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Hilbert modular forms from orthogonal modular forms on binary lattices
Haochen Wu, Dartmouth College
We show the explicit connection between Hilbert modular forms and orthogonal modular forms arising from positive definite binary lattices over the ring of integers of a totally real number field. Our work uses the even Clifford algebra to generalize Gauss composition in a categorical way over a Dedekind domain. This allows us to classify the class sets of different types of genera in terms of the class groups of the associated quadratic algebras. This is joint work with John Voight.
14:00–15:00 Combinatorics Seminar, Kemeny 307
Patterns in Rectangulations: T-like patterns, Inversion Sequences, and Dyck Paths
Michaela Polley, Dartmouth
Pattern avoidance is a foundational concept in combinatorics. In this talk I will define the notion of pattern avoidance in rectangulations (a decomposition of a rectangle into a finite number of rectangles such that no four of them meet at a corner), and then focus in on the case where the patterns we are avoiding are T shapes (in some rotation). This leads to a number of surprising connections to inversion sequences and Dyck paths, including a solution to a conjecture about the Wilf-equivalence of two classes of inversion sequences. This talk is based on joint work with Andrei Asinowski (University of Klagenfurt) and our preprint https://arxiv.org/abs/2501.11781.
14:30–15:30 Applied and Computational Mathematics Seminar, Zoom
Mixture of Experts in Large-scale and Multimodal Models
Huy Minh Nguyen, University of Texas Austin
Mixture of experts (MoE) framework has recently emerged as an effective approach to
enhancing the efficiency and scalability of machine learning models by aggregating the power
of multiple sub-models, called experts, through an adaptive gating network. In this talk, I
will present our investigation into two MoE components that are central to the success of the
DeepSeek-V3 language model. In particular, I will first demonstrate the benefits of the normalized
sigmoid gating mechanism over the conventional softmax gating. Then, I will examine the effects
of the shared expert structure on the expert convergence behavior. Next, I will introduce the
connection between MoE and the self-attention mechanism in the Transformer model architecture
as well as its applications parameter-efficient fine-tuning methods. Finally, in the context of
multimodal learning where data consist of different modalities, including time series, text, and
images, I will highlight our MoE router designs for integrating these data modalities, followed by
an empirical comparison with previous methods in the literature.
We consider a Hamiltonian action of a compact Lie group G on a complete Kaehler manifold M with a proper moment map. I define a regularized version of the Dolbeault cohomology of a G-equivariant holomorphic vector bundle, called the background cohomology. I show that the background cohomology of a prequantum line bundle over M `commutes with reduction', i.e. the invariant part of the background cohomology is isomorphic to the usual Dolbeault cohomology of the symplectic reduction.
Wednesday, April 23
18:00 Talks for Undergraduate Students, Kemeny 007
Min-Max Geometry
Juliette Bruce, Dartmouth College
Abstract: We will discuss a strange notation of arithmetic where we replace our standard operations of addition and multiplication with what some call "tropical" addition and "tropical" multiplication. We will explore how making these changes provides connections between polynomials, combinatorics, and applied mathematics.
Thursday, April 24
15:15 Math Colloquium, 006 Kemeny Hall
Domino tilings beyond 2D
Caroline Klivans, Brown University
There is a rich history of domino tilings in two dimensions. Through a variety of techniques we can answer questions such as: how many tilings are there of a given region or what does a random tiling look like? These questions and their answers become significantly more difficult in dimension three and above. Despite this curse of dimensionality, I will discuss recent advances in the theory. I will also highlight problems that still remain open.
