### Hi There!

I have recently started my Ph.D studies at the mathematics department at the Hebrew University of Jerusalem, Israel.
My advisor is the one and only Jake Solomon, with whom I (try to) work on problems in differential and symplectic geometry, related to mathematical physics.

If for some reason you feel like looking at my curriculum vitae, you are probably ill and should see a doctor (a real doctor!).

Oh, and if you are in Jerusalem on a Wednesday morning, you should come to the topology and geometry seminar. It's fun!

### Contact information

Amit Solomon
Einstein Institute of Mathematics
Edmond J. Safra Campus, Givat Ram
The Hebrew University of Jerusalem
Jerusalem, 91904, Israel.
Office: Ross 37.
Phone: +972-2-5494025.
Email: $\mathtt{amit}@\mathtt{math}{.}\mathtt{huji}{.}\mathtt{ac}{.}\mathtt{il}$
Office hours: Wednesdays, 17:00-18:00

Apologies for the low quality of the picture, it was cloudy that day.

:כ"א טבת, תשע"ה
יום הלשון העברית

## Research

Generally speaking, I am interested in geometry. More particularly, I am mostly interested in differential and symplectic geometry.
Currently, I am working on the construction of a Morse-Witten complex for non-Morse functions, with hope to apply the methods we develop to FJRW theory.
In short, given a smooth function with a degenerate Hessian on a closed manifold which behaves "like a homogeneous function" near its critical points, we try to construct a chain complex whose homology recovers the singular homology of the manifold. This in turn will enable us to define the invariants of FJRW theory (a mathematical theory related to high energy physics) without the need for perturbation. We also hope to be able to use our methods to prove mirror-symmetry results in this context.

#### Morse Theory

Morse theory illustrates the intimate relationship between the critical points of a smooth function on a manifold and the topology of the manifold:
Given a generic function with non-degenerate critical points, one can construct a chain complex, known as the Morse-Witten complex, whose homology equals the singular homology of the manifold. Namely, the graded abelian group is the free abelian group generated by the critical points, and the boundary operator is given by counting (with sign) the isolated trajectories of the gradient flow connecting pairs of critical points.

#### What goes wrong in the degenerate case

Unfortunately, when the critical points of the function are degenerate, the Morse complex construction fails miserably: A single degenerate critical point may contribute more than one generator to the homology of the manifold, so we can no longer use a chain complex generated by critical points to calculate the homology of the manifold. More importantly, the boundary operator is not well defined -- while the space of unparametrized connecting orbits is usually still compact, it might not be finite since it may fail to be a manifold. Moreover, stable trajectories do not "glue together" in a unique way with unstable trajectories when they meet at a common critical point. Hence, a challenge lies in showing that the square of the boundary map is zero.

#### So what can we do?

We consider a special class of functions with degenerate singularities, which behave like a homogeneous function near each critical point. These functions are called "semihomogeneous" (I should note that we hope to also apply our methods for semiquasihomogeneous functions).
Jake and I were able to make the first step towards extending the Morse complex for such functions. Namely, we endow the stable set of a degenerate critical point with a natural stratification generalizing the concept of the stable manifold.

We are now working on the next step, which is to develop a gluing theory for the situation at hand, and then defining the chain complex and boundary operator. Finally, we will have to show that the homology of the chain complex so obtained equals the singular homology of the manifold.

#### Why should we consider semi(quasi)homogeneous functions, and how does it relate to FJRW theory?

First, let's say a few words about FJRW theory: It is the A-model for Landau-Ginzburg theory, which is a quantum field theory with a unique classical vacuum state and a potential energy with a degenerate critical point. So that's where physics comes in.
FJRW theory was developed recently by Fan, Jarvis and Ruan (obviously, the "W" stands for Witten), and is conjecturally dual to Gromov-Witten theory, i.e., the A-model for the Calabi-Yau theory (Moreover, this duality interacts with mirror symmetry.). This is a rapidly growing field with many fascinating directions, and I refer the reader to the original papers by Fan, Jarvis and Ruan: [1], [2].

As you probably imagined, the problem lies in the degeneration of the singularity (of the potential energy) involved in the Landau-Ginzburg theory.
At the moment, in order to define the virtual fundamental class (and hence the associated cohomological field theory, from which one extracts the invariants of FJRW theory), one needs to use a Morse deformation of the degenerate singularity. However, there's a bright side to the story: the singularity is semiquasihomogeneous. Hence, if we successfully construct a Morse-Witten complex for such functions, we will be able to construct the virtual fundamental class without need of a deformation.
This is important for several reasons: First, a Morse-deformation bifurcates a degenerate singularity into possibly many non-degenerate ones, which makes computations practically impossible unless special conditions are satisfied (e.g., contributions from solutions to the W-spin equation are trivial). Furthermore, a Morsification cannot always be done equivariantly, while in many cases (the orbifold case) it is important to be able to preserve the action of a finite group of diffeomorphisms.

## Teaching

I enjoy teaching and find it very important to try and explain the nuances and formalism to students on the one hand, while unravelling definitions and technicalities to see the ideas on the other hand.
I have had the pleasure of TA-ing unfortunate students in the following courses:

Picture taken by Konstantin Golubev, the first of his name.

## Book on Lattices

Together with Tsachik Gelander from the Weizmann Institute of Science, I am writing a book on the subject of lattices in locally compact groups.

This is an ongoing long term project, so please do not expect frequent updates or rapid progress.

A draft of the first two chapters will be available here soon. The aim of these chapters is to introduce the reader to the basic ideas and results in the theory:

1. Chapter 1: Basic Definitions and results
• Space of closed subgroups
• The Chabauty topology.
• The space of closed subgroups.
• Measures on homogeneous spaces
• Invariant measures on groups.
• Invariant measures on quotients.
• Cofinite and cocompact subgroups.
2. Chapter 2: Lattices — a first encounter
• Definition.
• Basic properties of lattices
• Intersection of a lattice with other subgroups.
• Lattices inside the space of closed subgroups.
• Fixed point properties of lattices vs. the ambient group.
• The inclusion of $\text{SL}(n,\mathbb{Z})$ in $\text{SL}(n,\mathbb{R})$
• A simple group with no lattices
• Settings.
• Some notations.
• The proof of the proposition.

In the next chapter we delve into the reach theory of lattices in solvable and nilpotent Lie groups, so stay tuned!

## Hebrew Dictionary for Geometry

Together with Sara Tukachinsky, and with the encouragement of Jake, we translate geometric terms to Hebrew, with the hope that soon discussions on geometry in Israel will be in Hebrew, without having to suddenly use English terms.

We will be delighted to hear any suggestions and ideas, as well as terms in need of translation. Please feel free to email me.

## מילון למונחי הגאומטריה

בעידודו של יעקב, שרה טוקצ'ינסקי ואני שוקדים על כתיבת מילון עברי למונחי הגאומטריה. זאת, מתוך תקווה כי המילון יסייע לקיומו של שיח עברי פורה ושוטף בגאומטריה. על אחת כמה וכמה, אנו מקווים כי מילון זה יקל על לימוד גאומטריה (ומתמטיקה ככלל) בשפה העברית.

אנו נשמח לכל הארה והצעה, ונגיל על כל בקשה לתרגומו של מונח הזקוק לתרגום. אנא הרגישו חופשיים לשלוח לי דוא"ל בנושא.

הגרסה העדכנית של המילון זמינה בקישור זה.
נפנה את תשומת לבכם לקישורים השלובים במילון עצמו. היאח!

### Wise words by Dr. Seuss

(Drawn by yours truly.)

Also, this comics.