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29 M.Sc. Public Lecture
M.Sc. Public Lecture Speaker: Jacob Ender (Western) "Fast Computations of Discrete Homology" Time: 13:00 - 14:00 Room: MC 107 We study discrete cubical homology and algorithms that compute discrete homology groups of graphs. We first survey the existing state of the art algorithms and discuss applications of discrete homology. We then develop the fastest known algorithms for computing the first and second discrete homology groups of graphs and outline a future research program to develop similar algorithms in higher homology degrees. |
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2 Ph.D. Public Lecture
Ph.D. Public Lecture Speaker: Anif Shikder (Western) "Opening the Black Box: An Exactly Solvable Neural Network That Remembers, Predicts, and Remains Mathematically Tractable" Time: 13:00 - 14:00 Room: ZOOM Two problems sit at the heart of machine learning and computational neuroscience: associative memory, how a system recalls patterns from partial input, and sequence modelling, the long-range processing behind modern language models. The dominant architectures, Hopfield networks for memory, transformers and state-space models for sequences, work well but resist analysis. Their computation lives in implicit recurrent updates or all-pairs attention, and we understand it mostly through post hoc interpretability rather than first principles. My thesis addresses that transparency gap by building models whose computation admits an exact, closed-form description. The unifying object is a complex-valued nonlinear oscillator network, a generalization of the Kuramoto model. A nonlinear coordinate transformation linearizes it and collapses the dynamics into a single closed-form propagator, the matrix exponential of K times t. One operator, used in three studies. First, memory. Patterns are stored as eigenmodes of a complex connectivity matrix, and recall becomes a complex-walk filtration: phase-coherent interference along walks through the network reconstructs the occluded input. There are no spurious attractors, and it outperforms modern Hopfield networks on occluded-MNIST, including unseen digits. Second, sequences. Here, I focus on S4, a leading alternative to the transformer for long sequences where attention becomes prohibitively expensive, which makes understanding it especially worthwhile. I establish a formal correspondence between diagonal state-space models and the oscillator network and derive an exact operator for the forward pass of S4. A low-order truncation recovers roughly 94% of full performance on a real-world task. Third, a new architecture, S1, whose entire forward pass is a closed-form operator parameterized by the eigenvalues and eigenvectors of K. It's competitive on several benchmarks, and because the spectrum is explicit, it supports causal control through targeted modal intervention. Together, these show that recall and sequence processing are two regimes of one operator. Memory is the truncated walk-order expansion of the propagator; sequence processing is the same propagator sampled in discrete time, distinguished by the spectrum of K. Mathematical transparency and competitive performance are compatible, with implications for trustworthy sequence models in high-stakes domains. |
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other Speaker: Matthias Franz (Western) "Allocation-free collections in Julia" Time: 10:00 - 10:40 Room: MC 108 I am going to present a Julia package that provides variable-length vectors, sets and dictionaries which in their immutable versions don't allocate memory. Using these types often results in significant speed-ups for performance-critical code. This is often useful for computations with a combinatorial flavour. |
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24 Transformation Groups Seminar
Transformation Groups Seminar Speaker: Matthias Franz (Western) "Formal linear combinations in Julia" Time: 15:00 - 15:30 Room: MC 107 I am going to present a Julia package for computations with formal linear combinations, tensors and (multi)linear maps. The terms appearing in a linear combination can be of any type, and coefficients can be in any commutative ring with unit. The overall aim of the package is to provide functions that are efficient and easy to use. |
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28 Ph.D. Candidacy Exam Lecture
Ph.D. Candidacy Exam Lecture Speaker: Saleh Ahmed (Western) "Failure of Strong Approximation for the Moduli Stack of Elliptic Curves" Time: 13:00 - 14:00 Room: MC 108 A classical problem in arithmetic geometry is determining when the global properties of a space over a number field are determined by its local behavior at all completions. This local-to-global relation can be topologically formulated via strong approximation, which asks whether the rational points of a space are dense in its adelic points away from a specified finite set of places. To make this topological notion of density meaningful, one requires a functorial topology on adelic points that behaves compatibly with the geometric nature of the objects of study. Conrad provides a construction of such a topology on the adelic points of separated and finite type algebraic spaces. In a recent work, Dhillon extended this framework to certain algebraic stacks, enabling the study of strong approximation in the stack-theoretic setting. In this talk, we investigate strong approximation for the moduli stack of elliptic curves $\mathcal{M}_{1,1}$. Since its coarse moduli space is the affine line $\mathbb{A}^1$, a space which satisfies strong approximation, one might expect $\mathcal{M}_{1,1}$ to inherit this property. We show that this intuition is false. Using a quotient stack presentation of $\mathcal{M}_{1,1}$ and analyzing the image of the $j$-invariant, we construct a non-empty open subset of the adelic points of $\mathcal{M}_{1,1}$ containing no rational points, hence proving that $\mathcal{M}_{1,1}$ fails to satisfy strong approximation away from the places $\{2, 3, \infty\}$. |
29 PhD Thesis Defence
PhD Thesis Defence Speaker: Jeremy Gamble (Western) "Random Matrices and Schwinger-Dyson Equations for Dirac Ensembles with Fermions" Time: 10:00 - 11:00 Room: MC 107 Dirac ensembles lie at the intersection of random matrix theory and noncommutative geometry. In particular, we look at the (0,1) fuzzy Dirac ensemble with fermion in the case where the fermion mass is positive. We give an introduction to random matrix models, both analytic and formal, including the relationship between matrix integrals and generating functions of maps in the formal case. We also motivate and introduce the concept of a spectral triple from noncommutative geometry. One application of spectral triples is Dirac ensembles, consisting of a certain finite spectral triple with a probability distribution on the space of Dirac operators. Dirac ensembles describe a toy model of quantum gravity. We describe Dirac ensembles and their underlying hermitian matrix models, giving special attention to the (0,1) fuzzy Dirac ensemble with a fermion. The Schwinger-Dyson equations for this model can be found through complex analytic techniques, and allows for a perturbative solution of the moments. In the case where the potential is Gaussian, an exact solution for moments can be obtained using complex analytic techniques and elliptic integrals. |
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