SEC Geometry-Topology 2025

SEC Geometry and Topology Workshop

May 9-11, 2025 University of Alabama

Goal

Welcome to the second SEC Geometry and Topology Workshop. This workshop aims to bring together active researchers, ranging from leading experts to recent PhDs and graduate students to discuss some common research in modern geometry and topology. The workshop will include two mini lecture series, problem sessions, research talks and professional development activities.    

Mini Lectures

• Joe Breen (Iowa)
• Josh Sabloff (Haverford)

Research Talks

• John Etnyre (Ga Tech)
• Shunyu Wan (Ga Tech)
• Sumeyra Sakalli (South Florida)
• Scott Baldridge (Louisiana State)
• Tanushree Shah (Chennai Math Institute)
• Eduardo Fernandez (Georgia)

Registration and support

Some funding is available for workshop participants. To register and request funding please email Bulent Tosun at btosun@ua.edu. The deadline to request funding is April 10.   

The workshop is sponsored by the National Science Foundation Grants DMS 2144363, DMS 2105525 and the Simons Foundation.

Local Information and Venue

Accommodations:

• Capstone Inn–a few minutes walking distance to the math department.
• Hampton Inn — 15-20 minutes walking distance to the math department.

Restaurants:

• Many restaurant options in the strip area which is in 10-15 minutes walking distance from the math department. 
• Archibald’s and Dreamland are well-known/authentic BBQ options in Tuscaloosa
• River, Chuck’s Fish, Five, Central Mesa, Avenue Pub, Evangeline’s, Antojitos Izcalli are some good restaurants in Tuscaloosa.  

Venue: All the talks will be in Gordon Palmer (GP 208) where the mathematics department is located. See the campus map

Schedule (tentative)

Participants

• Bulent Tosun (Alabama)
• Bruce Trace (Alabama)
• Lawrence Roberts (Alabama)
• Jon Carson (Alabama)
• Thomas Polstra (Alabama)
• Kyungyong Lee (Alabama)
• Martyn Dixon (Alabama)
• Martin Evans (Alabama)
• Uly Alvarez (Alabama)
• Tucker Ervin (Alabama)
• Icarus Harrison (Alabama)
• Nicole Bruner (Alabama)
• Evan Lee (Alabama)
• Alex Seidel (Alabama)
• Alex Carlson (Alabama)
• Scott Baldridge (LSU)
• Remi Mandal (LSU)
• Benjamin Armokyi Appiah (LSU)
• Joe Breen (Iowa)
• Josh Sabloff (Haverford)
• Sumeyra Sakalli (South Florida)
• Shunyu Wan (Georgia Tech)
• John Etnyre (Georgia Tech)
• Thomas Rodewald (Georgia Tech)
• Eli Sean (Georgia Tech)
• Tanushree Shah (Chennai Math Institute)
• Wei Zhou (Complutense University of Madrid/UGA)
• Nur Saglam (UGA)
• Yukun Du (UGA)
• Sam Lisi (Ole Miss)
• Alex Thomas (UArk)
• Kevin Tuttle (UArk)
• Nilangshu Bhattacharya (LSU)
• Edu Fernandez (UGA)
• Sayani Mukherjee (LSU)
• Krishnendu Kar (LSU)
• Adithyan Pandikkadan (LSU)
• Abhijeet Ghanwat (UGA)
• Alexander Tepper (UGA)
• Joe Lopez (UTK)
• Veronica King (Randolph High School)

Titles and Abstracts

Josh Sabloff

Title: Perspectives on the Geometry of Lagrangian Cobordisms
​
Abstract: In the first talk, we will introduce several types of Lagrangian cobordisms: the classical Arnol’d definition, cylindrical-at-infinity cobordisms between Legendrians, and Lagrangian tangles, which, in some sense, generalize both of the earlier notions.  We will discuss what questions one can ask about the symplectic geometry of Lagrangian cobordisms, and describe several sample results for each type of Lagrangian cobordism.

In the second talk, we will examine more closely the symplectic geometry of Lagrangian tangle links in the product of a surface with the complex numbers.  The main tool is a novel Floer theory for Lagrangian tangles inspired by Morse theory for manifolds with gradient field tangent to the boundary.  This is joint work with Ipsita Datta (ETH Zurich).\

Joe Breen

Title: Open books and contact handle calculus

Abstract: The Giroux correspondence between open books and contact structures, a cornerstone result of contact topology in dimension 3, arose from Giroux’s development of Morse theory on contact manifolds. Morse theory and handle calculus is also a fundamental tool in, for example, 4-dimensional smooth and symplectic topology, but its use in contact topology in dimensions beyond 3 has been limited until recently. In these talks, I will explain some recently developed contact handle calculus techniques, along with applications (e.g. the Giroux correspondence in all dimensions) and future directions.  

Shunyu Wan

Title: Legendrian surgery, LOSS invariant and contact invariant.

Abstract: In this talk we will discuss some effects of Legendrian surgery on Legendrian knot in S^3 using Heegaard Floer contact invariant and LOSS invariants. More specifically, we study when map from the LOSS invariant of the dual Legendrian to the contact invariant of the surgery manifold is injective. We will see how injectivity allows us to conclude some interesting results about Legendrian surgery on non-simple knots and non-loose knots. This is joint work with Hugo Zhou.

Sumeyra Sakalli

Title: Topological Approach to Algebraic Genus Two Singular Fibers

Abstract: Kodaira’s classification of singular fibers in elliptic fibrations and its translation into the language of monodromies by Harer, Kas, Kirby was one of the main tools in constructions of exotic 4-manifolds. Later, Namikawa and Ueno classified singular fibers of genus two families of algebraic curves. A subset of those fibers has finite order monodromies, which, in this sense, is analogous to Kodaira’s genus one fibers, and has received attention in the literature. In this talk, first we will translate between singular fibers of genus two families of algebraic curves with finite-order monodromies, and the positive Dehn twist factorizations of Lefschetz fibrations. Our result can be thought as analogous to Harer, Kas, and Kirby’s result on Kodaira’s singular fibers. Then we will construct some of these fibers with a different approach and talk about the classification problem for algebraically exotic fibers. This is based on joint works with J. Van Horn-Morris, arXiv:2303.01554 and arXiv:2311.00264.

John Etnyre

Title:  Symplectic rational Homology ball fillings of Seifert fibered spaces

Abstract: There has been a lot of work towards studying when a small Seifert fibered space bounds a rational homology ball. Not only is this inherently interesting, it is related to generalizing rational blowdowns that have been effectively used to build exotic 4-manifolds. In this talk, we will discuss various constructions, highlight the differences between the smooth and symplectic category, and completely characterize for most small Seifert fibered spaces whether or not they bound symplectic rational homology balls. This is joint work with Burak Ozbagci and Bülent Tosun.

Scott Baldridge

Title: State-Reducibility and a New Gauge-Theoretic Approach to the Four Color Theorem

Abstract: The Birkhoff diamond has played a central role in attempts to prove the Four Color Theorem since it was first identified over a century ago as the minimal nontrivial reducible configuration. In this talk, I will present a new proof of its reducibility using filtered $3$- and $4$-color homology, which arise from a (2+1)-dimensional topological quantum field theory inspired by Khovanov and Lee homologies. This approach avoids Kempe-switch techniques by introducing the notion of state-reducibility, which analyzes face colorings on ribbon graphs associated to the original graph. As an application of a broader TQFT framework developed in earlier work, this result provides an independent verification of a key configuration and suggests a potential pathway toward a non-computer-assisted proof of the Four Color Theorem. This is joint work with Ben McCarty.

Tanushree Shah

Title: Rationally null homologous knots in contact 3 Manifold

Abstract: In an orientable 3-manifold, one can enrich the topology with an additional geometric structure known as a contact structure. Within this setting, an important class of knots called Legendrian knots arise. Beyond their topological knot type, Legendrian knots are distinguished by two classical invariants: the Thurston–Bennequin number and the rotation number. A knot is said to be Legendrian simple if these two invariants, together with the topological knot type, completely classify its Legendrian isotopy class.

While Legendrian simplicity has been extensively studied for knots in the standard contact 3-sphere, much less is known about this property in more general 3-manifolds. In this talk, we explore how different invariants behave under the connected sum operation in arbitrary contact 3-manifolds, and how this affects Legendrian simplicity. We then examine specific examples in the setting of Seifert fibered spaces, where the interaction between topology and contact geometry produces rich families of non-simple Legendrian knots.

Eduardo Fernandez

Title: Non-flexible loops of loose Legendrians in 3 dimensions

Abstract: In recent years, significant progress has been made in understanding families of Legendrians in 3-dimensional contact topology. For instance, the homotopy type of the component of the max-tb Legendrian unknot (or any algebraic link) in the standard contact 3-sphere is now completely understood. However, the existence of non-flexible families of Legendrians has remained an open question. In this talk, we will explore this question in the context of loops of Legendrians. Specifically, we will show that in every closed overtwisted contact 3-manifold, there exists a non-contractible yet formally contractible loop of loose Legendrians. The proof relies on h-principles and parametrized Legendrian surgery, drawing inspiration from Dave Gay’s work on diffeomorphisms of the 4-sphere. This talk is based on joint work in progress with Fabio Gironella.

Workshop Photos