Projects

1. 1. Grading Luo-Tan identity
      for my student!!!

   2. A advanced course on SLE by Ilia Binder: www.fields.utoronto.ca/activities/21-22/function-course
      Other course by Chris Bishop, and Lectures by Yilin Wang: www.math.stonybrook.edu/~bishop/classes/math638.F20/ 
      
      

   3. Extending Wang-Xue's identity to the case of surfaces with cones (flat surfaces, orbifolds, ...)
      
      

   4. Apply Parlier's method to connect Basmajian and McShane identities in higher rank
      (Section 7) Basmajian’s identity in higher Teichmuller-Thurston theory
      
      

   5. Geometric meaning of Wang-Xue's identity in their paper.
      Youtube: 19/05/2025 Hausdorff Center for Mathematics
      Extremal length (Didac): https://www.mathtube.org/lecture/video/extremal-length-systole-bolza-surface
      
      Q0: Relate this to a kind of model surface in BPT paper?
      
      Q1: For the modular torus and flat torus, what can we say about the multiplicity of the length spectrum of closed geodesics homotopic to the given one on the flat torus? Is it infinite, meaning that we can use the reflection trick to generate an arbitrarily large number of geodesics of the same length? I asked this question because in the case of  McShane's identity on the modular torus, the multiplicity of oriented simple closed geodesics is conjectured to be bounded by 12.
      Can we arrange the index set of Wang-Xue's identity in the set of complementary regions of a Markov-like tree?
      Counting function of the index set with respect to the hyperbolic length? Maybe e^L/L^2?
      
      Q2: Is it possible to find the gap associated with each closed geodesic of the form a^{s_1}b^{t_1}...a^{s_n}b^{t_n}? where a = [[2,1],[1,1]] and b = [[1,6],[0,1]] = aca^-1c^-1. Note that c = [[1,1],[1,2]] and ab^{-1} and a are in the same conjugacy class. Thus we need an algorithm to detect different conjugacy classes.
      
      Q3: It seems for the index set of Wang-Xue's identity, there is no gap since there is a convergence phenomeno, for example a closed geodesic W homotopic to a in the original surface, we take the sequence (W^n)(a^{-n+1})(b) which homotopic to a in the original surface, then the endpoints of this sequence converge to the endpoint of W. But I guess we can regroup them in a certain way so that there are gaps.
      
      Q4:  How about the  Brownian loop measure formula in the case of orbifolds or flat surfaces?
      
      Q5: Calculate the measure for different contexts: From boundary (cone) to boundary (cone). Using chordal or radical SLE:  https://en.m.wikipedia.org/wiki/Schramm–Loewner_evolution
      
      Q6: My combinatorial proof for Basmajian identity also give a probabilistic nature for the Basmajian identity.
      
      

   6. References: Good intro about the Stochastic process, Markov chain, and Martigale: Choongbum Lee
      
      

   7. Geometric inner product (with Tsukasa Ishibashi and Jessica Purcell)
      (Discussing)
      Gram & Ptolemy
      Cluster Algebras and Dilogarithm Identities (Tomoki Nakanishi's book)
      Dylan G.L. Allegretti: The Geometry of Cluster Varieties from Surfaces
      Volume of ideal tetrahedron (Danny Calegary's blog)
      Pseudomanifolds
      Ptolemy relation in 3-dim
      If you want to find the formula for x15 in its correct orientation, choose the biggest solution of the equation. Note that there are at most 4 solutions. Similarly, do the same for x25, x35, x45 in their correct orientations.
      https://www.wolframcloud.com/env/9607cb1c-a729-4704-8c4d-ee5a9c9da1f8
      
      

   8. Chris-Ser's suggestion about free group
      On groups generated by two positive multi-twists: Teichmüller curves and Lehmer's number,  
      
      

   9. Cohn's polynomial
      overleaf link
      
      

   10. Thurston's spine. Do OI surfaces lie on it? How about infinite type surfaces? What is the shape of the spine for small surfaces (one-holed tori, one-puncture tori)?
       NI AN, FERDINAND IHRINGER, AND INGRID IRMER
       The code 
       Teichmuller space for infinite-type surfaces
       
       

   11. Grading Brigeman (with Ara, Hugo, Ser)
       (Writing, deadline before June)
       
       

   12. Congruence subgroup Gamma_0(p^n) (DKLT)
       (Writing, deadline before August)
       
       

   13. Geometric meaning of Hu-Tan-Zhang's identities?
       Hurwitz equation
       and it's special case when n=3
       when n=4 (Norbury-Huang)
       
       

   14. Counting low-lying orthogeodesics (with many)
       (Postpone)
       
       

   15. DKLT function for surfaces with boundary, Belyi function technique to give the minimum of the DKLT function.
       The minima of the geodesic length functions of uniform filling curves
       
       

   16. New low-lying identity (Debattam and Hanh Vo)
       (Discussing, deadline before September)
       
       

   17. Pseudomodular group (with Ser Peow Tan and Daren Wei)
       (Discussing, deadline before October)
       https://irma.math.unistra.fr/~billon/pseudomodular_surfaces.pdf#:~:text=De%1Cnition%202,Γ2
       Phase transition 1/7, is it related to Ising model in 2D? SLE_3
       Is there a way to extend continuously the notion of killer interval to the case when alpha is not in R? Then we can study the phase transition phenomenon.
       
       

   18. Geometric topograph and Gauss product (with Didac and Ser Peow Tan)
       (Discussing)
       
       

   19. BMCG (with Didac,Hoang,Khoi)
       (Discussing)
       Topological entropy of random walks on MCGs (Masai)
       Estimate of the entropy (Lemma 4.8)
       Extention of Thurston-Veech's construction to infinite-type surfaces: LOXODROMIC ELEMENTS IN BIG MAPPING CLASS GROUPS, these elements lie in the closure of pA elements in finite-type subsurfaces
       Also see this book Infinite Translation Surfaces in the Wild Volume 1: Geometry and Symmetries (Delecroix-Hubert-Valdez)
       For family of elements not in the closure: Infinite-type loxodromic isometries of the relative arc graph 

Questions: for an arbitrarily undistorted element, is it a limit of elements constructed in the two papers above?

1. 20. Arithmetic of cosh-length orthospectrum (with Khanh Le)
       (Discussing)
       
       

   21. Length equivalent orthogeodesics (with Khanh Le)
       (Discussing)
       McShane-Parlier paper
       BaikChoiKim paper
       Hanh-Hugo-Binbin
       Idea: Length changing together under a mapping class, there is a formula for the ratio of lengths related to Wolpert's formula
       
       

   22. Do there exist subgroups of SL(2, Z) that exhibit reciprocity obstruction?
       Reciprocity obstructions in semigroup orbits in SL(2, Z)
       APOLLONIAN PACKINGS: THE RISE AND FALL OF THE LOCAL TO GLOBAL CONJECTURE
       
       

   23. Encoding arcs using the punctured cover H_0 of the model surface
       Infinite-type loxodromic isometries of the relative arc graph 
       (CAROLYN ABBOTT, NICHOLAS MILLER, PRIYAM PATEL)
       
       

   24. Good pants JEREMY KAHN AND VLADIMIR MARKOVIC
       
       

   25. Masai's work on extremal length https://arxiv.org/pdf/2505.12400
       
       

   26. BMCG (with Hoang Thanh Nguyen, The Khoi Vu, Didac)
       
       

   27. Rational lambda length problem: My old draft
       
       

   28. The conjecture by Calegari in his paper: https://arxiv.org/pdf/2505.07137 related to efficient universal ReLU neural networks
       
       

   29. Green metric, THE ERGODIC THEORY OF HYPERBOLIC GROUPS - Danny Calegari
       
       

   30. Markoff-type equations for (2,2,2): a^2+b^2+26-27(a+b+c)+27(ab+bc+ca)-25abc=0, study the unicity conjecture.
       Relate to Dao Quang Duc paper
       
       

   31. A combinatorial proof of McShane's identity in higher genus, using orthotree, and the recursive formula.
       
       

   32. Using the orthospectrum associated with a river to understand the randomness of horocycles associated with killer intervals.
       
       

   33. Some questions of Alan Reid and Mathew Stover: COMPLEX HYPERBOLIC 2-ORBIFOLDS WITH ISOLATED SINGULARITIES
       
       

   34. Geometric topograph in higher rank PSL(3,Z)\PSL(3,R), periodic tori, orthospectrum. Density of shapes of periodic tori in the cubic case (Nguyen-Thi Dang, Nihar Gargava, Jialun Li)
       
       

   35. Lagrange spectrum and Ian Agol's question (with Didac)
       
       

   36. Complex Ptolemy relation for geodesics (with Ser, and Dan)
       (will do)
       
       

   37. Green metric, geodesic current, McShane identity for self-intersection numbers.
       
       

   38. More number theory of OI surfaces (local-global). In a primitive integral Apollonian circle packing, the curvatures that appear must fall into one of six or eight residue classes modulo 24. The local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing.
       
       

   39. Hasse Principle Doan's quadratic polynomials
       
       

   40. OI 3-manifolds
       
       

   41. OI surface with cusp and cone (integral inner product problem)
       
       

   42. Study the length equivalent conjecture for punctured surfaces suggested by Didac. Then we can talk about the multiplicity of orthogeodesics (up to this equivalent). We expect that if the conjecture is true then the multiplicity is exactly 1 for general surfaces.
       
       

   43. Code curves or arcs on surfaces that detect bigons and monogons? Study the relation between the self-intersection number of arc and its accompanying closed geodesic?
       
       

   44. Self-intersection Zaremba conjecture for 2-low-lying reciprocal orthogeodesics.
       
       

   45. Randomly gluing OI (2,2,2), what can we say about the ortho-length spectrum https://www.sciencedirect.com/science/article/pii/S0001870825000568