# Lectures, seminars and dissertations

* Dates within the next 7 days are marked by a star.

Sergej Monavari (EPFL)

**Partitions, motives and Hilbert schemes**

*** ** Wednesday 16 October 2024, 14:15, M2 (M233)

Counting the number of higher dimensional partitions is a hard classical problem. Computing the motive of the Hilbert schemes of points is even harder, and should be seen as the geometric counterpart of the classical combinatorial problem. I will discuss some structural formulas for the generating series of both problems, their stabilisation properties when the dimension grows very large and how to apply all of this to obtain (infinite) new examples of motives of singular Hilbert schemes. This is joint work with M. Graffeo, R. Moschetti and A. Ricolfi.

ADM seminar

Tapani Matala-aho

**An analogue of Siegel's determinant**

*** ** Wednesday 16 October 2024, 16:15, M3 (M234)

The Siegel-Shidlovskii theory is a powerful method for studying transcendence and algebraic independence questions of analytic functions, in particular, of E-functions including entire hypergeometric series.
A crucial step in this method involves a non-vanishing proof for the determinants attached to the linear forms, derivatives of an auxiliary function L(t). Instead of the usual derivative D we use the derivative tD.
We give a short proof for the non-vanishing of modified determinants for a class of differential equations including a subclass of hypergeometric differential equations. As a corollary we get an irreducible criterion for the corresponding differential operator. Further, by some basics from differential modules we prove a converse statement.

ANTA Seminar / Hollanti et al.

Sari Rogovin

**Linear dilatation and absolute continuity**

Wednesday 23 October 2024, 10:15, M3 (M234)

Seminar on analysis and geometry

Petteri Kaski

** A universal sequence of tensors for the asymptotic rank conjecture**

Tuesday 29 October 2024, 15:15, M2 (M233)

The exponent $\sigma(T)$ of a tensor $T\in\mathbb{F}^d\otimes\mathbb{F}^d\otimes\mathbb{F}^d$ over a field $\mathbb{F}$ captures the base of the exponential growth rate of the tensor rank of $T$ under Kronecker powers. Tensor exponents are fundamental from the standpoint of algorithms and computational complexity theory; for example, the exponent $\omega$ of matrix multiplication can be characterized as $\omega=2\sigma(\mathrm{MM}_2)$, where $\mathrm{MM}_2\in\mathbb{F}^4\otimes\mathbb{F}^4\otimes\mathbb{F}^4$ is the tensor that represents $2\times 2$ matrix multiplication.
Our main result is an explicit construction of a sequence $\mathcal{U}_d$ of zero-one-valued tensors that is universal for the worst-case tensor exponent; more precisely, we show that $\sigma(\mathcal{U}_d)=\sigma(d)$ where $\sigma(d)=\sup_{T\in\mathbb{F}^d\otimes\mathbb{F}^d\otimes\mathbb{F}^d}\sigma(T)$. We also supply an explicit universal sequence $\mathcal{U}_\Delta$ localised to capture the worst-case exponent $\sigma(\Delta)$ of tensors with support contained in $\Delta\subseteq [d]\times[d]\times [d]$; by combining such sequences, we obtain a universal sequence $\mathcal{T}_d$ such that $\sigma(\mathcal{T}_d)=1$ holds if and only if Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)] holds for $d$. Finally, we show that the limit $\lim_{d\rightarrow\infty}\sigma(d)$ exists and can be captured as $\lim_{d\rightarrow\infty} \sigma(D_d)$ for an explicit sequence $(D_d)_{d=1}^\infty$ of tensors obtained by diagonalisation of the sequences $\mathcal{U}_d$. As our second result we relate the absence of polynomials of fixed degree vanishing on tensors of low rank, or more generally asymptotic rank, with upper bounds on the exponent $\sigma(d)$. Using this technique, one may bound asymptotic rank for all tensors of a given format, knowing enough specific tensors of low asymptotic rank.
Joint work with Mateusz Michałek (U. Konstanz).
arXiv: https://arxiv.org/abs/2404.06427

ADM seminar

Henri Lahdelma

**TBA**

Wednesday 30 October 2024, 10:15, M3 (M234)

Seminar on analysis and geometry

Lizao Ye

**TBA**

Tuesday 05 November 2024, 15:15, M2 (M233)

TBA

ADM seminar

Leah Schätzler

**TBA**

Wednesday 06 November 2024, 10:15, M3 (M234)

Seminar on analysis and geometry

Prof. Anne-Maria Ernvall-Hytönen (U. Helsinki)

**TBA**

Wednesday 06 November 2024, 16:15, M3 (M234)

ANTA Seminar / Hollanti et al.

**Matematiikan kandiseminaari (Bachelor thesis seminar in Math.)**

Friday 15 November 2024, 09:00, M3 (M234)

Further information

Milo Orlich

**TBA**

Tuesday 19 November 2024, 15:15, M2 (M233)

TBA

ADM seminar

Alex Takeda

**TBA**

Tuesday 26 November 2024, 15:15, M2 (M233)

TBA

ADM seminar

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