My research program is dedicated to the study of the geometry and topology of the moduli spaces of Higgs bundles, integrable systems and decorated bundles, and the geometric structures they parametrize. In particular, I am interested in understanding of branes of Higgs bundles, dualities within quiver varieties in general, and within generalized hyperpolygons in particular, with views towards applications to the Langlands program for wild Hitchin systems. In a different direction, I am interested in the appearances of geometric structures and symmetries within different areas of sciences, which has led to some publications in applied mathematics. You can see my work in each area by clicking on the links below.
all
geometry
interdisciplinary
My 45+ pieces both within geometry as well as on other topics are listed below in reverse chronological order by year. Note that authors on all of my publications appear alphabetically except in our Nature Scientific Reports paper, where authors are by contribution.
Citations to my papers can be found on Google Scholar.
Paper tags are colored as follows:
journal article conference article editorial work manuscript .
2022

applied
2022
Varun Mittal and
Laura P. Schaposnik
2022
Reliable forecasting of the housing market can provide salient insights into housing investments. Through the reinterpretation of housing data as candlesticks, we are able to utilize some of the most prominent technical indicators from the stock market to estimate future changes in the housing market. By providing an analysis of MACD, RSI, and Candlestick indicators (Bullish Engulfing, Bearish Engulfing, Hanging Man, and Hammer), we exhibit their statistical significance in making predictions for USA data sets (using Zillow Housing data), as well as for a stable housing market, a volatile housing market, and a saturated market by considering the datasets of Germany, Japan, and Canada. Moreover, we show that bearish indicators have a much higher statistical significance then bullish indicators, and we further illustrate how in less stable or more populated countries, bearish trends are only slightly more statistically present compared to bullish trends. Finally, we show how the insights gained from our trend study can help consumers save significant amounts of money.

applied
2022
Paula Bergero,
Laura P. Schaposnik and Grace Wang
2022
A dramatic increase in the number of outbreaks of Dengue has recently been reported, and climate change is likely to extend the geographical spread of the disease. In this context, this paper shows how a neural network approach can incorporate Dengue and COVID19 data as well as external factors (such as social behaviour or climate variables), to develop predictive models that could improve our knowledge and provide useful tools for health policy makers. Through the use of neural networks with different social and natural parameters, in this paper we define a Correlation Model through which we show that the number of cases of COVID19 and Dengue have very similar trends. We then illustrate the relevance of our model by extending it to a Long shortterm memory model (LSTM) that incorporates both diseases, and using this to estimate Dengue infections by using COVID19 data in countries that lack sufficient Dengue data.

geometry
2022
Laura P. Schaposnik
2022
This short note gives an overview of generalized hyperpolygons introduced by the author and Steven Rayan, and some directions of interest related to them with particular attention given to their relation to Higgs bundles and their interpretation within Ftheory. The manuscript has been prepared for the ZAG volume, is based on the authors’ talk in the ZAG seminar series during the COVID19 pandemic, and contains no new results.

applied
2022
Nature Scientific Reports, volume 12, Article number: 14536 (2022) 2022
We create a novel Physarum Steiner algorithm designed to solve the Euclidean Steiner tree problem. Physarum is a unicellular slime mold with the ability to form networks and fuse with other Physarum organisms. We use the simplicity and fusion of Physarum to create large swarms which independently operate to solve the Steiner problem. The Physarum Steiner tree algorithm then utilizes a swarm of Physarum organisms which gradually find terminals and fuse with each other, sharing intelligence. The algorithm is also highly capable of solving the obstacle avoidance Steiner tree problem and is a strong alternative to the current leading algorithm. The algorithm is of particular interest due to its novel approach, rectilinear properties, and ability to run on varying shapes and topological surfaces.

applied
2022
Journal of the Royal Society Interface 2022
Physarum polycephalum is a unicellular slime mold that has been intensely studied due to its ability to solve mazes, find shortest paths, generate Steiner trees, share knowledge, remember past events, and its applications to unconventional computing. The CELL model is a unicellular automaton introduced in the recent work of Gunji et al. in 2008, that models Physarum’s amoeboid motion, tentacle formation, maze solving, and network creation. In the present paper, we extend the CELL model by spawning multiple CELLs, allowing us to understand the interactions between multiple cells, and in particular, their mobility, merge speed, and cytoplasm mixing. We conclude the paper with some notes about applications of our work to modeling the rise of present day civilization from the early nomadic humans and the spread of trends and information around the world. Our study of the interactions of this unicellular organism should further the understanding of how Physarum polycephalum communicates and shares information.

applied
2022
Laura P. Schaposnik
ISBN 9798781457595 2022
An introductory set of lecture notes for a class on how to write about mathematics. The PDF of the file can be found here, where new updated and corrected versions are uploaded regularly.
2021


geometry
2021
Research in the Mathematical Sciences 2021
Given a smooth complex projective variety M and a smooth closed curve X in M such that the homomorphism of fundamental groups of X to that of M is surjective, we study the restriction map of Higgs bundles, namely from the Higgs bundles on M to those on X. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on M and X. We also consider the setup where a finite group is acting on M via holomorphic automorphisms or antiholomorphic involutions, and the curve X is preserved by this action. Branes are studied in this context.

geometry
2021
Annals of Global Analysis and Geometry 2021
Since their introduction by Beilinson–Drinfeld (Opers, 1993. arXiv math/0501398; Quantization of Hitchin’s integrable system and Hecke eigensheaves, 1991), opers have seen several generalizations. In Biswas et al. (SIGMA Symmetry Integr Geom Methods Appl 16:041, 2020), a higher rank analog was studied, named generalized Bopers, where the successive quotients of the oper filtration are allowed to have higher rank and the underlying holomorphic vector bundle is endowed with a bilinear form which is compatible with both the filtration and the oper connection. Since the definition did not encompass the even orthogonal groups, we dedicate this paper to study generalized Bopers whose structure group is SO(2n,C) and show their close relationship with geometric structures on a Riemann surface.

geometry
2021
Laura P. Schaposnik
Park City Mathematics Series 2021
These notes have been prepared as reading material for the minicourse given by the author at the "2019 Graduate Summer School" at Park City Mathematics Institute  Institute for Advanced Study. We begin by introducing Higgs bundles and their main properties (Lecture 1), and then we discuss the Hitchin fibration and its different uses (Lecture 2). The second half of the course is dedicated to studying different types of subspaces (branes) of the moduli space of complex Higgs bundles, their appearances in terms of flat connections and representations (Lecture 3), as well as correspondences between them (Lecture 4).

applied
2021
Paula Caporal,
Laura P. Schaposnik and Daniela Bozzacchi
Editorial Medica Panamericana 2021

geometry
2021
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 2021
Through the triality of SO(8,C), we study three interrelated homogeneous basis of the ring of invariant polynomials of Lie algebras, which give the basis of three Hitchin fibrations, and identify the explicit automorphisms that relate them.

geometry
2021
The Quarterly Journal of Mathematics 2021
We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a cometshaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such data, we associate an explicit meromorphic Higgs bundle on a genusg Riemann surface, where g is the number of loops in the comet, thereby embedding the Nakajima quiver variety into a Hitchin system on a punctured genusg Riemann surface (generally with positive codimension). We show that, under certain assumptions on flag types, the space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system of GelfandTsetlin type, inherited from the reduction of partial flag varieties. In the case where all flags are complete, we present the Hamiltonians explicitly. We also remark upon the discretization of the Hitchin equations given by hyperpolygons, the construction of triple branes (in the sense of KapustinWitten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents).

applied
2021
Nature Scientific Reports 2021
We present a modified agestructured SIR model based on known patterns of social contact and distancing measures within Washington, USA. We find that population agedistribution has a significant effect on disease spread and mortality rate, and contribute to the efficacy of agespecific contact and treatment measures. We consider the effect of relaxing restrictions across less vulnerable agebrackets, comparing results across selected groups of varying population parameters. Moreover, we analyze the mitigating effects of vaccinations and examine the effectiveness of agetargeted distributions. Lastly, we explore how our model can applied to other states to reflect socialdistancing policy based on different parameters and metrics.
2020

applied
2020
Alessandra Frabetti, Vladimir Salnikov and
Laura P. Schaposnik
EMS Newsletter 2020

applied
2020
Physical Review R 2020
By means of an experimental dataset, we use deep learning to implement an RGB extrapolation of emotions associated to color, and do a mathematical study of the results obtained through this neural network. In particular, we see that males typically associate a given emotion with darker colors while females with brighter colors. A similar trend was observed with older people and associations to lighter colors. Moreover, through our classification matrix, we identify which colors have weak associations to emotions and which colors are typically confused with other colors.

geometry
2020
Laura P. Schaposnik
Notices of the AMS 2020
This note is dedicated to introducing Higgs bundles and the Hitchin fibration, with a view towards their appearance within different branches of mathematics and physics, focusing in particular on the role played by the integrable system structure carried by their moduli spaces.

geometry
2020
Snapshots of modern mathematics 2020
Higgs bundles appeared a few decades ago as solutions to certain equations from physics and have attracted much attention in geometry as well as other areas of mathematics and physics. Here, we take a very informal stroll through some aspects of linear algebra that anticipate the deeper structure in the moduli space of Higgs bundles.

geometry
2020
Communications in Analysis and Geometry 2020
We examine Higgs bundles for noncompact real forms of SO(4, C) and the isogenous complex group SL(2, C) × SL(2, C). This involves a study of nonregular fibers in the corresponding Hitchin fibrations and provides interesting examples of nonabelian spectral data.

applied
2020
Physical Review Research 2020
This paper is dedicated to the study of the interaction between dynamical systems and percolation models, with views towards the study of viral infections whose virus mutate with time. Recall that rbootstrap percolation describes a deterministic process where vertices of a graph are infected once r neighbors of it are infected. We generalize this by introducing F(t)bootstrap percolation, a timedependent process where the number of neighbouring vertices which need to be infected for a disease to be transmitted is determined by a percolation function F(t) at each time t. After studying some of the basic properties of the model, we consider smallest percolating sets and construct a polynomialtimed algorithm to find one smallest minimal percolating set on finite trees for certain F(t)bootstrap percolation models.

applied
2020
Proceedings of the Royal Society A 2020
We introduce here a multitype bootstrap percolation model, which we call TBootstrap Percolation (TBP), and apply it to study information propagation in social networks. In this model, a social network is represented by a graph G whose vertices have different labels corresponding to the type of role the person plays in the network (e.g. a student, an educator, etc.). Once an initial set of vertices of G is randomly selected to be carrying a gossip (e.g. to be infected), the gossip propagates to a new vertex provided it is transmitted by a minimum threshold of vertices with different labels. By considering random graphs, which have been shown to closely represent social networks, we study different properties of the TBP model through numerical simulations, and describe its implications when applied to rumour spread, fake news, and marketing strategies.

geometry
2020
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 2020
Opers were introduced by BeilinsonDrinfeld arXiv:math.AG/0501398. In J. Math. Pures Appl. 82 (2003), 142 a higher rank analog was considered, where the successive quotients of the oper filtration are allowed to have higher rank. We dedicate this paper to introducing and studying generalized Bopers (where "B" stands for "bilinear"), obtained by endowing the underlying vector bundle with a bilinear form which is compatible with both the filtration and the connection. In particular, we study the structure of these Bopers, by considering the relationship of these structures with jet bundles and with geometric structures on a Riemann surface.
2019



geometry
2019
Transactions of the American Mathematical Society 2019
Through Cayley and Langlands type correspondences, we give a geometric description of the moduli spaces of real orthogonal and symplectic Higgs bundles of any signature in the regular fibres of the Hitchin fibration. As applications of our methods, we complete the concrete abelianization of real slices corresponding to all quasisplit real forms, and describe how extra components emerge naturally from the spectral data point of view.

geometry
2019
Oberwolfach Reports 2019
This workshop focused on interactions between the various perspectives on the moduli space of Higgs bundles over a Riemann surface. This subject draws on algebraic geometry, geometric topology, geometric analysis and mathematical physics, and the goal was to promote interactions between these various branches of the subject. The main current directions of research were well represented by the participants, and the talks included many from both senior and junior participants.

geometry
2019
Lara B. Anderson and
Laura P. Schaposnik
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 2019
This volume is based on the minicourses given during the graduate workshops organized by L. Schaposnik as part of the “Geometry and Physics of Higgs Bundles” series which began in 2015 (and on several occasions coorganized by L. Anderson). The latest of these meetings was held at the Simons Center for Geometry and Physics at Stony Brook University as part of a semesterlong program on the “Geometry and Physics of Hitchin Systems” in Spring 2019, coorganized by L. Anderson and L. Schaposnik. The goal of these workshop series has been to bring together researchers at all career stages to address open problems and to introduce young researchers to a rich and complex field that spans diverse areas of mathematics and physics. In addition, it has been a goal of the workshops to highlight the important work being done by female and minority researchers in the field and the workshops were organized in cooperation with the Association for Women in Mathematics.
2018

geometry
2018
Laura P. Schaposnik
European Journal of Mathematics 2018
We give a geometric characterisation of the topological invariants associated to SO(m,m+1)Higgs bundles through KOtheory and the Langlands correspondence between orthogonal and symplectic Hitchin systems. By defining the split orthogonal spectral data, we obtain a natural grading of the moduli space of SO(m,m+1)Higgs bundles.

geometry
2018
Journal of Geometry and Physics 2018
Middimensional (A,B,A) and (B,B,B)branes in the moduli space of flat GCconnections appearing from finite group actions on compact Riemann surfaces are studied. The geometry and topology of these spaces is then described via the corresponding Higgs bundles and Hitchin fibrations.

geometry
2018
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 2018
This brief survey aims to set the stage and summarize some of the ideas under discussion at the Workshop on Singular Geometry and Higgs Bundles in String Theory, to be held at the American Institute of Mathematics from October 30th to November 3rd, 2017. One of the most interesting aspects of the duality revolution in string theory is the understanding that gauge fields and matter representations can be described by intersection of branes. Since gauge theory is at the heart of our description of physical interactions, it has opened the door to the geometric engineering of many physical systems, and in particular those involving Higgs bundles. This note presents a curated overview of some current advances and open problems in the area, with no intention of being a complete review of the whole subject.

geometry
2018
Behaviour 2018
The development of mobile phones has largely increased human interactions. Whilst the use of these devices for communication has received significant attention, there has been little analysis of more passive interactions. Through census data on casual social groups, this work suggests a clear pattern of mobile phones being carried in people’s hands, without the person using it (that is, not looking at it). Moreover, this study suggests that when individuals join members of the opposite sex there is a clear tendency to stop holding mobile phones whilst walking. Although it is not clear why people hold their phones whilst walking in such large proportions (38 percent of solitary women, and 31 percent of solitary men), we highlight several possible explanation for holding the device, including the need to advertise status and affluence, to maintain immediate connection with friends and family, and to mitigate feelings related to anxiety and security.

geometry
2018
A Festschrift in Honour of Nigel Hitchin 2018
Given a compact Riemann surface X and a semisimple affine algebraic group G defined over C, there are moduli spaces of Higgs bundles and of connections associated to (X, G). The chapter computes the Brauer group of the smooth locus of these varieties.

applied
2018
Letters in Biomathematics 2018
Since social interactions have been shown to lead to symmetric clusters, we propose here that symmetries play a key role in epidemic modelling. Mathematical models on dary tree graphs were recently shown to be particularly effective for modelling epidemics in simple networks. To account for symmetric relations, we generalize this to a new type of networks modelled on dcliqued tree graphs, which are obtained by adding edges to regular dtrees to form dcliques. This setting gives a more realistic model for epidemic outbreaks originating within a family or classroom and which could reach a population by transmission via children in schools. Specifically, we quantify how an infection starting in a clique (e.g. family) can reach other cliques through the body of the graph (e.g. public places). Moreover, we propose and study the notion of a safe zone, a subset that has a negligible probability of infection.
2017

geometry
2017
Laura P. Schaposnik
Lecture Notes Series NUS 2017
These notes have been prepared as reading material for the minicourse that the author gave at IMS, National University of Singapore, as part of the "Summer school on the moduli space of Higgs bundles"

geometry
2017
Transactions of the AMS 2017
We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of Ltwisted GHiggs bundles, for the groups G=GL(2,C), SL(2,C) and PSL(2,C). We also determine the twisted Chern class of the regular locus, which obstructs the existence of a section of the moduli space of Ltwisted Higgs bundles of rank 2 and degree deg(L)+1. By counting orbits of the monodromy action with Z2coefficients, we obtain in a unified manner the number of components of the character varieties for the real groups G=GL(2,R), SL(2,R), PGL(2,R), PSL(2,R), as well as the number of components of the Sp(4,R)character variety with maximal Toledo invariant. We also use our results for GL(2,R) to compute the monodromy of the SO(2,2) Hitchin map and determine the components of the SO(2,2) character variety.

geometry
2017
Journal of High Energy Physics 2017
Singular limits of 6D Ftheory compactifications are often captured by Tbranes, namely a nonabelian configuration of intersecting 7branes with a nilpotent matrix of normal deformations. The long distance approximation of such 7branes is a Hitchinlike system in which simple and irregular poles emerge at marked points of the geometry. When multiple matter fields localize at the same point in the geometry, the associated Higgs field can exhibit irregular behavior, namely poles of order greater than one. This provides a geometric mechanism to engineer wild Higgs bundles. Physical constraints such as anomaly cancellation and consistent coupling to gravity also limit the order of such poles. Using this geometric formulation, we unify seemingly different wild Hitchin systems in a single framework in which orders of poles become adjustable parameters dictated by tuning gauge singlet moduli of the Ftheory model.

applied
2017
Journal of Structural Biology 2017
Large icosahedral virus capsids are composed of symmetrons, organized arrangements of capsomers. There are three types of symmetrons: disymmetrons, trisymmetrons, and pentasymmetrons, which have different shapes and are centered on the icosahedral 2fold, 3fold and 5fold axes of symmetry, respectively. Sinkovits and Baker (2010) gave a classification of all possible ways of building an icosahedral structure solely from trisymmetrons and pentasymmetrons, which requires the triangulation number T to be odd. In the present paper we incorporate disymmetrons to obtain a geometric classification of large icosahedral viruses formed by regular penta,tri, and disymmetrons, giving all mathematically consistent and theoretically possible solutions. For every class of solutions, we further provide formulas for symmetron sizes and parity restrictions on h, k, and T numbers. We also present several methods in which invariants may be used to classify a given configuration
2016

geometry
2016
Laura P. Schaposnik
AIMatters 2016 2016


geometry
2016
Advances in Theoretical and Mathematical Physics. 2016
We construct triples of commuting real structures on the moduli space of Higgs bundles, whose fixed loci are branes of type (B, A, A), (A, B, A) and (A, A, B). We study the real points through the associated spectral data and describe the topological invariants involved using KO, KR and equivariant Ktheory.

geometry
2016
Research in the Mathematical Sciences 2016
We explore relations between Higgs bundles that result from isogenies between lowdimensional Lie groups, with special attention to the spectral data for the Higgs bundles. We focus on isogenies onto SO(4,C) and SO(6,C) and their split real forms. Using fiber products of spectral curves, we obtain directly the desingularizations of the (necessarily singular) spectral curves associated to orthogonal Higgs bundles. In the case of SO(6,C) our construction can be interpreted as a new description of Recillas’ trigonal construction.

applied
2016
Physical Review E 2016
The growth of snow crystals is dependent on the temperature and saturation of the environment. In the case of dendrites, Reiter’s local twodimensional model provides a realistic approach to the study of dendrite growth. In this paper we obtain a new geometric rule that incorporates interface control, a basic mechanism of crystallization that is not taken into account in the original Reiter model. By defining two new variables, growth latency and growth direction, our improved model gives a realistic model not only for dendrite but also for plate forms.

geometry
2016
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 2016
We compute the automorphism groups of the Dolbeault, de Rham and Betti moduli spaces for the multiplicative group C star associated to a compact connected Riemann surface.
2015

geometry
2015
Laura P. Schaposnik
International Mathematics Research Notices 2015
We define and study spectral data associated to U(m,m)Higgs bundles through the Hitchin fibration. We give a new interpretation of the topological invariants involved, as well as a geometric description of the moduli space.

geometry
2015
Laura P. Schaposnik
Oberwolfach Reports 2015
This short note gives an overview of how a few conjectures and theorems of the author and collaborators fit together. It was prepared for Oberwolfach’s workshop Differentialgeometrie im Grossen, 28 June  4 July 2015, and contains no new results.
2014

geometry
2014
Communications in Mathematical Physics 2014
Through the action of antiholomorphic involutions on a compact Riemann surface Σ we construct families of (A, B, A)branes in the moduli spaces of Gc Higgs bundles on the Riemann surface. We study the geometry of these (A, B, A)branes in terms of spectral data and show they have the structure of real integrable systems.

geometry
2014
Journal of Differential Geometry 2014
We consider the integrable system on the moduli space of Higgs bundles restricted to the subvariety corresponding to representations of a surface group into certain noncompact real forms, and in doing so encounter as the fiber moduli spaces of rank bundles on a spectral curve. This contrasts with the general case where the fiber is an abelian variety.
2013

geometry
2013
Laura P. Schaposnik
International Journal of Mathematics 2013
We calculate the monodromy action on the mod 2 cohomology for SL(2, ℂ) Hitchin systems and give an application of our results in terms of the moduli space of semistable SL(2, ℝ)Higgs bundles.

geometry
2013
Laura P. Schaposnik
Oxford University DPhil 2013
We develop a new geometric method of understanding principal GHiggs bundles through their spectral data, for G a real form of a complex Lie group. In particular, we consider the case of G a split real form, as well as G = SL(2,R), U(p,p), SU(p,p), and Sp(2p,2p). Further, we give some applications of our results, and discuss open questions.