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Texas A&M University
Mathematics

Topology Seminar

Fall 2021

 

Date:October 6, 2021
Time:4:00pm
Location:Zoom
Speaker:Shu Shen, Sorbonne University
Title:The Fried conjecture for admissible twists
Abstract:The relation between the spectrum of the Laplacian and the closed geodesics on a closed Riemannian manifold is one of the central themes in differential geometry. Fried conjectured that the analytic torsion, which is an alternating product of regularized determinants of the Laplacians, equals the zero value of the dynamical zeta function. In this talk, I will show the Fried conjecture on locally symmetric spaces twisted by an acyclic flat vector bundle obtained by the restriction of a representation of the underlying Lie group. This generalises the results of myself for unitary twists, and the results of Brocker, Muller, and Wotzker on closed hyperbolic manifolds.

Date:October 13, 2021
Time:4:00pm
Location:Zoom
Speaker:Mark Shoemaker, Colorado State University
Title:Enumerative geometry and mirror symmetry
Abstract:The goal of enumerative geometry is to study a geometric space by counting certain subspaces within it. The first result in enumerative geometry is Euclid's observation that, given 2 distinct points in the plane, there is a single line through these points. A harder question is, given 2 points and 3 random lines in the plane how many conics (degree 2 curves) pass through both points and are tangent to each of the lines. These types of questions have interested geometers since the 1800's and earlier, but they are famously difficult. However, a breakthrough occurred in the 1990's when a surprising connection was made with physics. It was discovered that techniques and intuitions from string theory could be used to answer longstanding questions in enumerative geometry. The phenomenon behind this remarkable connection came to be known as mirror symmetry. In this talk I will give an introduction to mirror symmetry and its connection to enumerative geometry. At the end of the talk I will mention some current directions of inquiry and open questions.

Date:October 20, 2021
Time:4:00pm
Location:Zoom
Speaker:Chris Gerig, Simons Center for Geometry and Physics, Stony Brook University
Title:Studying 4-spheres using near-symplectic geometry and ECH
Abstract:I will introduce certain 2-forms that are "nearly symplectic" and use ECH (embedded contact homology) to probe homotopy 4-spheres. The “invariants” built for homotopy 4-spheres will count pseudoholomorphic curves in the complement of circles, but at the moment they are not sensitive enough to distinguish/detect 4-spheres: I will discuss ideas to possibly refine these “invariants”.

Date:November 3, 2021
Time:4:00pm
Location:Zoom
Speaker:Cheuk Yu Mak, University of Edinburgh
Title:Lagrangian Floer theory and a simplicity problem
Abstract:It is a classical and fundamental problem to study algebraic properties (e.g. simplicity, perfectness) of the automorphism group of an object. Building on the foundational works of Kirby, Thurston, Fathi and many others, there are a lot of studies for the automorphism group of compact (smooth) manifolds, possibly equipped with additional structures. When it comes to the simplicity of volume preserving homeomorphism groups, surprisingly, the higher dimensional cases are well-understood and the 2 dimensional case is more mysterious. In this talk, I will explain how to combine ideas from Lagrangian Floer theory and Hofer geometry to completely resolve this 40-year-old question. The technical heart of the proof is an extension of Calabi homomorphism, which answers a question of Ghys at his 2006 ICM talk on knots and dynamics. This is based on a joint work with Daniel Cristofaro-Gardiner, Vincent Humili`ere, Sobhan Seyfaddini and Ivan Smith.

Date:November 10, 2021
Time:4:00pm
Location:zoom
Speaker:Dylan Wilson, Harvard University
Title:Higher Bott periodicities in Algebraic K-theory
Abstract:Algebraic K-theory is a powerful invariant that encodes a lot of information in number theory and geometric topology. One of the deepest theorems about the algebraic K-theory of number rings is that it approximately behaves like complex topological K-theory, which famously satisfies Bott periodicity (this is the Lichtenbaum-Quillen conjecture, now resolved by work of Voevodsky and many others). I will describe joint work with Jeremy Hahn where we exhibit an analog of this result involving periodicities of larger and larger 'wavelength', thus affirming the Redshift Conjecture of Ausoni-Rognes for a large class of examples.

Date:November 17, 2021
Time:4:00pm
Location:Zoom
Speaker:Erkao Bao, University of Minnesota
Title:An invitation to contact homology
Abstract:Contact homology is an invariant of the contact structure, which is an odd-dimensional counterpart of a symplectic structure. It was proposed by Eliashberg, Givental and Hofer in 2000. The application of contact homology and its variants include distinguishing contact structures, knot invariants, the Weinstein conjecture and generalization, and calculating Gromov-Witten invariants. In this talk, I will start with the notion of contact structures, then give a heuristic definition of the contact homology as an infinite dimensional Morse homology, and explain the major difficulties to make the definition rigorous. In the very end, I will talk about the chain homotopy type of contact differential graded algebra. This is a joint work with Ko Honda.

Date:December 1, 2021
Time:4:00pm
Location:Zoom
Speaker:Carissa Slone, University of Kentucky
Title: Characterizing 2-slices over C_2 and K_4
Abstract:The slice filtration focuses on producing certain irreducible spectra, called slices, from a genuine G-spectrum X over a finite group G. We have a complete characterization of all 1-, 0-, and (-1)-slices for any such G. We will characterize 2-slices over $C_2$ and expand this characterization to $K_4 = C_2 \times C_2$.

Date:December 8, 2021
Time:4:00pm
Location:Zoom
Speaker:William Balderrama, University of Virginia
Title:Deformations of homotopy theories and spectral sequences
Abstract:A standard technique in homotopy theory is to compute something homotopical by relating it to something algebraic. A standard example is the Künneth theorem, relating the homology H_*(X x Y) to the Tor groups Tor(H_*X, H_*Y). This reflects a more general homotopical construction, namely the Künneth spectral sequence for computing tensor products of modules over ring spectra. I will describe a novel method of producing ``algebra-to-homotopy'' spectral sequences such as the latter, which proceeds by considering certain deformations of homotopy theories, both stably and unstably.