Geometry Meets Physics: Finiteness, Tameness, and Complexity

Dates: November 12–14, 2025

Location: CMSA G10, 20 Garden Street, Cambridge MA 02138

(note: this event is in-person only)

Finiteness is a fundamental property in consistent physical theories. From the earliest days of quantum field theory and string theory, the drive to eliminate unphysical infinities has been a guiding principle. More recently, finiteness has emerged as a key criterion for constraining effective theories that can be embedded in quantum gravity.  Formulating and testing these constraints remains a central challenge in current research.

In parallel, mathematics has made remarkable advanced in addressing finiteness questions using tame geometry. Built on the framework of o-minimal structures, tame geometry offers a precise language for describing objects of finite geometric complexity. Recent developments, such as sharp o-minimality, go further by introducing a quantitative notion of complexity, opening new directions for analyzing finiteness in mathematics and physics alike.

This workshop brings together mathematicians and physicists to exchange ideas, explore new perspectives, and spark collaborations at the interface of geometry, logic, and fundamental physics.

Invited Speakers

• Vijay Balasubramanian (UPenn)
• Gregorio Baldi (CNRS, IMJ-PRG & IAS)
• Gal Binyamini (Weizmann Institute & IAS)
• Raf Cluckers (Lille, France)
• Matilda Delgado (Max Planck Institute Munich)
• Bruno Klingler (Humboldt University, Berlin & IAS)
• Adele Padgett (Vienna)
• David Prieto (Utrecht)
• Washington Taylor (MIT)
• David Urbanik (IHES, France & IAS)
• Cumrun Vafa (Harvard)
• Mick van Vliet (Utrecht)
• Benny Zak (Weizmann Institute & IAS)

Organizers: Thomas Grimm, Harvard CMSA & Utrecht University | Gal Binyamini, Weizmann Institute & IAS | Bruno Klingler, Humboldt University, Berlin & IAS

Schedule

(download pdf)

Wednesday Nov. 12, 2025

8:30–8:55 am
Morning refreshments (Common Room)

8:55–9:00 am
Introductions

9:00–10:30 am
Lecture
Speaker: Gal Binyamini, Weizmann Institute & IAS
Title: O-minimality: finiteness and complexity
Abstract: O-minimality is a mathematical formalism of “tame geometry”: a geometry where every set has finite geometric complexity. I will give an introduction to o-minimality in general, and to quantitative variants where one measures the complexity of sets in terms of some natural parameters. I’ll try to focus on the main examples that potentially come up in the interaction with physics, and describe the state of the art and some conjectures.

10:30–11:00 am
Break

11:00 am–12:00 pm
Speaker: Benny Zak, Weizmann Institute & IAS
Title: Analytic tameness – complex cells
Abstract: Complex cells are a complex anayltic version of cells from o-minimality, invented by Binyamini and Novikov. We aim to introduce complex cells, and demonstrate their usefullness in quantifying the analytic information present in a complex set. If time permits, we will discuss applications of this theory.

12:00–1:00 pm
Catered Lunch (Common Room)

1:00–2:30 pm
Lecture
Speakers: David Prieto and Mick van Vliet, Utrecht
Title: Tameness and Complexity in Physical Theories
Abstract: We give an introductory overview of recent applications of o-minimality to physics, focusing on quantum field theories and quantum gravity. In the first part of the lecture we explain how o-minimality makes a first appearance in physical theories when considering amplitudes in quantum field theory. In the second part, we concentrate on a class of theories where finiteness principles seem to be essential, namely the quantum field theories which are consistent with quantum gravity. We review some of these finiteness principles and interpret them through the lens of the o-minimal framework. Along the way, we highlight recent progress in this direction, as well as open questions to explore in the future.

2:30–3:00 pm
Break with refreshments (Common Room)

3:00–4:00 pm
Speaker: Matilda Delgado, Max Planck Institute Munich
Title: Dualities and the Compactifiability of Moduli Space
Abstract:  After introducing (self-)dualities in string theory and their action on the field content & spectrum of the theory, I will present the notion of compactifiability for the moduli space of massless fields as the condition that its volume is finite or grows no faster than Euclidean space. I will argue that compactifiability generically implies the existence of non-trivial dualities by providing evidence from string theory. Moreover, I will explain how one can connect compactifiability to the condition that the spectrum of objects charged under the duality group transform in a semisimple representation. Finally, I will provide a bottom-up argument for compactifiability, and argue that it (at least in supersymmetric cases) can be explained by the finiteness of the number of massless states upon compactification to 1D. Based on arXiv:2412.03640.

5:00 PM
Millennium Lecture and Reception: Pierre Deligne (IAS) (Science Center Hall D)
Title: What is the Hodge conjecture?

Thursday, Nov. 13, 2025

8:30–9:00 am
Morning refreshments (Common Room)

9:00–10:30 am
Lecture
Speaker: Bruno Klingler, Humboldt University, Berlin & IAS
Title: Tame geometry and Hodge theory
Abstract: I will give an introduction to applications of o-minimality in complex geometry, in particular in Hodge theory.

10:30–11:00 am
Break

11:00 am–12:00 pm
Speaker: Cumrun Vafa, Harvard
Title: The Swampland Program

12:00–1:30 pm
Catered Lunch (Common Room)

1:30–2:30 pm
Speaker: Gregorio Baldi, CNRS, IMJ-PRG & IAS
Title: The Hodge locus
Abstract: We will survey various recent results around the distribution of the Hodge locus of a (mixed) variation of Hodge structures. Various concrete applications to moduli spaces will also be presented.

2:30–3:00 pm
Break with refreshments (Common Room)

3:00–4:00 pm
Speaker: Vijay Balasubramanian, U Penn
Title: Chaos and complexity in quantum dynamics

4:30–5:30
Discussion/Q&A session

6:30 PM
Dinner: Changsho Restaurant, 1712 Massachusetts Ave., Cambridge, MA 02138

Friday Nov. 14, 2025

8:30–9:00 am
Morning refreshments (Common Room)

9:00–10:00 am
Speaker: Washington Taylor, MIT
Title: Finiteness, connectivity, and the power of fibrations in the Calabi-Yau landscape

10:00–10:30am
Break

10:30–11:30 am
Speaker: Adele Padgett, Vienna
Title: Tameness of multisummable series
Abstract: There are sophisticated theories of summability that map divergent series solutions of differential or functional equations to solutions that are holomorphic in sector-like domains. Van den Dries and Speissegger proved that functions obtained from real multisummable power series have tame geometric behavior when restricted to the real numbers. It would be desirable to know that these functions are also tame on their whole sector-like domains, but recently Speissegger and I proved that these functions are in general only tame on part of their domains. I will present this result and discuss the domains on which some examples are tame, including the Stirling series which appears in the asymptotic expansion of the Gamma function. In this talk, “tame” means definable in an o-minimal structure.

11:30 am–1:00 pm
Catered Lunch (Common Room)

1:00–2:00 pm
Speaker: Raf Cluckers, Lille, France
Title:  Finiteness and tameness in (non-archimedean) geometry
Abstract: Non-archimedean geometry work with orders of magnitude rather than with precise measurements. The former works for example with orders of vanishing of functions, and the latter typically works with real or complex numbers. I will discuss recent progress on non-archimedean tame geometry. I will present analogues of o-minimality, of Pila-Wilkie’s o-minimal counting results, and of other finiteness results, in non-archimedean settings.

2:00–2:30 pm
Break with refreshments (Common Room)

2:30–3:30 pm
Speaker: David Urbanik, IHES, France & IAS
Title: Degrees of Hodge Loci