This Week in Mathematics


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Today:

Topology seminar
String topology and graph cobordisms
- Andrea Bianchi, University of Bologna
Time: 3:05PM Room: Zoom
Abstract/Desc: String topology, introduced by Chas and Sullivan, is the study of the homology of mapping spaces of the form M^X, where M is a closed oriented d-dimensional manifold and X is a space. Fixing M and letting X vary, the homology groups H_*(M^X) carry additional algebraic structure coming from (contravariant) functoriality in X and from Poincare' duality of M; the most famous example is the Chas-Sullivan product. We introduce a symmetric monoidal infty-category GrCob of "graph cobordisms between spaces"; we define compatible local coefficient systems xi_d on the morphism spaces of GrCob, and use the twisted homology of the morphism space GrCob(Y,X) to define higher string operations H_*(M^X)--->H_*(M^Y). We assemble all such operations into a "graph field theory" associated with M, i.e. a contravariant symmetric monoidal functor out of the linearisation of GrCob given by xi_d. We recover some basic operations, including the Chas-Sullivan product, as special cases.The construction of the graph field theory can in fact be carried out for any oriented Poincare' duality space M, and is natural in M with respect to orientation-preserving equivalences; in particular all string operations we obtain are automatically homotopy invariant. The main technical input to the construction is a recent result by Barkan-Steinebrunner, giving a universal property for the category of graph cobordisms between finite sets in terms of commutative Frobenius algebras.

Entries for this week: 7

Monday March 30, 2026

Colloquium
Structure in group-labeled graphs and its applications
- Youngho Yoo, Alaska Fairbanks
Time: 3:05 Room: LOV 101
Abstract/Desc: Can a cycle of length L modulo M be found in a given graph in polynomial time? This problem was first posed by Arkin, Papadimitriou, and Yannakakis (1991), motivated by periods of Markov chains, and reiterated in the study of graph databases and extremal graph theory. I will discuss recent work on the structure of group-labeled graphs that resolves this problem in a far more general form. Our work also provides a characterization of the topological obstructions to an approximate packing-covering duality for cycles of length L modulo M, resolving a problem of Dejter and Neumann-Lara (1988). I will further discuss applications of our work to other group-expressible length constraints.

Tuesday March 31, 2026

Topology seminar
String topology and graph cobordisms
- Andrea Bianchi, University of Bologna
Time: 3:05PM Room: Zoom
Abstract/Desc: String topology, introduced by Chas and Sullivan, is the study of the homology of mapping spaces of the form M^X, where M is a closed oriented d-dimensional manifold and X is a space. Fixing M and letting X vary, the homology groups H_*(M^X) carry additional algebraic structure coming from (contravariant) functoriality in X and from Poincare' duality of M; the most famous example is the Chas-Sullivan product. We introduce a symmetric monoidal infty-category GrCob of "graph cobordisms between spaces"; we define compatible local coefficient systems xi_d on the morphism spaces of GrCob, and use the twisted homology of the morphism space GrCob(Y,X) to define higher string operations H_*(M^X)--->H_*(M^Y). We assemble all such operations into a "graph field theory" associated with M, i.e. a contravariant symmetric monoidal functor out of the linearisation of GrCob given by xi_d. We recover some basic operations, including the Chas-Sullivan product, as special cases.The construction of the graph field theory can in fact be carried out for any oriented Poincare' duality space M, and is natural in M with respect to orientation-preserving equivalences; in particular all string operations we obtain are automatically homotopy invariant. The main technical input to the construction is a recent result by Barkan-Steinebrunner, giving a universal property for the category of graph cobordisms between finite sets in terms of commutative Frobenius algebras.

Wednesday April 01, 2026

Biomathematics Seminar
Spectral reduction methods for complex networked systems
- Shiyi Lyu, FSU
Time: 3:05 Room: Love 232
Abstract/Desc: Low-dimensional reductions provide an important tool for understanding spreading dynamics on complex networks. We study nonlinear SIS-type dynamics on undirected scale-freenetworks, where strong degree heterogeneity can drive sharp transitions. We focus on a reversible spreading model with nonlinear incidence on a weighted adjacency network and use forward and backward parameter sweeps to numerically resolve bistability and identify the transition points. Building on the Gao–Barzel–Barab´asi (GBB) mean-field reduction, we use the GBB formulation as a baseline and then refine the reduction to better capture the heterogeneous core structure typical of scale-free graphs. This refinement reduces systematic offsets in predicted thresholds and more accurately reproduces the branch structure observed near explosive transitions. The resulting reduced description enables improved prediction of critical points and transition behavior compared with the standard GBB reduction. By contrast, in Erd˝os–R´enyi networks, where heterogeneity is much weaker, the standard GBB reduction already provides an accurate description and no refinement is required.

Thursday April 02, 2026

Algebra seminar
Bands, Idylls, Valuated Matroids and Linear Spaces
- Jeffery Liu, FSU
Time: 3:05pm Room: LOV 0232
Abstract/Desc: Matroids encode dependencies among elements of a finite set. One class of examples are "realizable" matroids, those which come from linear dependence among an arrangement of vectors in a vector space. However, most matroids are not realizable. We survey the development of ring-like and field-like objects called "bands" and "idylls" respectively, and matroids with coefficients valued in a band or idyll. In particular, all matroids can be valued in the Krasner hyperfield, which is the terminal object in the category of idylls. Then, following the band/idyll-theoretic model we introduce a vector space-like object over an idyll whose vector arrangements realize matroids valued in that idyll.

Friday April 03, 2026

Mathematics Colloquium [url]
Graduate Student Flash Talks
- FSU Grad Students, FSU
Time: 3:05 Room: Lov 101
Abstract/Desc: Participating graduate students will give flash talks about their research in this competition.

Data Science and Machine Learning Seminar [url]
AI for Scientific Discovery
- Zavala Romero group, FSU
Time: 12:00 Room: DSL/SC-499
Abstract/Desc: This talk examines how machine learning and artificial intelligence are being used not only in healthcare applications, but also as tools for scientific discovery in biomedicine and medicine. It traces the field from early predictive models and image-based deep learning to modern foundation models, multimodal systems, and AI-enabled software used in clinical and research settings. Rather than focusing on heavy mathematics, the emphasis is on how these methods support discovery, hypothesis generation, pattern finding, and decision-making across biological, medical, and healthcare data.

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Department of Mathematics

This Week in Mathematics