Travis Mandel

Department of Mathematics
University of Oklahoma
Norman, OK 73019-3103

Office: PHSC 918

Email: tmandel@ou.edu

About me: I am an assistant professor in the Mathematics Department at the University of Oklahoma. I was previously a postdoc at the University of Edinburgh from 2018-2020, at the University of Utah from 2015-2018, and at the Center for Quantum Geometry of Moduli Spaces (QGM) in Aarhus, Denmark from 2014-2015. 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 research generally lies in the realm of algebraic geometry, but aspects of my work also touch on ideas from algebraic combinatorics, representation theory, symplectic geometry, and mathematical physics.

I am especially interested in tools coming from mirror symmetry, a phenomenon first observed by string theorists which relates the algebro-geometric structure of one space to the symplectic structure of a "mirror" space. The Gross-Siebert program uses tropical geometry (a sort of piecewise-linear skeleton of geometry) to create a combinatorial framework for understanding the structures arising in mirror symmetry. Much of my work is based on applying these tools from the Gross-Siebert program towards problems concerning cluster algebras.

Cluster algebras are a class of combinatorially defined commutative algebras which admit many different local coordinate systems, called clusters, related to each other via certain birational maps called mutations. They were defined by Fomin-Zelevinsky with the goal of better understanding certain canonical bases and positivity phenomena observed by Lusztig in the context of quantum groups and representation theory. Work of Gross-Hacking-Keel-Kontsevich applied the Gross-Siebert program to construct canonical "theta bases" for cluster algebras, thus settling many important conjectures in cluster theory. Most of my research is centered around better understanding every aspect of these theta bases, incluing their general properties, quantization, tropicalization, and connections to Gromov-Witten theory (counting holomorphic curves) and quiver DT-theory (e.g., Euler characteristics of certain spaces of quiver representations).


Papers

Also see my pages on arXiv and Google Scholar.

Publsihed: Preprints:

Notes from some talks I've given


Teaching

Current teaching (Spring 2022): Differential and Integral Calculus III [MATH 2934-004 and 007 (honors)]


Past courses at University of Oklahoma:
  • Fall 2021: Topics in Algebra: Toric varieties and related topics (MATH-6393-001). Lecture notes on ``Introduction to toric varieties and algebraic geometry'' available here.
  • Spring 2021: Calculus and Analytic Geometry IV (Math 2443 003,004)
  • Fall 2020: Calculus and Analytic Geometry II (Math 2423 001,003)

    Here is a list of courses I have taught in the past at other universities:

    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).