Mathematical Explorations of Dr. Katrin Wehrheim

Three Classic Boundary Conditions

• Dirichlet: Fix the temperature at the boundary (like holding the edge of a metal plate at 0°C)
• Neumann: Fix the heat flow at the boundary (like insulating the edge so no heat escapes)
• Mixed: Different conditions on different parts of the boundary

Wehrheim's work extends these ideas to Lagrangian boundary conditions in gauge theory—geometric generalizations where the boundary values must lie on special submanifolds.

Why Symplectic Geometry?

Dr. Wehrheim studies symplectic manifolds because they're the natural home for Hamiltonian mechanics, but also because they have rich topological invariants (Floer homology) that connect to quantum theory. The symplectic form $\omega$ determines which submanifolds are "Lagrangian" (where $\omega$ vanishes)—these are like the classical configurations in quantum mechanics.

From Physics to Mathematics

Dr. Wehrheim's book Uhlenbeck Compactness studies what happens to sequences of gauge fields (connections) when their curvature is bounded. This is crucial because solutions to physical equations (like Yang-Mills) come in infinite-dimensional families, and we need compactness to extract convergent subsequences.

Why Correspondences?

Correspondences compose more flexibly than functions. Given $L_{01} \subset M_0^- \times M_1$ and $L_{12} \subset M_1^- \times M_2$, we can "compose" them by looking at the fiber product and taking the Lagrangian connecting $M_0$ to $M_2$. This is the functoriality that Wehrheim proves!

At the undergraduate level, we've explored:

• How boundary conditions profoundly affect PDE solutions
• What symplectic structures are (area-preserving geometry)
• How gauge theory describes parallel transport and curvature
• The difference between functions and correspondences
• Why manifolds with boundary require special care

These ideas form the foundation for Dr. Wehrheim's advanced work. As you progress to the Graduate level, we'll see how these concepts combine in powerful ways!

From Morse to Floer

Morse homology for a function $f: M \to \mathbb{R}$:

• Chains: $C_k = \mathbb{Z}\langle \text{critical points of index } k \rangle$
• Boundary map: $\partial x = \sum_y n(x,y) \cdot y$ where $n(x,y)$ counts gradient flow lines from $x$ to $y$
• Homology: $HM_*(M) = \ker \partial / \text{im } \partial$ recovers $H_*(M)$!

Floer homology for a symplectic manifold $(M, \omega)$ and Hamiltonian $H: M \times S^1 \to \mathbb{R}$:

• Chains: $CF_* = \mathbb{Z}\langle \text{periodic orbits of } X_H \rangle$
• Boundary map: $\partial x = \sum_y n(x,y) \cdot y$ where $n(x,y)$ counts pseudoholomorphic strips $u: \mathbb{R} \times [0,1] \to M$ connecting $x$ to $y$
• Homology: $HF_*(M, H)$ is a symplectic invariant!

The Quilted Solution

Instead of composing geometric correspondences directly, Wehrheim-Woodward define quilted Floer homology:

• A quilt is a pseudoholomorphic map from a surface with seams (not just boundary)
• Each "patch" of the quilt maps to a different symplectic manifold
• Along seams, the map must satisfy seam conditions: it maps to a Lagrangian correspondence

The quilted Floer chain complex has generators: tuples of Hamiltonian orbits $(x_0, x_1, x_2)$ related by the correspondences. The boundary operator counts quilted pseudoholomorphic surfaces!

Uhlenbeck Compactness for Manifolds with Boundary

Dr. Wehrheim's book Uhlenbeck Compactness proves that sequences of connections with bounded curvature on manifolds with boundary admit a convergent subsequence (after gauge transformation) if and only if the boundary values converge in an appropriate sense.

This required developing new gauge-fixing procedures compatible with Lagrangian boundary conditions—a major technical innovation!

Interactive Example: Heat Equation on a Manifold with Boundary

This simulation shows the solution to a boundary value problem—the heat equation on a disk with prescribed temperature on the boundary. This illustrates the type of analysis needed for gauge theory and Floer theory.

What Does Functoriality Mean?

Given Lagrangian correspondences:

• $L_{01}: M_0 \to M_1$ and $L_{12}: M_1 \to M_2$
• Their geometric composition $L_{01} \circ L_{12}$ (via fiber product)
• Quilted Floer homology defines functors $\Phi_{01}: HF(M_0) \to HF(M_1)$ and $\Phi_{12}: HF(M_1) \to HF(M_2)$

Functoriality: There exists a canonical isomorphism: $$\Phi_{12} \circ \Phi_{01} \cong \Phi_{02}$$ where $\Phi_{02}$ is the quilted Floer functor for $L_{01} \circ L_{12}$.

Key Applications

• Symplectic Field Theory: Provides rigorous foundations for SFT, allowing gluing of holomorphic curves in various geometries
• Fukaya Categories: Lagrangian correspondences give morphisms between Fukaya categories of different manifolds
• Mirror Symmetry: The functorial structure helps formalize homological mirror symmetry conjectures
• Geometric Quantization: Connects classical (symplectic) and quantum (Hilbert space) mechanics via categorical structures

Essential Reading

• Wehrheim: Uhlenbeck Compactness (2004) — The definitive treatment of gauge theory on manifolds with boundary
• Wehrheim-Woodward: "Functoriality for Lagrangian correspondences in Floer theory" (2010) — Introduces quilts and proves functoriality
• Wehrheim-Woodward: "Quilted Floer cohomology" (2010) — Full technical details of quilted theory
• Ma'u-Wehrheim-Woodward: "A∞-functors for Lagrangian correspondences" (2018) — Higher categorical structures

Background references: McDuff-Salamon's J-holomorphic Curves and Symplectic Topology, Seidel's Fukaya Categories and Picard-Lefschetz Theory

At the graduate level, we've explored:

• Floer homology as infinite-dimensional Morse theory for symplectic manifolds
• Pseudoholomorphic quilts and how they solve the composition problem for correspondences
• Analytical foundations: compactness, transversality, and gluing theorems
• Functoriality: the categorical structure of Lagrangian correspondences and Floer homology
• Applications to symplectic field theory, Fukaya categories, and mirror symmetry

These tools are at the forefront of modern symplectic topology. Dr. Wehrheim's work has provided the rigorous foundations needed to make many heuristic ideas in the field mathematically precise.

From 1-Category to A∞-Category

The issue: Floer homology is only the degree-0 part of a richer structure. The full Fukaya category has:

• Objects: Lagrangian submanifolds $L_i \subset M$
• Morphisms: $\text{hom}(L_0, L_1) = CF^*(L_0, L_1)$ (Floer cochain complex)
• Compositions: $m_d: \text{hom}(L_{d-1}, L_d) \otimes \cdots \otimes \text{hom}(L_0, L_1) \to \text{hom}(L_0, L_d)[2-d]$

These satisfy the A∞-relations: $$\sum_{i+j=d+1} (-1)^* m_i(\text{id}^{\otimes a} \otimes m_j \otimes \text{id}^{\otimes b}) = 0$$

A Lagrangian correspondence $L_{01} \subset M_0^- \times M_1$ should give an A∞-functor $\Phi_{01}: \mathcal{F}(M_0) \to \mathcal{F}(M_1)$.

2-Categorical Structure

• 0-morphisms (objects): Symplectic manifolds $(M, \omega)$
• 1-morphisms: Lagrangian correspondences $L_{01} \subset M_0^- \times M_1$
• 2-morphisms: Quilted Floer continuations (morphisms between correspondences)

Composition of 1-morphisms is given by the quilted fiber product, and composition of 2-morphisms involves counting higher-dimensional quilts.

Uhlenbeck Compactness on Manifolds with Boundary

Given a sequence of connections $A_k$ on a bundle over a manifold $X$ with boundary, suppose:

• $\|F_{A_k}\|_{L^2} < C$ (bounded curvature)
• $A_k|_{\partial X}$ satisfy Lagrangian boundary conditions

Theorem (Wehrheim): There exists a subsequence converging (after gauge transformation) to a limit connection $A_\infty$, possibly with finitely many "bubble points" where energy concentrates.

The key innovation: compatible gauge-fixing near the boundary that respects the Lagrangian condition.

Current Research Questions

1. Symplectic (∞,n)-Categories

Can we extend the 2-categorical picture to a full (∞,n)-category? This would require:

• Higher morphisms (k-morphisms for all k)
• Coherence conditions ensuring all compositions are compatible
• Connection to extended TQFTs and the cobordism hypothesis

2. Lagrangian Floer Theory for Singular Lagrangians

Many interesting Lagrangians have singularities (like Lagrangian skeleta in mirror symmetry). Can quilted Floer theory handle singular correspondences? Progress requires:

• Understanding how pseudoholomorphic curves interact with singularities
• Developing appropriate transversality and compactness theorems
• Applications to homological mirror symmetry

3. Relation to Physics

Lagrangian correspondences appear naturally in:

• Geometric quantization (relating classical and quantum mechanics)
• Topological field theories (TQFT partition functions)
• String theory (D-branes as Lagrangian submanifolds)

Can the categorical structure be made physically rigorous?

4. Computational Methods

Floer homology is notoriously hard to compute explicitly. Can quilted techniques lead to:

• Spectral sequences relating quilted and standard Floer homology
• Combinatorial models for special classes of correspondences
• Explicit calculations for important examples (e.g., monotone tori)

Essential Papers and Monographs

Foundational Works:

• K. Wehrheim, Uhlenbeck Compactness, EMS Series of Lectures in Mathematics (2004)
• K. Wehrheim & C. Woodward, "Functoriality for Lagrangian correspondences in Floer theory," Quantum Topology (2010)
• K. Wehrheim & C. Woodward, "Quilted Floer cohomology," Geometry & Topology (2012)

Recent Developments:

• S. Ma'u, K. Wehrheim & C. Woodward, "A∞-functors for Lagrangian correspondences," Selecta Mathematica (2018)
• K. Wehrheim & C. Woodward, "Floer field theory for coprime rank and degree," in preparation
• K. Wehrheim, "Functorial Lagrangian field theory," in preparation

Related Work:

• M. Abouzaid, "A geometric criterion for generating the Fukaya category," Publications mathématiques de l'IHÉS (2010)
• P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, Zurich Lectures in Advanced Mathematics (2008)
• D. Auroux, "A beginner's introduction to Fukaya categories," arXiv:1301.7056
• K. Fukaya et al., Lagrangian Intersection Floer Theory, AMS (2009-2010)

At the research level, we've explored:

• A∞-structures: Extending functoriality to full Fukaya categories
• 2-categories and TQFTs: The categorical framework for symplectic topology
• Gauge theory: Analytical foundations via Uhlenbeck compactness
• Open problems: Higher categories, singular Lagrangians, and physical applications

Current research directions in Wehrheim's program include:

• Developing symplectic (∞,n)-categories rigorously
• Understanding Floer theory for singular correspondences
• Making homological mirror symmetry explicit via quilts
• Connecting to geometric quantization and TQFT
• Computational tools for quilted Floer homology

These are active areas where fundamental questions remain open. Dr. Wehrheim's work has provided the technical foundations, but much remains to be discovered!

🔗 Theme 1: Functoriality—Making the Algebra Work

Perhaps the most central theme in Dr. Wehrheim's work is functoriality: ensuring that geometric compositions correspond to algebraic compositions. This isn't just a technical detail—it's fundamental to making symplectic topology into a coherent mathematical theory.

Why it matters:

• Consistency: Without functoriality, the theory is internally inconsistent—different paths to the same result might give different answers
• Computability: Functoriality allows us to break complex problems into simpler pieces
• Categorification: It's the foundation for organizing mathematics into categories and higher categories
• Physical applications: In quantum field theory, functoriality corresponds to locality—the principle that physics is determined by local data

Wehrheim's contributions:

• The functoriality theorem for quilted Floer homology proves that composing Lagrangian correspondences induces the expected composition on Floer homology
• This required solving deep analytical problems about moduli spaces of quilted pseudoholomorphic curves
• The result: Lagrangian correspondences form a genuine 2-category, not just a "almost-category"

Visual Example: Why Functoriality Can Fail

Imagine you want to show that $(L_{01} \circ L_{12}) \circ L_{23} = L_{01} \circ (L_{12} \circ L_{23})$. Geometrically, both sides are defined by fiber products, but these involve different limiting processes. Without careful analysis, the two sides might not even be diffeomorphic!

🎯 Theme 2: Boundary Conditions—The Interface Challenge

Much of mathematics deals with "nice" objects—smooth manifolds without boundary, compact spaces, etc. But many interesting phenomena happen at boundaries and interfaces. Dr. Wehrheim has pioneered methods for handling these challenging situations.

The challenge: When you have a manifold with boundary, standard techniques often break down:

• Integration by parts produces boundary terms
• Elliptic PDE theory requires boundary conditions
• Compactness theorems need control at the boundary
• Gauge transformations must respect boundary data

Wehrheim's approach:

• Lagrangian boundary conditions: Use Lagrangian submanifolds to specify boundary data in a geometrically natural way
• Compatible gauge fixing: Develop gauge conditions that work well with Lagrangian boundaries (solving Uhlenbeck compactness)
• Seam conditions in quilts: At interfaces between patches, require the map to land in a Lagrangian correspondence

Example: Uhlenbeck Compactness

Standard Uhlenbeck compactness says: sequences of connections with bounded curvature have convergent subsequences (after gauge transformation). But on manifolds with boundary, this fails unless you:

1. Specify appropriate boundary conditions
2. Find gauge transformations that respect those conditions
3. Prove the boundary values converge in the right topology

Wehrheim's monograph Uhlenbeck Compactness solves all three problems for Lagrangian boundary conditions.

📐 Theme 3: Bridging Analysis and Geometry

Modern symplectic topology relies heavily on analysis—specifically, the theory of elliptic partial differential equations. Dr. Wehrheim's work exemplifies how deep analytical techniques are essential for geometric results.

The analytical toolkit:

• Elliptic regularity: Solutions to elliptic PDEs (like $\bar{\partial}_J u = 0$) are smooth, even if only defined weakly
• Sobolev spaces: Function spaces that measure both size and differentiability, crucial for weak solutions
• Fredholm theory: Understanding when linear operators have finite-dimensional kernels and cokernels
• Compactness theorems: Conditions ensuring sequences have convergent subsequences
• Gluing maps: Reconstructing smooth objects from pieces

Key insight: The geometric objects we care about (pseudoholomorphic curves, instantons, etc.) are solutions to PDEs. To understand their moduli spaces, we need functional analysis.

Example: Transversality

To define Floer homology, we need moduli spaces of pseudoholomorphic curves to be smooth manifolds. This requires:

1. Linearizing the equation $\bar{\partial}_J u = 0$ at a solution
2. Showing the linearization is surjective (Fredholm theory)
3. Applying the implicit function theorem in Banach spaces
4. Generically perturbing to achieve transversality

Each step involves sophisticated analysis, but the payoff is geometric: we can count curves and define homological invariants.

🎨 Theme 4: Categorification—From Numbers to Categories

Categorification is the process of replacing numbers, sets, or algebraic structures with richer categorical objects. Instead of asking "what is the invariant?", we ask "what category does it come from?"

Example progression:

Why categorify?

• More information: A category contains more data than a single number or vector space
• Natural operations: Functors between categories capture natural transformations
• Coherence: Higher categorical structures ensure everything fits together consistently
• Connections: Different areas of mathematics are related via functors

Wehrheim's categorification:

• Traditional: Lagrangian intersections give numbers (Euler characteristic)
• Floer: Lagrangian intersections give homology groups (graded vector spaces)
• Fukaya: Lagrangians form a category, with morphisms given by Floer complexes
• Wehrheim-Woodward: Lagrangian correspondences give a 2-category, with Floer homology as 2-morphisms

The Categorification Ladder

🌐 Theme 5: Interdisciplinary Connections

Dr. Wehrheim's work connects multiple areas of mathematics and physics. Understanding these connections reveals the broader significance of her contributions.

Connections to Other Fields:

1. Gauge Theory and Physics

• Yang-Mills theory describes fundamental forces in particle physics
• Instantons (solutions to Yang-Mills equations) give topological invariants
• Wehrheim's work on gauge theory with boundary conditions applies to physical systems with interfaces

2. Mirror Symmetry

• Mirror symmetry (from string theory) relates symplectic and complex geometry
• Fukaya categories (symplectic side) should be equivalent to derived categories (complex side)
• Lagrangian correspondences may provide explicit equivalences

3. Topological Quantum Field Theory

• TQFTs assign algebraic data to manifolds in a way that respects gluing
• Extended TQFTs use higher categories to capture more structure
• The Wehrheim-Woodward 2-category provides a symplectic version of (1+1)-dimensional TQFT

4. Algebraic Topology

• Floer homology is an infinite-dimensional generalization of Morse homology
• Spectral sequences connect Floer homology to classical homology
• Higher categorical structures relate to homotopy theory

5. Geometric Quantization

• Quantization seeks to construct quantum theories from classical (symplectic) data
• Lagrangian submanifolds correspond to quantum states
• Lagrangian correspondences may give quantum operators

⚖️ Theme 6: Rigorous Foundations

A hallmark of Dr. Wehrheim's work is the emphasis on rigorous foundations. Many ideas in symplectic topology were initially understood heuristically or through examples. Wehrheim has provided the technical machinery to make them precise.

Areas where foundations were needed:

• Gauge theory on manifolds with boundary: Solved by the Uhlenbeck compactness monograph
• Composition of Lagrangian correspondences: Solved by quilted Floer theory
• Functoriality in Floer homology: Solved by the Wehrheim-Woodward functoriality theorem
• A∞-structures for correspondences: Addressed in the Ma'u-Wehrheim-Woodward work

The role of rigor:

• Reliability: Results are trustworthy and can be built upon
• Generality: Precise statements reveal the true scope of results
• New insights: Careful proofs often reveal unexpected phenomena
• Progress: Solid foundations enable further development

📚 Dr. Wehrheim's Key Publications

Books and Monographs

• Uhlenbeck Compactness (2004)
  EMS Series of Lectures in Mathematics
  The definitive treatment of compactness for connections on manifolds with boundary. Essential for understanding gauge-theoretic foundations.

Foundational Papers on Quilted Floer Theory

• "Functoriality for Lagrangian correspondences in Floer theory" (2010)
  K. Wehrheim & C. Woodward, Quantum Topology 1(2), 129-170
  Introduces quilted pseudoholomorphic curves and proves functoriality.
• "Quilted Floer cohomology" (2012)
  K. Wehrheim & C. Woodward, Geometry & Topology 16, 927-1026
  Full technical details of the quilted theory with complete proofs.
• "Pseudoholomorphic quilts" (2015)
  K. Wehrheim & C. Woodward, arXiv:1905.09334
  Comprehensive treatment of the analytical foundations.

A∞-Structures and Higher Categories

• "A∞ functors for Lagrangian correspondences" (2018)
  S. Ma'u, K. Wehrheim & C. Woodward, Selecta Mathematica 24, 1913-2002
  Extends functoriality to the A∞-categorical setting.

Other Significant Papers

• "Floer cohomology and geometric composition of Lagrangian correspondences" (2010)
  K. Wehrheim & C. Woodward, Advances in Mathematics 230(1), 177-228
• "Orientations for pseudoholomorphic quilts" (2015)
  K. Wehrheim & C. Woodward, arXiv:1503.07803

📖 Background Reading by Topic

Symplectic Geometry (Prerequisites)

• Ana Cannas da Silva, "Lectures on Symplectic Geometry" (2008)
  Lecture Notes in Mathematics, Springer
  Excellent introduction covering basic symplectic manifolds, Hamiltonian mechanics, and moment maps. Free PDF available.
• Dusa McDuff & Dietmar Salamon, "Introduction to Symplectic Topology" (3rd ed., 2017)
  Oxford Mathematical Monographs
  The standard comprehensive textbook. Chapters 1-6 provide essential background.

Floer Homology

• Dusa McDuff & Dietmar Salamon, "J-holomorphic Curves and Symplectic Topology" (2nd ed., 2012)
  AMS Colloquium Publications
  The bible of pseudoholomorphic curve theory. Chapters 5-12 cover Floer homology in detail.
• Michèle Audin & Mihai Damian, "Morse Theory and Floer Homology" (2014)
  Springer
  Connects Morse theory to Floer theory, with clear explanations of the infinite-dimensional perspective.
• Yong-Geun Oh, "Symplectic Topology and Floer Homology" (2015)
  Cambridge Studies in Advanced Mathematics
  Two-volume comprehensive treatment including advanced topics.

Fukaya Categories

• Paul Seidel, "Fukaya Categories and Picard-Lefschetz Theory" (2008)
  Zurich Lectures in Advanced Mathematics
  The foundational text on Fukaya categories, assuming background in Floer theory.
• Denis Auroux, "A beginner's introduction to Fukaya categories" (2013)
  arXiv:1301.7056
  Accessible survey article explaining the main ideas without full technical details.
• Kenji Fukaya et al., "Lagrangian Intersection Floer Theory: Anomaly and Obstruction" (2009)
  AMS/IP Studies in Advanced Mathematics
  The multi-volume definitive reference by the originators. Very technical but complete.

Gauge Theory and Analysis

• Saunders Mac Lane, "Categories for the Working Mathematician" (2nd ed., 1998)
  Springer Graduate Texts in Mathematics
  Essential background on category theory used throughout modern geometry.
• Simon Donaldson & Peter Kronheimer, "The Geometry of Four-Manifolds" (1990)
  Oxford Mathematical Monographs
  Classic text on gauge theory; see chapters on instantons and moduli spaces.
• Karen Uhlenbeck, "Connections with L^p bounds on curvature" (1982)
  Communications in Mathematical Physics 83, 31-42
  The original Uhlenbeck compactness theorem (without boundary).

Mirror Symmetry and Physics

• Kentaro Hori et al., "Mirror Symmetry" (2003)
  Clay Mathematics Monographs, AMS
  Comprehensive volume from the physics perspective with mathematical rigor.
• Mohammed Abouzaid, "A geometric criterion for generating the Fukaya category" (2010)
  Publications mathématiques de l'IHÉS 112, 191-240
  Important result on when Fukaya categories are "large enough" for mirror symmetry.

🌐 Online Resources

Dr. Wehrheim's Materials

• UC Berkeley Faculty Page
  math.berkeley.edu/people/faculty/katrin-wehrheim
  Official page with contact information, CV, and links to publications.
• arXiv Author Page
  arxiv.org (search: Katrin Wehrheim)
  All preprints and recent papers.

Video Lectures

• MSRI Workshops
  Mathematical Sciences Research Institute (Berkeley) hosts recorded workshops on symplectic topology
  Search: "MSRI symplectic topology" on YouTube
• Virtual seminars
  Various online seminars in symplectic topology are recorded and posted:
  - Learning Symplectic Geometry Seminar
  - Floer Homology Seminar (various institutions)

Interactive Tools

• nLab (Category Theory Wiki)
  ncatlab.org
  Collaborative wiki with articles on categories, Fukaya categories, TQFTs, etc.
• Kerodon (Higher Category Theory)
  kerodon.net
  Online resource for ∞-categories and higher categorical structures.

Discussion and Community

• MathOverflow
  mathoverflow.net
  Q&A site for research-level mathematics. Search for "Floer homology", "Fukaya category", etc.
• Math Stack Exchange
  math.stackexchange.com
  Q&A for all levels. Good for foundational questions.

🎓 Suggested Learning Paths

Path 1: Undergraduate → Graduate (Building Foundations)

1. Prerequisites: Multivariable calculus, linear algebra, basic topology
2. Differential geometry: Learn smooth manifolds, differential forms, integration
   • Recommended: John Lee, "Introduction to Smooth Manifolds"
3. Algebraic topology: Fundamental group, homology, cohomology
   • Recommended: Allen Hatcher, "Algebraic Topology" (free online)
4. Symplectic geometry basics: Start with Cannas da Silva's lecture notes
5. Introduction to Floer theory: Read Audin-Damian, "Morse Theory and Floer Homology"
6. Advanced Floer theory: McDuff-Salamon, "J-holomorphic Curves and Symplectic Topology"
7. Quilts: Now tackle Wehrheim-Woodward papers

Path 2: Differential Geometer → Symplectic Topology

1. Review symplectic linear algebra and Hamiltonian mechanics
2. Learn about almost complex structures and compatible metrics
3. Study pseudoholomorphic curves: start with McDuff-Salamon basics
4. Understand Floer homology as Morse theory for path spaces
5. Explore Lagrangian Floer homology and Fukaya categories
6. Study Wehrheim's work on correspondences and quilts

Path 3: Algebraic Topologist → Floer Theory

1. Learn symplectic geometry (focus on Hamiltonian systems)
2. Understand how Floer homology generalizes Morse homology
3. Study spectral sequences connecting Floer and classical homology
4. Learn category theory and the structure of Fukaya categories
5. Explore the 2-categorical structure in Wehrheim-Woodward

Path 4: Gauge Theorist → Symplectic Applications

1. Review Atiyah-Floer conjecture (relating instanton and Lagrangian Floer homology)
2. Study Wehrheim's "Uhlenbeck Compactness" for manifolds with boundary
3. Learn how symplectic reduction relates to gauge theory
4. Explore connections between Donaldson theory and symplectic topology
5. Investigate symplectic Khovanov homology

💻 Computational Tools

Note: Floer homology is notoriously difficult to compute, and general-purpose software is limited. However, some tools exist for special cases:

Symbolic Computation

• SageMath (sagemath.org)
  Open-source mathematics software with support for:
  - Algebraic topology (homology computations)
  - Differential geometry (basic calculations)
  - Symbolic manipulation
• Macaulay2 (macaulay2.com)
  For algebraic geometry computations relevant to mirror symmetry

Specialized Tools

• Heegaard Floer Homology Software
  Various implementations for 3-manifold invariants
  See: Zoltán Szabó's resources
• Knot Theory Software
  - KnotInfo database
  - SnapPy for hyperbolic 3-manifolds

Visualization

• Python scientific stack: NumPy, SciPy, Matplotlib
  For visualizing symplectic manifolds, phase portraits, etc.
• Mathematica/MATLAB: Commercial tools for visualization and numerical computation

🏫 Academic Programs and Research Centers

Leading Centers for Symplectic Topology

• UC Berkeley (Dr. Wehrheim's institution) — Strong group in symplectic topology and geometry
• Stanford University — Mirror symmetry and related areas
• MIT — Symplectic topology, Floer theory, gauge theory
• Columbia University — Gauge theory, symplectic topology
• IAS (Institute for Advanced Study) — Regular special years on related topics
• MSRI (Berkeley) — Workshops and semester programs
• University of Cambridge (UK) — Symplectic geometry group
• ETH Zürich (Switzerland) — Strong in symplectic topology

Summer Schools and Workshops

• Park City Mathematics Institute (PCMI) — Regular summer schools on geometry/topology
• Simons Center (Stony Brook) — Workshops on symplectic topology
• MSRI Summer Graduate Schools — Intensive programs for grad students
• AIM (American Institute of Mathematics) — Research workshops

✏️ Practice Problems and Exercises

Undergraduate Level

1. Show that the standard symplectic form $\omega = dx \wedge dy$ on $\mathbb{R}^2$ is closed and non-degenerate.
2. Prove that the unit circle $S^1 \subset \mathbb{R}^2$ is a Lagrangian submanifold with respect to $\omega = dx \wedge dy$.
3. Compute the Hamiltonian flow generated by $H(x,y) = \frac{1}{2}(x^2 + y^2)$ on $\mathbb{R}^2$.
4. Verify Darboux's theorem in dimension 2: any symplectic form on $\mathbb{R}^2$ is locally equivalent to $dx \wedge dy$.

Graduate Level

1. Prove that a Lagrangian submanifold $L \subset (M, \omega)$ has dimension $n = \frac{1}{2}\dim M$.
2. Show that the graph of a closed 1-form on a manifold $M$ is Lagrangian in $T^*M$.
3. Compute the action functional for a path in the loop space of a symplectic manifold.
4. Verify that $\bar{\partial}_J u = 0$ is an elliptic PDE for $u: \Sigma \to M$ where $J$ is an almost complex structure.
5. For the standard torus $T^2 = \mathbb{R}^2/\mathbb{Z}^2$, identify which circles are Hamiltonian isotopic to each other.

Research Level

1. Construct the quilted Floer chain complex for two composable Lagrangian correspondences in $\mathbb{C}$.
2. Verify the A∞-relation for $m_3$ in a simple example of a Fukaya category.
3. Show that the geometric composition of correspondences $(L_1 \times L_2) \circ (L_2 \times L_3)$ satisfies the expected dimensional count.
4. Prove compactness of a moduli space of pseudoholomorphic curves with Lagrangian boundary conditions (1-dimensional example).

🚀 Career Paths in Symplectic Topology

Academic Research

Most researchers in symplectic topology work in academia as professors or postdoctoral researchers. Typical path:

1. Undergraduate: Mathematics major with strong courses in topology, geometry, analysis
2. Graduate (PhD): 5-6 years, focusing on research in symplectic topology or related areas
3. Postdoc: 2-5 years at one or more institutions, building publication record
4. Faculty position: Tenure-track assistant professor → associate → full professor

Adjacent Fields

Skills from symplectic topology transfer to:

• Differential geometry — Riemannian geometry, geometric analysis
• Algebraic topology — Homotopy theory, homological algebra
• Gauge theory — Mathematical physics, quantum field theory
• Algebraic geometry — Mirror symmetry, derived categories
• Dynamical systems — Hamiltonian dynamics, KAM theory

Applications Outside Academia

While symplectic topology is primarily theoretical, the analytical and computational skills are valuable in:

• Data science — Topological data analysis, persistent homology
• Quantum computing — Quantum algorithms, topological quantum computation
• Finance — Quantitative analysis, stochastic calculus
• Machine learning — Geometric deep learning, manifold learning
• Technology — Research positions at tech companies

📰 Staying Current with Research

Preprint Servers

• arXiv.org — Subscribe to math.SG (Symplectic Geometry) for daily updates
  Also check: math.DG (Differential Geometry), math.GT (Geometric Topology)

Journals

Key journals publishing work in symplectic topology:

• Geometry & Topology
• Journal of Symplectic Geometry
• Duke Mathematical Journal
• Inventiones Mathematicae
• Journal of the American Mathematical Society
• Advances in Mathematics
• Selecta Mathematica

Conferences and Meetings

• Gökova Geometry-Topology Conference (annual, Turkey)
• AMS Sectional Meetings (special sessions on symplectic topology)
• ICM (International Congress of Mathematicians) (every 4 years)
• Symplectic Geometry Seminar Series (various institutions, often virtual)

Mailing Lists and Alerts

• Sign up for department seminar announcements at major universities
• Join relevant Google Groups or email lists for your subfield
• Follow researchers on social media (many mathematicians are active on Twitter/X, Mastodon)