Organizers: Richard Wentworth (Math), Tristan Hubsch (Physics, Howard Univ.), Jonathan Rosenberg (Math), Amin Gholampour (Math, on leave)
Other Faculty Participants: Joel Cohen (Math, on leave), Paul Green (Math, emeritus)
When: Thursdays @ 3:30pm-4:30pm
Where: PHY 1117 (This is the Center for Particle & String Theory, located in what used to be the physics chair's office in the Toll Physics Building adjacent to the Mathematics department.)

This interdisciplinary RIT will aim to foster interactions between mathematicians and physicists on topics of mutual interest.  It will roughly follow the example of a similar RIT from 2010-2011 and from the last three years. The topic for 2016-17 was mirror symmetry.

It is not assumed that participants already be knowledgeable in both math and physics, just in some aspect of one or the other. Relevant math topics are differential geometry, representation theory, algebraic topology, and algebraic geometry. Relevant physics topics are classical and quantum field theories, and supersymmetry.

The organization meeting for Fall 2017 is scheduled for September 7th. Students (advanced undergraduates or graduate students) who want to participate can get credit as MATH 489 (undergrad) or 689 (graduate) if they wish, by contacting the organizers.

We have a wiki where participants can exchange comments and revise notes.  Contact the math/physics computer helpdesk if you need a login id and password.

 Topics from previous years:

  1. Supermanifolds, topology, and integration --- a few references:
    • S. J. Gates, Ectoplasm has no topology, hep-th/9709104 and hep-th/9809056.
    • E. Witten, Notes On Supermanifolds and Integration1209.2199.
    • The Berezin integral.
    • S. J. Gates and G. Tartaglino-Mazzucchelli, Ectoplasm and superspace integration measure for 2D supergravity with four spinorial supercurrents, 0907.5264.
    • S. J. Gates and A. Morrison, A Derivation of an Off-Shell N = (2,2) Supergravity Chiral Projection Operator, 0901.4165.
  2. Dimensonal reduction in supersymmetry
    • For example, S. J. Gates and T. Hubsch, On dimensional extension of supersymmetry: From worldlines to worldsheets, 1104.0722, deals with reduction from 1+1 to 0+1 dimensions.
  3. Adinkras and combinatorics--- a few references:
    • Yan Zhang, The combinatorics of adinkras.
    • Yan Zhang, Adinkras for mathematicians, 1111.6055.
    • Greg Landweber, Bibliography on adinkras.
    • C. Doran, K. Iga, G. Landweber, and S. Mendez-Diez, Geometrization of N-extended 1-dimensional supersymmetry algebras, 1311.3736.
    • T. Hübsch and G.A. Katona, On the Construction and the Structure of Off-Shell Supermultiplet Quotients, Int. J. Mod. Phys. A27 (2012) 1250173, 1202.4342.
    • C.F. Doran, T. Hübsch, K.M. Iga and G.D. Landweber,  On General Off-Shell Representations of Worldline (1D) Supersymmetry, Symmetry 6 no. 1, (2014) 67–88, 1310.3258.
  4. Super-Riemann surfaces and physical applications
  5. The Haag-Łopuszański-Sohnius Theorem and its variants. This is the supersymmetric analogue of the better-known Coleman-Mandula Theorem.
  6. Mirror symmetry (2016-2017).  In the fall, we followed a somewhat ad hoc approach based on looking at a lot of examples (e.g., elliptic curves and the quintic Calabi-Yau).  In the spring, we followed the multi-author book published by AMS, Dirichlet Branes and Mirror Symmetry.  The preface and Chapter 1 can be downloaded from the AMS website; Chapter 2 is at arXiv:hep-th/0609042.  An electronic version of the whole book is available at http://www.claymath.org/publications/online-books

Topics and references for 2017-2018

The topic for fall 2017 is topological states of matter. In no particular order, here is a list of references:

  1. Emil Prodan and Hermann Schulz-Baldes, "Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics", arXiv:1510.08724.
  2. Daniel Freed and Greg Moore, Twisted equivariant matter, Ann. Henri Poincaré 14 (2013), no. 8, 1927–2023, arXiv:1208.5055.
  3. A. Kitaev, Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009). doi:10.1063/1.3149495, arXiv:0901.2686
  4. F.D.M. Haldane, Model for a quantum Hall effect without Landau levels: condensed-matter realization of the "parity anomaly". Phys. Rev. Lett. 61, 2015–2018 (1988).
  5. C.L. Kane and E.J. Mele, Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005), arXiv:cond-mat/0411737
  6. C.L. Kane and E.J. Mele, Z2 Topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005), arXiv:cond-mat/0506581.
  7. C.L. Kane and E.J. Mele, Topological Mirror Superconductivity, Phys. Rev. Lett. 111, 056403 (2013), arXiv:1303.4144.
  8. Jean Bellissard, Noncommutative Geometry and the Quantum Hall Effect, Proceedings of the International Conference of Mathematicians (Zürich 94), Birkhäuser (1995).
  9. J. Bellissard, A. van Elst, H. Schulz-Baldes, The Non Commutative Geometry of the Quantum Hall Effect (longer version of #8 above), arXiv:cond-mat/9411052.
  10. David Tong: Lectures on the Quantum Hall Effect, Univ. of Cambridge, arXiv:1606.06687.
  11. Edward Witten, Three Lectures On Topological Phases Of Matter, arXiv:1510.07698.
  12. Ralph M. Kaufmann, Dan Li, and Birgit Wehefritz-Kaufmann, Notes on topological insulators, Rev. Math. Phys., 28(10), 1630003, 2016, arXiv:1501.02874.
  13. Anton Akhmerov, Jay Sau, et al., Topological condensed matter, an online course.

The topic for spring 2018 is generalized geometry and its applications to physics (especially supersymmetry and string theory). We will begin with the first reference listed, Hitchin's notes. Here is a short list of basic references:

  1. N. Hitchin, Lectures on Generalized Geometry, arXiv:1008.0973.
  2. M. Zabzine, Lectures on Generalized Complex Geometry and Supersymmetry, arXiv:hep-th/0605148.
  3. Dimitrios Tsimpis, Generalized geometry lectures on type II backgrounds, arXiv:1606.08674.
  4. N. Hitchin, Generalized Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281–308, arXiv:math/0209099.
  5. M. Gualtieri, Generalized Kahler geometry, arXiv:1007.3485.
  6. G. Cavalcantri and M. Gualtieri, Generalized complex geometry and T-duality, arXiv:1106.1747.

Detailed schedule posted below.

Organizers: Ricardo Nochetto, Wujun Zhang
When: Wednesdays @ 5pm-6pm, starting the first week of February
Where: room TBA
Subtitle: PDE theory and numerical analysis

Fully nonlinear second order elliptic PDEs arise naturally from differential geometry, stochastic control theory, optimal transport and other fields in science and engineering. In this RIT, we will discuss the concept of viscosity solutions and regularity theory of these PDEs. Some possible topics include:

  • fully nonlinear elliptic equations and viscosity solutions
  • Alexandroff-Bakelman-Pucci estimates
  • Harnack inequality and Hölder regularity
  • Uniqueness of solutions
  • W2,p estimates
  • C1, α estimates
  • C2, α estimates

In contrast to an extensive PDE literature, the numerical approximation reduces to a few papers. We would like to discuss some criteria for designing convergent numerical methods and some tools developed recently which are useful to obtain rates of convergence. Some possible topics include:

  • criterion for convergence of numerical methods
  • discrete version of the Alexandroff-Bakelman-Pucci estimate
  • finite element method for linear elliptic equations in non-divergence form
  • numerical method for Monge-Ampere equations

Organizers: Niranjan Ramachandran, Dio Margetis, and Leo Koralov
When: 
Wednesday @ 3:15pm, Tea 2:45pm - 3:15 pm in room 3201
Where:
Math 3206
From time to time special colloquia are held on other days, sometimes as part of conferences.
Other special colloquia are the Aziz Lectures and Avron Douglis Memorial Lectures.

Location: Math 3206
Day: Wednesday (with occasional talks Friday)
Time: 3:15pm
Tea: 2:45pm in 3201
Organizers: Giovanni Forni, Harry Tamvakis, Konstantina Trivisa

2011 - 2012 Colloquium Schedule

Sept 14

Wonderful compactifications of groups as moduli spaces of principal bundles
Michael Thaddeus (Columbia University),

Sept 28

An introduction to essential dimension
Patrick Brosnan(University of Maryland),

Oct 21

Rigidity of group actions, cohomology and compactness
David Fisher (Indiana University, Bloomington),

Oct 28

A Filtration of Open/Closed Topological Field Theory
Ezra Getzler (Northwestern University),

Nov 2

Rational billiards and the SL(2,R) action on moduli space
Alex Eskin (University of Chicago),

Nov 9

Why and how do we use wavelets to study turbulence?
Marie Farge(Directrice de Recherche CNRS)

Nov 16

Mixed volume computation and solving polynomial systems
Tien-Yien Li (Michigan State University),

Dec 2

Optimal and Practical Algebraic Solvers for Discretized PDEs - Aziz Lecture
Jinchao Xu (Pennsylvania State University),

Dec 7

DMS to THRIVE
Sastry G. Pantula(National Science Foundation)

Feb 8

On the rigidity of black holes - Douglis Lecture
Sergiu Klainerman(Princeton University )

Feb 17

Product formulas for positive measures and applications - February Fourier Talks
Peter Jones(Yale University),

Feb 22

Semismooth Newton Methods: Theory, Numerics and Applications - Aziz Lecture
Michael Hintermüller (Humboldt University, Berlin),

Feb 29

On the size of the Navier - Stokes singular set
Walter Craig (McMaster University),

Mar 14

De Giorgi methods applied to regularity issues in Fluid Mechanics
Alexis Vasseur (UT Austin )

Mar 30

Birkhoff Normal Form and a problem of Herman - Dynamics Conference
Hakan Eliasson (University of Paris-6 and IAS),

Apr 25

Lyapunov Functions: Towards an Aubry-Mather theory for homeomorphisms?
Albert Fathi (ENS-Lyon),

May 2

Contractions of Lie Groups and Representation Theory
Nigel Higson (Penn State University), cancelled

May 9

The Pfaffian-Grassmannian Derived Equivalence
Andrei Caldărăru (University of Wisconsin, Madison),

This area includes information on research done in the department, seminars and conferences hosted by the department, as well as access to electronic research resources.

Subcategories

Archives: 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018

  • Aziz Lecture

    Speaker: Aziz Lecture () -

    When: Wed, September 12, 2018 - 3:15pm
    Where: Kirwan Hall 3206
  • Scholze's Fields medal

    Speaker: Michael Rapoport (UMD and University of Bonn) - http://www.math.uni-bonn.de/ag/alggeom/rapoport

    When: Wed, September 26, 2018 - 3:15pm
    Where: Kirwan Hall 3206