The following papers were published with the support of this RTG.

Spring 2017: Carolyn Abbott is again spending the semester as a visiting research scholar at CUNY Graduate Center working with Jason Behrstock. They semester they completed the project they began the previous spring on universal acylindrical actions and have begun a new project on bounding the length of conjugators in hierarchically hyperbolic groups.

Fall 2016: Carolyn Abbott is spending the semester as a Research Fellow at MSRI in Berkeley, CA, as part of a semester-long program on geometric group theory.

Spring 2016: Carolyn Abbott is spending the semester in New York working with Jason Behrstock at the CUNY graduate center on a project related to universal actions of acylindrically hyperbolic groups.

Summer 2016: With the RTG funding, I have been able to spend the semester in New York. I am a visiting research scholar at CUNY Graduate Center, visiting Jason Behrstock. He and I have started working on a project that would not have possible otherwise. In addition, since I have not been teaching, I have been able to attend the Young Geometric Group Theorists conference in Germany and the CIMPA school on hyperbolic groups and their representations in Uruguay. While the funding for the conferences did not come from the RTG, they wouldn’t have been possible if I did not have the RA-ship. In addition, I have had the time to complete projects with two collaborators, both of which should be appearing on the arXiv shortly.

Fall 2016: With the help of the RTG funding, I have been able to spend the fall semester at MSRI for their semester-long program on geometric group theory. I have been able to complete a project with a collaborator, as well as begin multiple new projects with new people.

Spring 2016: I worked on computing the moments of the distribution of unramified G-extensions of imaginary quadratic fields for various G, verifying a conjecture of Melanie Wood.

Summer 2016: I worked on my paper. I also participated in the RTG lunch on Mondays, and some RTG funds went to my trip to the Arizona winter school and my trip to Warwick for a conference on Cohen-Lenstra Heuristics and Arithmetic statistics.

Fall 2016: I posted two papers to the ArXiV. The first paper is titled "Cohen-Lenstra Moments for Some Nonabelian Groups", which I had been working on for the past year. The second paper written collaboratively with Jack Klys is titled "The Distribution of H_8-extensions of Quadratic Fields". I also used some RTG funds to attend and present the results of the first paper at the Quebec-Maine number theory conference.

Spring 2017: I used RTG funds to travel to the 2017 Atkin workshop and Arizona winter school.

Summer 2017: I posted a third paper to the arxiv on unramified metabelian extensions using lemmermeyer factorizations. I also used RTG funds to travel to the CAAATQuafs conference and a conference and summer school at the ICTP.

Fall 2017: In addition to editting my papers based on reviewer suggestions, I began applying for postdoctoral positions. I used RTG funds to travel and present at the Maine-Quebec number theory conference and the Southern California Number Theory Day.

Spring 2016: Zach Charles is working on various applications of algebra, including to the Belgian Chocolate Problem in control theory, the Optimal Power Flow Problem in power systems, and the Pooling Problem as studied by the oil industry.

Summer 2016: The RTG funding has helped me accomplish quite a few varied things. First, it gave me time to work on the Belgian Chocolate Problem and (working with you) obtain the world record value for the parameter delta. I then presented the work on the BCP at the Optimization Research Consortium in the fall semester. There I was able to connect with other researchers and people from industry. The funding also gave me time to prepare for my specialty exam and become a dissertator. The RTG funding has also allowed me to devote time to co-organizing the applied algebra seminar this year and performing interdisciplinary research in power systems.

Spring 2016: His research concerns moduli spaces of Higgs bundles in some low-dimensional examples.

Summer 2016:

Goal: Prove existence of certain moduli spaces of logarithmic Higgs Bundles on P1 and demonstrate isomorphisms between them. This should yield two dimensional examples of Hitchen Fibrations. Extend this result to the case of vector bundles with logarithmic connections on P1.

Results: Major progress was made on the Higgs Bundle case and a paper (my thesis) is in preparation. Specifically, I now understand, using results of Carlos Simpson and others, the C-points of my moduli problems and the associated isomorphisms. Further treatment is required to deal with points valued in non-reduced rings.

Future plans: By the end of Summer 2016 I plan to finish writing up the Higgs Bundle case and begin work on the case of connections.

Spring 2016: He is planning to graduate in Spring 2016. His thesis project is on characteristic classes of cameral covers; his preprint on the subject has been posted to arxiv.org website http://arxiv.org/abs/1511.07410

Summer 2016: RTG funding hass been very helpful by giving me extra time to work on my thesis. It has also given me time to discover that certain approaches to a problem that I am working on don't work, which is at least useful knowledge to me.

**Summer 2018:** With the time made available by RTG funding, I have put two papers up on the ArXiv. One is titled “Algebraic Intersection Spaces” for which I am the sole author. It offers a potential extension to the theory of intersection spaces - an alternative way to achieve duality for singular spaces - that applies to a wider class of singular spaces than previously available. I intend to soon submit this paper to the Journal of Topology and Analysis. The other paper is titled “On the Signed Euler Characteristic Property for Subvarieties of Abelian Varieties” and is coauthored by Eva Elduque and Laurentiu Maxim. It provides a topological proof using stratified Morse theory for the signed Euler characteristic property that closed subvarieties of abelian varieties are known to satisfy.

I have also received RTG funding to cover a portion of a trip to the Institute of Mathematics of the Romanian Academy, where I gave a talk on the first paper “Algebraic Intersection Spaces”.

**Summer 2016:** Because I did not have to teach this spring owing to RTG funding and having just selected Laurentiu Maxim as my advisor at the start of the semester, I was able to dedicate a significant portion of my time to gathering the background knowledge necessary to understand and contribute to my advisor’s research. Since Professor Maxim was on sabbatical, this involved progressing through several books on the subject of singularity theory (Dimca’s Sheaves in Topology and Singularities and Topology of Hypersurfaces) and on occasion givings talks on the material in the weekly topology/singularities seminar. At the same time, the absence of teaching duties allowed me to spend more time with my courses, one of which (Topics in Ring Theory) has particular relevance to my field. The RTG funding allowed me to get the jumpstart necessary to quickly enter research level mathematics, something I would not have had had I been a teaching assistant.

Spring 2016: DANIEL HAST has a nearly-finished preprint with Matei on arithmetic statistics in function fields; he is now working on an ambitious project using Minhyong Kim’s non-abelian Chabauty method.

Summer 2016: This semester, I've been able to make significant progress on three of my research projects thanks to RTG funding: First, I've finished most of the remaining work on my project with Jordan Ellenberg on extending the p-adic Chabauty-Kim approach to Diophantine finiteness to hyperelliptic curves, and I expect we'll be able to have the paper written up before the end of the semester. Second, Vlad Matei and I have made some some final revisions to our paper on higher moments of arithmetic functions and will be ready to post the paper soon. Third, Vlad and I have made substantial progress in our joint project with Joseph Gunther on counting tetragonal curves over finite fields. In addition, I've been regularly participating in the RTG lunches

Spring 2016: She is computing Betti numbers of unordered configuration spaces of smooth, compact, complex varieties, and uncovering new phenomena in their stable and unstable values.

Summer 2016: I've given talks in GNTS and at the Graduate Student Geometry and Topology Conference on work that I've done while supported on the RTG. I've also been asked to speak about it at the Upper Midwest Commutative Algebra Colloquium. I'm also working on a preprint of this work that hopefully will be online within the next month.

Spring 2017: Last spring and summer I gave talks at the Graduate Student Topology and Geometry Conference, Upper Midwest Commutative Algebra Colloquium, Graduate Student Conference in Algebra, Geometry, and Topology, Topology Students' Workshop, and the Young Topologist's Meeting. I also participated in an AIM workshop and Representation Stability and the West Coast Algebraic Topology Summer School. I've been invited to speak at Purdue and the University of Oregon, and at the Spring 2017 AMS Sectionals at Indiana Unversity. I've also been invited to attend an Oberwolfach workshop in January 2018 on the Topology of Arrangements and Representation Stability.

I guess I should speak regarding some portion of my independent studies: I have made what feels like a substantial amount of progress in mastering Ravi Vakil's algebraic geometry notes, in the sense that I am roughly on track to have finished a first reading by the end of the term. Also, I have solved many exercises out of chapters II and III in Hartshorne. I've learned a lot of other algebra topics too, of course...

To say something really vague: The freedom that comes with being an RA means that I can devote long periods of unscheduled time towards focused thought, towards skepticism and technical thoroughness, and towards naive curiosity and adventurous questioning.

Spring 2016: ERIC RAMOS has proved several deep results in the theory of FI-modules and related categories; he has two papers already posted to the arXiv (one joint with Liping Li) and another fully written.

Summer 2016: Eric continues to work on structure theorems of FI-modules and related categories, with applications. Two of these works have been posted to the arXiv. Following this, he has begun studying configuration spaces of graphs, and stability phenomenom present in their homology groups.

Spring 2017: Eric expanded his studies on the configuration spaces of graphs. In his new work, he considers configurations of points on graphs, wherein points are allowed to collide on vertices. It is proven that the homology of these spaces provide the first non-trivial, naturally occuring, examples of FI_d-modules.

Jason is working on a projects on Borcherds product on a Picard modular surface who might developed into a thesis project.

Summer 2016:

Besides job applications and interviewing (which take me a lot of time), I have been polishing my paper, "Higher order invariants of hypersurface complements", and add some more examples. I have also converted this piece of work into my thesis. In addition, I spent some time with Tommy to discuss how more computations can be done on the case of complex hyperplane arrangements.

Summer 2016:

RTG has been a great help for me to focus on the followings:

1. sharpen up some of the results in two of my existing preprints

2. turn my work into a thesis

3. think about and work on a new project (which is a joint work with Yun Su)

Summer 2016: As for concrete results, the most notable the progress towards the completion of the paper "Random Toric Surfaces and a Threshold for Smoothness". I hope to have it on arxiv within the month, much of this semester has been spent improving this paper in various ways. This includes importantly fixing many of the minor issues that cropped up in close inspection. But as well as edits of that form, I have significantly refined the result that I had a the beginning of the semester. In addition to the actual work of preparing this paper for publication, I gave a talk this semester about the contents of this paper at the Algebraic Geometry Seminar.

In new work, my adviser Daniel Erman and I have also begun work on a project tackling the application of the results of the theory of random graphs to Stanley-Reisner ideals. We are hoping to get results about the expected values of betti table entries. In particular our preliminary work has yielded at least an idea for a proof that the expected values of the first row of the betti table follows a normal curve.

I have also used the time to move my further my studies. In additon to classes, in the past semester, I've learned the basics of tropical geometry, and gave a talk about it in the graduate algebraic geometry seminar.