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# RTG: Algebra, Geometry and Topology at the University of Utah

This NSF Research Training Grant award, RTG: Algebra, Geometry, and Topology at the University of Utah is designed to train a new generation of researchers in mathematics. It funds activities for undergraduates, graduate students and postdoctoral scholars as well as several activities in collaboration with the University of Utah chapter of the AWM (the Association for Women in Mathematics)

See the abstract here: NSF award #1840190.

This grant builds upon the previous grant, RTG: Algebraic Geometry and Topology at the University of Utah, NSF award #1246989.

Fellowship funds from this grant can only be used to support U.S. citizens, nationals, and permanent residents.

# People involved in this NSF RTG grant

#### Fall 2019

• Hanna Astephan
• Donald Chacon-Taylor
• Kristen Lee
• Faith Pearson

#### Spring 2020

• Kevin Childers
• Pinches Dirnfeld
• George Domat
• Matthew Goroff
• Hannah Hoganson
• Peter McDonald

#### Summer 2020

• Christian Klevdal
• Kristen Lee
• Peter McDonald

• Leo Herr (2019--2022): Leo's work concerns applications of Log Geometry in calculating Gromov-Witten Invariants. This extends the breadth to which these invariants may be applied to mildly singular spaces. Leo will work most directly with Y. P. Lee.
• Joshua Pollitz (2019--2022): Josh' primary research interests are studying and applying homological algebra in the context of commutative algebra and its related areas. While an RTG postdoc he plans on working with Srikanth Iyengar and other faculty members in the commutative algebra community at Utah.
• Emily Stark (2019--2020): Emily's research area is geometric group theory, and her work focuses on spaces with hyperbolic or non-positive curvature. She will work with Mladen Bestvina, Ken Bromberg, and Priyam Patel.

• Aaron Bertram: Aaron works within algebraic geometry; specifically on questions about moduli spaces related to mirror symmetry. Current interests include stable maps and Gromov-Witten theory, derived categories and Bridgeland stability conditions, and the relationship between classical Riemann surface theory and tropical curves (i.e., metric graphs).
• Mladen Bestvina: Mladen studies geometric group theory. Of particular interest is the geometry of Outer Space and other spaces associated to Out(F_n), as well as mapping class groups, Teichm\uller space and curve complexes. Other interests include: asymptotic dimension, bounded cohomology, limit groups.
• Ken Bromberg: Ken works on hyperbolic geometry and deformation spaces of Kleinian groups. He is particularly interested in the "bumping" phenomena leading to non-local connectivity of deformation spaces. Other interests: geometric group theory, actions on quasi-trees, asymptotic dimension.
• Jon Chaika: Jon works on ergodic theory, interval exchange transformations and flows on flat surfaces. He is particularly interested in diophantine approximation and also non-unique ergodicity. His other interests include properties of and actions on the parameterizing space of interval exchanges and flat surfaces, diophantine approximation in abstract settings and isomorphism problems.
• Tommaso de Fernex: Tommaso works in algebraic geometry, with focus on problems in birational geometry of higher dimensional varieties and singularity theory. Of particular interest are topics related to rationality, Fano varieties, log canonical thresholds and multiplier ideals, spaces of arcs, and non-archimedean geometry.
• Christopher Hacon: Christopher's research is focused on the birational geometry of varieties and especially on questions related to the minimal model program, both in characteristic 0 and in positive characteristics. In particular he studies global questions such as the geometry of pluricanonical maps of projective varieties and the geometry of irregular varieties, as well as local questions regarding singularities of quasi-projective varieties and their invariants such as the log canonical thresholds and minimal log discrepancies.
• Sean Howe: Sean works at the crossroads of algebraic geometry, number theory, and representation theory. His current research focuses on: 1) p-adic automorphic forms and their role in the p-adic Langlands program, and 2) arithmetic and motivic statistics. Most of his work is on problems that connect representation theoretic structures to the geometry and topology of moduli spaces.
• Srikanth Iyengar: Srikanth is interested in commutative algebra and its manifestations in other fields, especially the representation theory of finite groups, and dimensional algebras. Some of the topics he likes to think about are free resolutions (finite and infinite), structure of morphisms, Hochschild and Andre-Quillen (co)homology, and various triangulated categories linked to commutative rings.
• Y.P. Lee: Works on algebraic geometry and mathematical physics, with emphasis on Gromov-Witten theory and its relations with and applications to algebraic cobordism, birational geometry, K-theory, integrable systems, mirror symmetry, moduli of curves, and variation of Hodge structures.
• Dragan Milicic: Works in representation theory of Lie groups, specifically in analytic, algebraic and geometric aspects of representation theory of real reductive Lie groups. His current interests are focused on applications of the theory of holonomic D-modules to the study of categories of Harish-Chandra modules and related categories of representations.
• Priyam Patel: Priyam works in low-dimensional topology and geometry and geometric group theory. She is particularly interested in hyperbolic manifolds and their finite degree covering spaces, arc and curve complexes associated to hyperbolic surfaces, Teichmuller theory, right-angled Artin groups, and cube complexes. More recently, she has been studying surfaces of infinite type and their (big) mapping class groups.
• Stefan Patrikis: Stefan's research includes number theory, arithmetic geometry, and representation theory, with an emphasis on questions arising out of the Langlands program. Current work focuses on Galois representations: deformation theory, monodromy groups and independence-of-l questions, and their motivic origin (the Fontaine-Mazur conjecture).
• Gordan Savin: Works in representation theory and number theory, in particular, representations of p-adic groups and algebraic structures related to exceptional reductive groups. Current work includes research on the Gan-Gross-Prasad restriction program, and exceptional theta correspondences.
• Karl Schwede: Karl works on the border between algebraic geometry and commutative algebra. He is especially interested in singularities, their relation to the minimal model program and characteristic $p$ methods such as Frobenius splittings and tight closure theory. More recently, he has been working to develop a theory of singularities in mixed characteristic.
• Anurag Singh: Anurag studies commutative algebra, specifically questions related to local cohomology, tight closure theory, determinantal ideals, and invariant theory. Current projects include the study of local cohomology and differential operators in the integral setting.
• Peter Trapa: Peter studies representations of reductive Lie groups over local fields. Recent work focuses on relating the underlying geometry of the Local Langlands Conjectures (for example, singularities of spaces of Langlands parameters) to questions of harmonic analysis on reductive groups (for example, the classification of unitary representations).
• Kevin Wortman: Kevin works in geometric group theory, particularly on lattices in locally compact groups and their corresponding actions on nonpositively curved complexes and spaces, as well as some of their naturally occurring subspaces. Of particular interest are finiteness properties of groups and group cohomology.

We are collaborating with the University of Utah's AWM chapter. Learn more about what they are doing by clicking the image below

# Questions?

Below are the people involved in managing various aspects of this grant, and what they are involved in (what you should email them about).

### Lauren McComber

RTG Staff
General RTG management, Reimbursements, Fellowships, Insurance, Tuition

### Christopher Hacon

PI
General RTG management, RTG conferences and speaker series oversight

### Stefan Patrikis

PI
General RTG management, Pre-REU oversight, RTG Travel funding 2019-2020

### Priyam Patel

PI
General RTG management, RTG Mini-Conference winter 2020, RTG Travel funding 2019-2020

### Karl Schwede

PI
General RTG management, this website, research training seminar oversight

### Sabine Lang

AWM RTG lecture series, 2019-2020

### Janina Letz

AWM RTG Conference, May 2020

### Peter McDonald

AWM RTG Conference, May 2020

### Allechar Serrano López

AWM RTG lecture series, 2019-2020

### Aaron Bertram

Faculty
AWM RTG lecture series, 2019-2020

### Jon Chaika

Faculty
RTG mini conference, winter 2020

### Tommaso de Fernex

Faculty
RTG mini conference, winter 2020

### Sean Howe

Faculty
Pre-REU, summer 2020

### Srikanth Iyengar

Faculty
Research training seminar, 2019-2020
AWM RTG lecture series, 2019-2020