Travis Mandel

University of Edinburgh School of Mathematics
Room 5401, James Clerk Maxwell Building
King's Buildings
Edinburgh, EH9 3JZ

Email: Travis.Mandel@ed.ac.uk


About me: I am currently a postdoc at the University of Edinburgh, supported on Ben Davison's ERC grant. I was a postdoc at the University of Utah from July 2015-June 2018. I spent the 2014-2015 academic year as a postdoc at the Center for Quantum Geometry of Moduli Spaces (QGM) in Aarhus, Denmark. I earned my Ph.D. in math from UT Austin in May 2014, under the supervision of Sean Keel. I received a B.S. in math and physics from Tulane University in 2008.

Click here for my CV

Research Interests

My interests include:

In other words, I am interested in the Gross-Siebert program (especially as it applies to cluster varieties) along with some new extensions and refinements of the program.


More detail: Mirror symmetry is a phenomenon first observed by string theorists which relates algebro-gemetric structures (like vector bundles on subvarieties) for one space to symplectic features (like Lagrangian subspaces and holomorphic disks) for another. This geometric data for each space can be represented in terms of some combinatorial data like tropical curves. I am interested in using this tropical data to develop structures on the algebraic side that are mirror to various structures on the symplectic side.

Cluster varieties are a special type of space constructed by gluing together many copies of algebraic tori (ℂ*)n via certain birational maps called mutations. These spaces were motivated by structures arising in the study of representation theory and Teichmüller theory, and particularly by a desire to understand certain canonical bases and positivity properties arising in the study of quantum groups.

The tropical approach to mirror symmetry has been particularly fruitful when applied to cluster varieties. Here, certain generating functions for the tropical data can be used to build the canonical "theta" bases for cluster algebras. Ben Davison and I recently accomplished this for quantum cluster varieties, using ideas from the DT theory of quivers to prove the conjectured strong positivity properties.

According to mirror symmetry, these theta bases should be determined by certain Gromov-Witten numbers (i.e., counts of holomorphic curves) in the mirror/Langlands dual cluster variety. I have proved this result (the Frobenius structure conjecture) for the classical theta functions using new results (developed with Helge Ruddat) on the correspondence between counts of tropical curves and counts of holomorphic curves. One of my current goals is to prove a refined version of this, expressing the quantum theta functions in terms of certain open Gromov-Witten invariants.


Papers

Also see my pages on arXiv and Google Scholar.

Publsihed: Preprints:

Notes from some talks I've given


Teaching

I am not currently teaching, but here is a list of courses I have taught in the past:

At University of Utah:
  • Spring 2018: Calculus III (Math 2210-001). Syllabus
  • Fall 2017: Foundations of Analysis II (Math 3220-001). Syllabus. Notes
  • Spring 2017: Calculus III (2210-006). Syllabus
  • Fall 2016: Foundations of Analysis I (3210-001). Syllabus
  • Spring 2016: Calculus III (2210-003). Syllabus
  • Fall 2015: Calculus II (1220-004).

    At University of Aarhus (QGM):
  • Fall 2014: Graduate course on mirror symmetry and cluster algebras. Here are some (incomplete) notes meant to accompany that course.

    At University of Texas at Austin (teaching assistant and grader positions):
  • Spring 2014: Teaching assistant for Integral Calculus for Science (M 408S).
  • Fall 2013: Grading for Algebraic Structures (M 373K, 57500 and 57510) and for Curves and Surfaces (M 365G, 57475).
  • Summer 2012: Teaching assistant for Integral Calculus (M 408L).
  • Summer 2010: Grader for Linear Algebra and for Real Analysis.
  • Spring 2010 and Fall 2009: Supplemental instructor for Differential Calculus (M 408K).
  • Spring 2009 and Fall 2008: Teaching Assistant for Differential and Integral Calculus (M 408C).