must hit zero. After presenting some neat, elementary applications of this result, I will discuss how the Borsuk–Ulam theorem can be used to give a computable obstruction to embedding simplicial complexes into Euclidean space, with several examples in low dimensions. With a generalization of this result (joint work with Florian Frick), we show that no embedding of \(RP^2\) into \(R^4\) can be deformed (through embeddings) to its mirror image. This is a particular example of a more general result which seeks to quantify the richness of a space of embeddings in terms of its \(Z/2\)-coindex. I will briefly discuss several other applications of our techniques to the studies of coupled embeddings and \(k\)-regular embeddings.

Title: Symmetric knots and involutions of the 4-sphere Speaker: Keegan Boyle, University of British Columbia Time: 3:00pm–4:00pm Place: NES 104 Sponsor: N. Jonoska

In this talk I will discuss involutions of the 4-sphere. One pathway toward studying these involutions is via symmetric embedded surfaces, and restricting such a surface to an invariant 3-sphere leaves us with a symmetric knot. I will state several new theorems about symmetric knots and the symmetric surfaces which they bound in the 4-ball, and briefly describe some of the techniques used to prove these theorems. This talk covers results from several papers, which are joint work with Ahmad Issa, Antonio Alfieri, or Wenzhao Chen.

Title: 3-manifolds, knots, and instanton Floer homology Speaker: Zhenkun Li, Stanford University Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: N. Jonoska

Instanton Floer homology was introduced by Floer in 1980s. It is a powerful invariants for 3-manifolds and knots, which serves as a bridge to bring together different aspects of 3-dimensional topology. Many milestone results has been established with the help of instanton theory, such as the approval of Property P conjecture and the establishment of the fact that Khovanov homology detects the unknot. In this talk, I will first present of my work on study the basic aspects of instanton Floer homology and many new applications of instanton Floer homology, such as establishing new detection results for Khovanov homology and understanding the SU(2)-representations of the fundamental groups of 3-manifolds.

Title: Bridging applied and quantitative topology Speaker: Henry Adams, Colorado State University Time: 2:00pm–3:00pm Place: CMC 120 Sponsor: N. Jonoska

The Gromov-Hausdorff distance is a notion of dissimilarity between two datasets or between two metric spaces. It is an important tool in geometry, but notoriously difficult to compute. I will show how to provide new lower bounds on the Gromov-Hausdorff distance between unit spheres of different dimensions by combining Borsuk-Ulam theorems with Vietoris-Rips complexes. This joint work with 15 coauthors stems from a large collaboration that I organized, and is available at https://arxiv.org/abs/2301.00246. Many questions remain open!